Solutions for these following international mathematical problems used to select the top mathematical students in the world and nations are in the book titled
Narrative Approaches to the International Mathematical Problems
by Steve Dinh, a.k.a. Vo Duc Dien
Available at Amazon.com and other online outlets.
http://www.amazon.com/Narrative-Approaches-International-Mathematical-Problems/dp/1468568515
At the time of this book’s publication, the solutions to many of these problems were not yet available.
Problem 1 of the United States Mathematical Olympiad 1973
Two points, P and Q, lie in the interior of a regular tetrahedron ABCD. Prove that angle PAQ < 60°.
Problem 1 of the United States Mathematical Olympiad 2010
Let AXYZB be a convex pentagon inscribed in a semicircle of diameter AB. Denote by P, Q, R, S the feet of the perpendiculars from Y onto lines AX, BX, AZ, BZ, respectively. Prove that the acute angle formed by lines PQ and RS is half the size of ∠XOZ, where O is the midpoint of segment AB.
Problem 1 of the International Mathematical Olympiad 2006
Let ABC be a triangle with incenter I. A point P in the interior of the triangle satisfies ∠PBA + ∠PCA = ∠PBC + ∠PCB. Show that AP > AI, and that equality holds if and only if P = I.
Problem 4 of the United States Mathematical Olympiad 1975
Two given circles intersect in two points P and Q. Show how to construct a segment AB passing through P and terminating on the two circles such that AP×PB is a maximum.
Problem 4 of the United States Mathematical Olympiad 1979
Show how to construct a chord FPE of a given angle A through a fixed point P within the angle A such that + is a maximum.
Problem 4 of the United States Mathematical Olympiad 2010
Let ABC be a triangle with ∠A = 90°. Points D and E lie on sides AC and AB, respectively, such that ∠ABD = ∠DBC and ∠ACE = ∠ECB. Segments BD and CE meet at I. Determine whether or not it is possible for segments AB, AC, BI, ID, CI, IE to all have integer lengths.
Problem 5 of the United States Mathematical Olympiad 1990
An acute-angled triangle ABC is given in the plane. The circle with diameter AB intersects altitude CC’ and its extension at points M and N, and the circle with diameter AC intersects altitude BB’ and its extensions at P and Q. Prove that the points M, N, P, and Q lie on a common circle.
Problem 5 of the United States Mathematical Olympiad 1996
Triangle ABC has the following property: there is an interior point P such that ∠PAB = 10°, ∠PBA = 20°, ∠PCA = 30°, and ∠PAC = 40°. Prove that triangle ABC is isosceles.
Problem 5 of the International Mathematical Olympiad 2004
In a convex quadrilateral ABCD the diagonal BD does not bisect the angles ABC and CDA. The point P lies inside ABCD and satisfies ∠PBC = ∠DBA and ∠PDC = ∠BDA. Prove that ABCD is a cyclic quadrilateral if and only if AP = CP.
Problem 2 of the International Mathematical Olympiad 2009
Let ABC be a triangle with circumcenter O. The points P and Q are interior points of the sides CA and AB, respectively. Let K, L and M be the midpoints of the segments BP, CQ and PQ, respectively, and let Г be the circle passing through K, L and M. Suppose that the line PQ is tangent to the circle Г. Prove that OP = OQ.
Problem 4 of the International Mathematical Olympiad 2007
In triangle ABC the bisector of angle BCA intersects the circumcircle again at R, the perpendicular bisector of BC at P, and the perpendicular bisector of AC at Q. The midpoint of BC is K and the midpoint of AC is L. Prove that the triangles RPK and RQL have the same area.
Problem 4 of the International Mathematical Olympiad 2009
Let ABC be a triangle with AB = AC. The angle bisectors of ∠CAB and ∠ABC meet the sides BC and CA at D and E, respectively. Let K be the incenter of triangle ADC. Suppose that ∠BEK = 45°. Find all possible values of ∠CAB.
Problem 7 of the Canadian Mathematical Olympiad 1971
Let n be a five digit number (whose first digit is non-zero) and let m be the four digit number formed from n by deleting its middle digit. Determine all n such that is an integer.
Problem 9 of the Irish Mathematical Olympiad 1994
Let w, a, b, c be distinct real numbers with the property that there exist real numbers x, y, z for which the following equations hold:
x + y + z = 1
xa² + yb² + zc² = w²
xa³ + yb³ + zc³ = w³
xa+ yb+ zc = w
Express w in terms of a, b, c.
Problem 9 of the Middle European Mathematical Olympiad 2009
Let ABCD be a parallelogram with ∠BAD = 60° and denote by E the intersection of its diagonals. The circumcircle of triangle ACD meets the line BA at K≠A, the line BD at P≠D and the line BC at L≠C. The line EP intersects the circumcircle of triangle CEL at points E and M. Prove that triangles KLM and CAP are congruent.
Problem 2 of the Ibero-American Mathematical Olympiad 1998
The circumference inscribed on the triangle ABC is tangent to the sides BC, CA and AB on the points D, E and F, respectively. AD intersect the circumference on the point Q. Show that the line EQ intersect the segment AF on its midpoint if and only if AC = BC.
Problem 2 of the Ibero-American Mathematical Olympiad 2001
The inscribed circumference of the triangle ABC has center at O and it is tangent to the sides BC, AC and AB at the points X, Y and Z, respectively. The lines BO and CO intersect the line Y Z at the points P and Q, respectively. Show that if the segments XP and XQ have the same length, then the triangle ABC is isosceles.
Problem 1 of International Mathematical Talent Search Round 8
Prove that there is no triangle whose altitudes are of lengths 4, 7 and 10 units.
Problem 4 of the International Mathematical Olympiad 2010
Let P be a point inside the triangle ABC. The lines AP, BP and CP intersect the circumcircle Г of triangle ABC again at the points K, L and M, respectively. The tangent to Г at C intersects the line AB at S. Suppose that SC = SP. Prove that MK = ML.
Problem 2 of the Korean Mathematical Olympiad 2007
ABCD is a convex quadrilateral, and AB ≠ CD. Show that there exists a point M such that = = .
Problem 3 of Hong Kong Mathematical Olympiad 2002
Two circles intersect at points A and B. Through the point B a straight line is drawn, intersecting the first circle at K and the second circle at M. A line parallel to AM is tangent to the first circle at Q. The line AQ intersects the second circle again at R.
a) Prove that the tangent to the second circle at R is parallel to AK.
b) Prove that these two tangents are concurrent with KM.
Problem 1 of Hong Kong Mathematical Olympiad 2002
Find the value of sin²1° + sin²2° + … + sin²89°.
Problem 4 of Austria Mathematical Olympiad 2000
In the acute, non-isosceles triangle ABC with angle C = 60° let U be the circumcenter, H be the orthocenter and D the intersection of the lines AH and BC (that is, the orthogonal projection of A onto BC). Show that the Euler line HU is the bisector of ∠BHD.
Problem 2 of the Irish Mathematical Olympiad 2010
Let ABC be a triangle and let P denote the midpoint of the side BC. Suppose that there exist two points M and N interior to the sides AB and AC, respectively such that |AD| = |DM| = 2|DN|, where D is the intersection point of the lines MN and AP. Show that |AC| = |BC|.
Problem 2 of Australia Mathematical Olympiad 2008
Let ABC be an acute triangle, and let D be the point on AB (extended if necessary) such that AB and CD are perpendicular. Further, let tA and tB be the tangents to the circumcircle of ABCthrough A and B, respectively, and let E and F be the points on tA and tB, respectively, such that CEis perpendicular to tA and CFis perpendicular to tB. Prove that = .
Problem 6 of the British Mathematical Olympiad 2009
Two circles, of different radius, with centers at B and C, touch externally at A. A common tangent, not through A, touches the first circle at D and the second at E. The line through A which is perpendicular to DE and the perpendicular bisector of BC meet at F. Prove that BC = 2AF.
Problem 4 of the British Mathematical Olympiad 1995
ABC is a triangle, right-angled at C. The internal bisectors of angles BAC and ABC meet BC and CA at P and Q, respectively. M and N are the feet of the perpendiculars from P and Q to AB. Find angle MCN.
Problem 3 of the British Mathematical Olympiad 1996
Let ABC be an acute triangle, and let O be its circumcenter. The circle through A, O and B is called S. The lines CA and CB meet the circle S again at P and Q, respectively. Prove that the lines CO and PQ are perpendicular.
Problem 5 of the British Mathematical Olympiad 1996
Let a, b and c be positive real numbers,
a) Prove that 4(a³ + b³) ≥ (a + b)³
b) Prove that 9(a³ + b³ + c³) ≥ (a + b + c)³
Problem 3 of Austria Mathematical Olympiad 2002
Let ABCD and AEFG be two similar cyclic quadrilaterals (labeled counter-clockwise). Let P be the second point of intersection of the circumcircles of the quadrilaterals. Show that P lies on the line BE.
Problem 8 of the Irish Mathematical Olympiad 1991
Let ABC be a triangle and L the line through C parallel to the side AB. Let the internal bisector of the angle at A meet the side BC at D and the line L at E, and let the internal bisector of the angle at B meet the side AC at F and the line L at G. If |GF| = |DE|, prove that |AC| = |BC|.
Problem 1 of the British Mathematical Olympiad 2000
Two intersecting circles C1 and C2 have a common tangent which touches C1 at P and C2 at Q. The two circles intersect at M and N, where N is nearer to PQ than M is. The line PN meets the circle C2 again at R. Prove that MQ bisects angle PMR.
Problem 8 of the British Mathematical Olympiad 2001
A triangle ABC has ∠ACB > ∠ABC. The internal bisector of ∠BAC meets BC at D. The point E on AB is such that ∠EDB = 90°. The point F on AC is such that ∠BED = ∠DEF. Show that ∠BAD = ∠FDC.
Problem 5 of Austria Mathematical Olympiad 1988
The bisectors of angles B and C of triangle ABC intersect the opposite sides at points B′ and C′, respectively. Show that the line B′C′ intersects the incircle of the triangle.
Problem 7 of the British Mathematical Olympiad 2003
Let ABC be a triangle and let D be a point on AB such that 4AD = AB. The half-line ℓ is drawn on the same side of AB as C, starting from D and making an angle of a with DA where a = ∠ACB. If the circumcircle of ABC meets the half-line ℓ at P, show that PB = 2PD.
Problem 1 of the British Mathematical Olympiad 1997
N is a four-digit integer, not ending in zero, and R(N) is the four-digit integer obtained by reversing the digits of N; for example, R(3275) = 5723. Determine all such integers N for which R(N) = 4N + 3.
Problem 8 of the Russian Mathematical Olympiad 2010
In a acute triangle ABC, the median, AM, is longer than side AB. Prove that you can cut triangle ABC into three parts out of which you can construct a rhombus.
Problem 3 of the Middle European Mathematical Olympiad 2010
We are given a cyclic quadrilateral ABCD with a point E on the diagonal AC such that AD = AE and CB = CE. Let M be the center of the circumcircle k of the triangle BDE. The circle k intersects the line AC at points E and F. Prove that the lines FM, AD and BC meet at one point.
Problem 1 of the Ibero-American Mathematical Olympiad 1999
Find all the positive integers less than 1000 such that the cube of the sum of its digits is equal to the square of such integer.
Problem 3 of Japan’s Hitotsubashi University Entrance Exam 2010
In the xyz space with O (0, 0, 0), take points A on the x-axis, B on the xy plane and C on the z-axis such that ∠OAC = ∠OBC = θ, ∠AOB = 2θ, OC = 3. Note that the x-coordinate of A, the y– coordinate of B and the z-coordinate of C are all positive. Denote H the point that is inside ΔABC and is the nearest to O. Express the z-coordinate of H in terms of θ.
Problem 24 of the Iranian Mathematical Olympiad 2003
In an acute triangle ABC points D, E, F are the feet of the altitudes from A, B and C, respectively. A line through D parallel to EF meets AC at Q and AB at R. Lines BC and EF intersect at P. Prove that the circumcircle of triangle PQR passes through the midpoint of BC.
Problem 5 of Taiwan Mathematical Olympiad 1999
The altitudes through the vertices A, B, C of an acute triangle ABC meet the opposite sides at D, E, F, respectively, and AB > AC. The line EF meets BC at P, and the line through D parallel to EF meets the lines AC and AB at Q and R, respectively. N is a point on the line BC such that ∠NQP + ∠NRP < 180º. Prove that BN > CN.
Problem 4 of Hong Kong Mathematical Olympiad 2009
In figure below, the sector OAB has radius 4 cm and ∠AOB is a right angle. Let the semi-circle with diameter OB be centered at I with IJ || OA, and IJ intersects the semi-circle at K. If the area of the shaded region is T cm², find the value of T.
Problem 1 of the Vietnamese Mathematical Olympiad 1992
Let ABCD be a tetrahedron satisfying
a) ∠ACD + ∠BCD = 180°, and
b) ∠BAC + ∠CAD + ∠DAB = ∠ABC + ∠CBD + ∠DBA = 180°.
Find value of (ABC) + (BCD) + (CDA) + (DAB) if we know AC + CB = k and ∠ACB = a. Note: (Ω) denotes the area of shape Ω.
Problem 2 of the British Mathematical Olympiad 2005
Let x and y be positive integers with no prime factors larger than 5. Find all such x and y which satisfy x² – y² = 2k for some non-negative integer k.
Proof of Carnot’s theorem for the obtuse triangle
Let ABC be an arbitrary obtuse triangle. Prove that DG + DH = R + r + DF, where r and R are the inradius and circumradius of triangle ABC, respectively, D the circumcenter of triangle ABC, DF, DG and DH the altitudes to the sides AC, AB and BC, respectively.
(Carnot’s theorem is used in a proof of the Japanese theorem for concyclic polygons.)
Problem 1 of Hong Kong Mathematical Olympiad 2007
Let D be a point on the side BC of triangle ABC such that AB + BD = AC + CD. The line segment AD cut the incircle of triangle ABC at X and Y with X closer to A. Let E be the point of contact of the incircle of triangle ABC on the side BC. Show that
a) EY is perpendicular to AD,
b) XD is 2×IM, where I is the incenter of the triangle ABC and M is the midpoint of BC.
Problem 4 of the Estonian Mathematical Olympiad 2007
In square ABCD, points E and F are chosen in the interior of sides BC and CD, respectively. The line drawn from F perpendicular to AE passes through the intersection point G of AE and diagonal BD. A point K is chosen on FG such that AK = EF. Find ∠EKF.
Problem 4 of Hong Kong MO Team Selection Test 2009
Two circles C1, C2 with different radii are given in the plane, they touch each other externally at T. Consider any points A∈C1 and B∈C2, both different from T, such that ∠ATB = 90°.
a) Show that all such lines AB are concurrent.
b) Find the locus of midpoints of all such segments AB.
Problem 3 of Tokyo University Entrance Exam 2006
Given the point P(0, p) on the y-axis and the line m: y = (tanq)x on the coordinate plane with the origin, where p > 1, 0 < q < p/2. Now by the symmetric transformation, the line l with slope a as the axis of symmetry, the origin O was mapped the point Q lying on the line y = 1 in the first quadrant and the point P on the y-axis was mapped the point R lying on the line m in the first quadrant.
a) Express tanq in terms of a and p.
b) Prove that there exist the point P satisfying the following condition, then find the value of p.
Condition: For any q (0 <q < p/2) the line passing through the origin and is perpendicular to the line l is y =[tan(q/3)]x.
Problem 5 of Korean Mathematical Olympiad 2006
In a convex hexagon ABCDEF triangles ABC, CDE, EFA are similar. Find conditions on these triangles under which triangle ACE is equilateral if and only if so is BDF.
Problem 5 of Taiwan Mathematical Olympiad 1995
Let P be a point on the circumscribed circle of ΔABC and H be the orthocenter of ΔABC. Also let D, E and F be the points of intersection of the perpendicular from P to BC, CA and AB, respectively. It is known that the three points D, E and F are colinear. Prove that the line DEF passes through the midpoint of the line segment PH.
Problem 4 of Taiwan Winter Camp 2001
Let O be the center of excircle of ΔABC touching the side BC externally. Let M be the midpoint of AC, P the intersection point of MO and BC. Prove that AB = BP, if ∠BAC = 2∠ACB.
Problem 9 of the British Mathematical Olympiad 1999
Consider all numbers of the form 3n² + n + 1, where n is a positive integer.
a) How small can the sum of the digits (in base 10) of such a number be?
b) Can such a number have the sum of its digits (in base 10) equal?
Problem 6 of Uruguay Mathematical Olympiad 2009
Is the sum 1+ 2+ 3+ … + 2008 divisible by 7?
Problem 3 of the Japanese Mathematical Olympiad 1995
In a convex pentagon ABCDE, let S, R, T, P and Q be the intersections of AC and BE, AD and BE, AC and BD, CE and BD, CE and AD, respectively. If all of ΔASR, ΔBTS, ΔCPT, ΔDQP and ΔERQ have the area of 1, then find the area of the following pentagons
a) The pentagon PQRST.
b) The pentagon ABCDE.
Problem 2 of the Czech and Slovak Mathematical Olympiad 2002
Consider an arbitrary equilateral triangle KLM, whose vertices K, L and M lie on the sides AB, BC and CD, respectively, of a given square ABCD. Find the locus of the midpoints of the sides KL of all such triangles KLM.
Iceland’s problem for International Mathematical Olympiad
For an acute triangle ABC, let H be the foot of the perpendicular from A to BC. Let M, N be the feet of the perpendicular from H to AB, AC, respectively. Define lA to be the line through A perpendicular to MN and similarly define lB and lC. Show that lA, lB and lC pass through a common point O. (This problem was proposed by Iceland and was never chosen for testing by the IMO organization.)
Problem 3 of Hong Kong Mathematical Olympiad 2008
For arbitrary real number x, define [x] to be the largest integer less than or equal to x. For instance, [2] = 2 and [3.4] = 3. Find the value of [1.008×100].
Problem 6 of Hong Kong Mathematical Olympiad 2007
If R is the remainder of 1 + 2 + 3 + 4 + 5 + 6 divided by 7, find the value of R.
Sample problem for the Irish Mathematical Olympiad
Prove that, for every positive integer n which ends in the digit 5, 20 + 15 + 8 + 6 is divisible by 2009. (This problem was just an example and has never yet been used in any competition.)
Problem 10 of Hong Kong Mathematical Olympiad 2008
Let [x] be the largest integer not greater than x. If a = [()] + 16, find the value of a.
Problem 3 of Hong Kong Mathematical Olympiad 2007
208208 = 8a + 8b + 8c + 8d + 8e + f, where a, b, c, d, e and f are integers and 0 ≤ a, b, c, d, e, f ≤ 7. Find the value of a×b×c + d×e×f.
Problem 8 of Hong Kong Mathematical Olympiad 2007
Amongst the seven numbers 3624, 36024, 360924, 3609924, 36099924, 360999924 and 3609999924, there are n of them that are divisible by 38. Find the value of n.
Problem 2 of the Iranian Mathematical Olympiad 2010
Let O be the center of the excircle C of triangle ABC opposite vertex A. Assume C touches AB and AC at E and F, respectively. Let OB and OC intersect EF at P and Q, respectively. Let M be the intersection of CP and BQ. Prove that the distance between M and the line BC is equal to the inradius of ΔABC.
Problem 1 of Belarus Mathematical Olympiad 2004 Category B
The diagonals AD, BE, CF of a convex hexagon ABCDEF meet at point O. Find the smallest possible area of this hexagon if the areas of the triangles AOB, COD, EOF are equal to 4, 6 and 9, respectively.
Problem 5 of Hong Kong Mathematical Olympiad 2007
AD, BE, and CF are the altitudes of an acute triangle ABC. Prove that the feet of the perpendiculars from F onto the segments AC, BC, BE and AD lie on the same straight line.
Problem 4 of the British Mathematical Olympiad 2006
Two touching circles S and T share a common tangent which meets S at A and T at B. Let AP be a diameter of S and let the tangent from P to T touch it at Q. Show that AP = PQ.
Problem 2 of the Estonian MO Team Selection Test 2004
Let O be the circumcenter of the acute triangle ABC and let lines AO and BC intersect at point K. On sides AB and AC, points L and M are chosen such that KL = KB and KM = KC. Prove that the segments LM and BC are parallel.
Problem 1 of Uruguay Mathematical Olympiad 2009
What is the highest 8-digit number ending in 2009 and is a multiple of 99?
Problem 4 of Hong Kong Mathematical Olympiad 2007
Given triangle ABCwith ∠A = 60°, AB = 2005, AC = 2006. Bob and Bill in turn (Bob is the first) cut the triangle along any straight line so that two new triangles with area more than or equal to 1 appear. After that an obtused-angled triangle (or any of two right-angled triangles) is deleted and the procedure is repeated with the remained triangle. The player loses if he cannot do the next cutting. Determine, which player wins if both play in the best way.
Problem 4 of the Czech-Polish-Slovak Math Competition 2009
Given a circle k and its chord AB which is not a diameter, let C be any point inside the longer arc AB of k. We denote by K and Lthe reflections of A and B with respect to the axes BC and AC. Prove that the distance of the midpoints of the line segments KL and AK is independent of the location of point C.
Problem 1 of the British Mathematical Olympiad 2006
Find four prime numbers less than 100 which are factors of 3 –2.
Problem 5 of the British Mathematical Olympiad 2006
For positive real numbers a, b, c, prove that (a² + b²)² ≥ (a + b + c)(a + b – c)(b + c – a)(c + a – b).
Problem 6 of the British Mathematical Olympiad 2006
Let n be an integer. Show that, if 2 + 2 is an integer, then it is a perfect square.
Problem 1 of the British Mathematical Olympiad 2007
Find the value of .
Problem 2 of Pan African Mathematical Competition 2004
Is 4 + an integer?
Problem 1 of the British Mathematical Olympiad 1993
Find, showing your method, a six-digit integer n with the following properties: (i) n is a perfect square, (ii) the number formed by the last three digits of n is exactly one greater than the number formed by the first three digits of n. (Thus n might look like 123124, although this is not a square.)
Problem 4 of the Czech and Slovak Mathematical Olympiad 2002
Find all pairs of real numbers a, b for which the equation in the domain of the real numbers x
= x
Problem 1 of the Brazilian Mathematical Olympiad 1995
ABCD is a quadrilateral with a circumcircle center O and an inscribed circle center I. The diagonals intersect at S. Show that if two of O, I, S coincide, then it must be a square.
Problem 4 of China Mathematical Olympiad 1997
Let quadrilateral ABCD be inscribed in a circle. Suppose lines AB and DC intersect at P and lines AD and BC intersect at Q. From Q construct the two tangents QE and QF to the circle where E and F are the points of tangency. Prove that the three points P, E, F are collinear.
Problem 5 of the Irish Mathematical Olympiad 1988
A person has seven friends and invites a different subset of three friends to dinner every night for one week (7 days). In how many ways can this be done so that all friends are invited at least once?
Problem 1 of the British Mathematical Olympiad 1996
Consider the pair of four-digit positive integers (M, N) = (3600, 2500).
Notice that M and N are both perfect squares, with equal digits in two places, and differing digits in the remaining two places. Moreover, when the digits differ, the digit in M is exactly one greater than the corresponding digit in N. Find all pairs of four-digit positive integers (M, N) with these properties.
Problem 1 of Poland Mathematical Olympiad 1997
Let ABCD be a tetrahedron with ∠BAD = 60°, ∠BAC = 40°, ∠ABD = 80°, ∠ABC = 70°. Prove that the lines AB and CD are perpendicular.
Problem 1 of British Mathematical Olympiad 1991
Prove that the number 3 + 2×17 where n is a non-negative integer, is never a perfect square.
Problem 4 of Poland Mathematical Olympiad 1996
ABCD is a tetrahedron with ∠BAC = ∠ACD, and ∠ABD = ∠BDC. Show that AB = CD.
Problem 6 of Hungary Mathematical Olympiad 1999
The midpoints of the edges of a tetrahedron lie on a sphere. What is the maximum volume of the tetrahedron?
Problem 5 of International Mathematical Talent Search Round 18
Let a and b be two lines in the plane, and let C be a point as shown in the figure below. Using only a compass and an unmarked straight edge, construct an isosceles right triangle ABC, so that A is on line a, B is on line b, and AB is the hypotenuse of triangle ABC.
Problem 2 of Austria Mathematical Olympiad 2004
Solve the equation
= .
(all the square roots are non-negative)
Problem 3 of the Vietnamese Mathematical Olympiad 1962
Let ABCD be a tetrahedron. Denote by A’, B’ the feet of the perpendiculars from A and B, respectively to the opposite faces. Show that AA’ and BB’ intersect if and only if AB is perpendicular to CD. Do they intersect if AC = AD = BC = BD?
Problem 8 of Georgia MO Team Selection Test 2005
In a convex quadrilateral ABCD the points P and Q are chosen on the sides BC and CD, respectively so that ∠BAP = ∠DAQ. Prove that the line, passing through the orthocenters of triangles ABP and ADQ, is perpendicular to AC if and only if the triangles ABP and ADQ have the same areas.
Problem 4 of Hong Kong MO Team Selection Test 1994
Suppose that yz + zx + xy = 1 and x, y, and z ≥ 0. Prove that
x(l – y²)(1 – z²) + y(l – z²)(1 – x²) + z(l – x²)(1 – y²) ≤ 4.
Problem 5 of the Iranian Mathematical Olympiad 2000
In a tetrahedron we know that the sum of angles of all vertices is 180°. (e.g., for vertex A, we have ∠BAC + ∠CAD + ∠DAB = 180°.) Prove that the faces of this tetrahedron are four congruent triangles.
Problem 3 of Moldova Mathematical Olympiad 2002
Consider an angle ∠DEF, and the fixed points B and C on the semi-line EF and the variable point A on ED. Determine the position of A on ED such that the sum AB + AC is minimum.
Problem 15 of Moldova Mathematical Olympiad 2002
In a triangle ABC, the bisectors of the angles at B and C meet the opposite sides B1 and C1, respectively. Let T be the midpoint AB1 Lines BT and B1C1 meet at E and lines AB and CE meet at L. Prove that the lines TL and B1C1 have a point in common.
Problem 7 of Moldova MO Team Selection Test 2003
The sides AB and AC of the triangle ABC are tangent to the incircle with center I of the ΔABC at the points M and N, respectively. The internal bisectors of the ΔABC drawn from B and C intersect the line MN at the points P and Q, respectively. Suppose that F is the intersection point of the lines CP and BQ. Prove that FI ⊥BC.
Problem 20 of Indonesia MO Team Selection Test 2009
Let ABCD be a convex quadrilateral. Let M, N be the midpoints of AB, AD, respectively. The foot of perpendicular from M to CD is K, and the foot of perpendicular from N to BC is L. Show that if AC, BD, MK and NL are concurrent, then KLMN is a cyclic quadrilateral.
Problem A5 Tournament of Towns 2009
Let XYZ be a triangle. The convex hexagon ABCDEF is such that AB, CD and EF are parallel and equal to XY, YZ and ZX, respectively. Prove that the area of triangle with vertices at the midpoints of BC, DE and FA is no less than the area of triangle XYZ.
Problem 16 of Moldova Mathematical Olympiad 2002
Let ABCD be a convex quadrilateral and let N on side AD and M on side BC be points such that = . The lines AM and BN intersect at P, while the lines CN and DM intersect at Q. Prove that if SABP + SCDQ = SMNPQ, then either AD || BC or N is the midpoint of DA.
Problem 3 of Hungary-Israel Binational 1994
Three given circles have the same radius and pass through a common point P. Their other points of pairwise intersections are A, B, C. We define triangle A’B’C’, each of whose sides is tangent to two of the three circles. The three circles are contained in triangle A’B’C’. Prove that the area of triangle A’B’C’ is at least nine times the area of triangle ABC.
Problem 21 of Moldova Mathematical Olympiad 2002
Let the triangle ADB1 such that ∠DAB1 ≠ 90°. On the sides of this triangle externally are constructed the squares ABCD and AB1C1D1 with centers O1 and O2, respectively. Prove that the circumcircles of the triangles BAB1, DAD1 and O1AO2 share a common point differs from A.
Problem 2 of Hungary-Israel Binational 2001
Points A, B, C and D lie on a line l, in that order. Find the locus of points P in the plane for which ∠APB = ∠CPD.
Problem 11 of Moldova Mathematical Olympiad 2002
Consider a circle Γ(O, R) and a point P found in the interior of this circle. Consider a chord AB of Γ that passes through P. Suppose that the tangents to Γ at the points A and B intersect at Q. Let M ∈QA and N ∈QB such that PM⊥QA and PN⊥QB. Prove that the value of + doesn’t depend of choosing the chord AB.
Problem 3 of Hitotsubashi University Entrance Exam 2010
In the xyz space with O(0, 0, 0), take points A on the x-axis, B on the xy plane and C on the z-axis such that ∠OAC = ∠OBC = q,∠AOB = 2q, OC = 3. Note that the x coordinate of A, the y coordinate of B and the z coordinate of C are all positive. Denote H the point that is inside ΔABC and is the nearest to O. Express the z coordinate of H in terms of q.
Problem 4 of Moldova Mathematical Olympiad 2006
Let ABCDE be a right quadranular pyramid with vertex E and height EO. Point S divides this height in the ratio ES:SO = m. In which ratio does the plane [ABS] divide the lateral area of triangle EDC of the pyramid.
Problem 4 of Tokyo University Entrance Exam 2010
In the coordinate plane with O (0, 0), consider the function C: y = x + and two distinct points P(x, y), P(x, y) on C.
a) Let H (i = 1, 2) be the intersection points of the line passing through P (i = 1, 2), parallel to x-axis and the line y = x.
Show that the area of ΔOPH and ΔOPH are equal.
b) Let x < x . Express the area of the figure bounded by the part of x < x < x for C and line segments PO, PO in terms of y, y .
Problem 1 of Tokyo University Entrance Exam 2010
Let the lengths of the sides of a cuboid be denoted a, b and c. Rotate the cuboid in 90° the side with length b as the axis of the cuboid. Denote by V the solid generated by sweeping the cuboid.
a) Express the volume of V in terms of a, b and c.
b) Find the range of the volume of V with a + b + c = 1.
Problem 3 of the Vietnamese Mathematical Olympiad 1990
A tetrahedron is to be cut by three planes which form a parallelepiped whose three faces and all vertices lie on the surface of the tetrahedron.
a) Can this be done so that the volume of the parallelepiped is at least of the volume of the tetrahedron?
b) Determine the common point of the three planes if the volume of the parallelepiped is of the volume of the tetrahedron.
Problem 3 of Spain Mathematical Olympiad 1994
A tourist office was investigating the numbers of sunny and rainy days in a year in each of six regions. The results are partly shown in the following table:
Region Sunny or rainy Unclassified
A 336 29
B 321 44
C 335 30
D 343 22
E 329 36
F 330 35
Looking at the detailed data, an officer observed that if one region is excluded, then the total number of rainy days in the other regions equals one third of the total number of sunny days in these regions. Determine which region is excluded.
Problem 26 of India Postal Coaching 2010
Let M be an interior point of a triangle ABC such that ∠AMB = 150°, ∠BMC = 120°, Let P, Q, R be the circumcenters of the triangles AMB, BMC, CMA, respectively. Prove that (PQR) ≥ (ABC).
Problem 4 of the International Zhautykov Olympiad 2010
Positive integers 1, 2, . . ., n are written on а blackboard (n > 2).
Every minute two numbers are erased and the least prime divisor of their sum is written. In the end only the number 97 remains. Find the least n for which it is possible.
Problem 6 of the Iranian Mathematical Olympiad 1995
In a quadrilateral ABCD let A’, B’, C’ and D’ be the circumcenters of the triangles BCD, CDA, DAB and ABC, respectively. Denote by S(X, YZ) the plane which passes through the point X and is perpendicular to the line YZ. Prove that if A’, B’, C’ and D’ don’t lie in a plane, then four planes S(A, C’D’), S(B, A’D’), S(C, A’B’) and S(D, B’C’) pass through a common point.
Problem 8 of Hong Kong Mathematical Olympiad 2008
Let Q = log(2 – ). Find the value of Q.
Problem 9 of Hong Kong Mathematical Olympiad 2008
Let F = 1 + 2 + 2 + 2 + … +2 and T = . Find the value of T.
Problem 2 of Netherlands Dutch Mathematical Olympiad 1998
Let TABCD be a pyramid with top vertex T, such that its base ABCD is a square of side length 4. It is given that, among the triangles TAB, TBC, TCD and TDA one can find an isosceles triangle and a right-angled triangle. Find all possible values for the volume of the pyramid.
Problem 6 of Austria Mathematical Olympiad 2001
We are given a semicircle with diameter AB. Points C and D are marked on the semicircle, such that AC = CD holds. The tangent of the semicircle in C and the line joining B and D intersect in a point E, and the line joining A and E intersects the semicircle in a point F. Show that FD > FC must hold.
Problem 3 of Tokyo University Entrance Exam 2008
A regular octahedron is placed on a horizontal rest. Draw the plan of top-view for the regular octahedron.
Let G1, G2 be the barycenters of the two faces of the regular octahedron parallel to each other. Find the volume of the solid by revolving the regular tetrahedron about the line G1G2 as the axis of rotation.
Problem 2 of the British Mathematical Olympiad 2007
Find all solutions in positive integers x, y, z to the simultaneous equations
x + y − z = 12
x² + y² − z² = 12.
Problem 6 of the Vietnamese Mathematical Olympiad 1982
Let ABCDA’B’C’D’ be a cube (where ABCD and A’B’C’D’ are faces and AA’, BB’, CC’, DD’ are edges). Consider the four lines AA’, BC, D’C’ and the line joining the midpoints of BB’ and DD’. Show that there is no line which cuts all the four lines.
Problem 1 of British Mathematical Olympiad 2011
Let ABC be a triangle and X be a point inside the triangle. The lines AX, BX and CX meet the circumcircle of triangle ABC again at P, Q and R, respectively. Choose a point U on XP which is between X and P. Suppose that the lines through U which are parallel to AB and CA meet XQ and XR at points V and W, respectively. Prove that the points W, R, Q and V lie on a circle.
Problem 3 of the Vietnamese Mathematical Olympiad 1981
A plane r and two points M, N outside it are given. Determine the point A on r for which is minimal.
Problem 5 of International Mathematical Talent Search Round 4
The sides of triangle ABC measure 11, 20, and 21 units. We fold it along PQ, QR, RP where P, Q, R are the midpoints of its sides until A, B, C coincide. What is the volume of the resulting tetrahedron?
Problem 4 of International Mathematical Talent Search Round 7
In an attempt to copy down from a board a sequence of six positive integers in arithmetic progression, a student wrote down the five numbers
113, 137, 149, 155, 173 accidentally omitting one. He later discovered that he also miscopied one of them. Can you help him and recover the original sequence?
Problem 1 of British Mathematical Olympiad 1990
Find a positive integer whose first digit is 1 and which has the property that, if this digit is transferred to the end of the number, the number is tripled.
Problem 2 of the British Mathematical Olympiad 2008
Find all real values of x, y and z such that
Problem 1 of International Mathematical Talent Search Round 15
Is it possible to pair off the positive integers 1, 2, 3, . . . , 50 in such a manner that the sum of each pair of numbers is a different prime number?
Problem 4 of International Mathematical Talent Search Round 15
Suppose that for positive integers a, b, c and x, y, z, the equations a² + b² = c² and x² + y² = z² are satisfied. Prove that
(a + x)² + (b + y)² ≤ (c + z)²,
and determine when equality holds.
Problem 1 of International Mathematical Talent Search Round 17
The 154-digit number, 19202122 . . . 939495, was obtained by listing the integers from 19 to 95 in succession. We are to remove 95 of its digits, so that the resulting number is as large as possible. What are the first 19 digits of this 59-digit number?
Problem 2 of International Mathematical Talent Search Round 17
Find all pairs of positive integers (m, n) for which m² – n² = 1995.
Problem 4 of International Mathematical Talent Search Round 17
A man is 6 years older than his wife. He noticed 4 years ago that he has been married to her exactly half of his life. How old will he be on their 50th anniversary if in 10 years she will have spent two-thirds of her life married to him?
Problem 3 of Spain Mathematical Olympiad 1985
Solve the equation tan²2x + 2tan2xtan3x = 1.
Problem 5 of International Mathematical Talent Search Round 8
Given that a, b, x and y are real numbers such that
a + b = 23,
ax + by = 79,
ax² + by² = 217,
ax³ + by³ = 691.
Determine ax + by.
Problem 1 of Yugoslav Mathematical Olympiad 2001
Vertices of a square ABCD of side 25/4 lie on a sphere. Parallel lines passing through points A, B, C and D intersect the sphere at points A’, B’, C’ and D’, respectively. Given that AA’ = 6, BB’ = 10, CC’ = 8, determine the length of the segment DD’.
Problem 16 of the Iranian Mathematical Olympiad 2010
In a triangle ABC, I is the incenter, BI and CI cut the circumcircle of ABC at E and F, respectively. M is the midpoint of EF. C is a circle with diameter EF. IM cuts C at two points L and K and the arc BC of circumcircle of ABC (not containing A) at D. Prove that = .
Problem 2 of the United States Mathematical Olympiad 1997
ABC is a triangle. Take points D, E and F on the perpendicular bisectors of BC, CA and AB, respectively. Show that the lines through A, B and C perpendicular to EF, FD and DE, respectively are concurrent.
Problem 7 of the British Mathematical Olympiad 1999
Let ABCDEF be a hexagon (which may not be regular), which circumscribes a circle S. (That is, S is tangent to each of the six sides of the hexagon.) The circle S touches AB, CD, EF at their midpoints P, Q, R, respectively. Let X, Y, Z be the points of contact of S with BC, DE, FA, respectively. Prove that PY, QZ and RX are concurrent.
Problem 1 of Hong Kong Mathematical Olympiad 2000
Let C be the circumcenter of a triangle ABC with AB > AC > BC. Let D be a point on the minor arc BC of the circumcircle, and let E and F be points on AD such that AB⊥OE and AC⊥OF. The lines BE and CF meet at P. Prove that if PB = PC + PO, then ∠BAC = 30°.
Problem 3 of British Mathematical Olympiad 1990
The angles A, B, C, D of a convex quadrilateral satisfy the relation
cosA + cosB + cosC + cosD = 0. Prove that ABCD is a trapezium (British for trapezoid) or is cyclic.
Problem 5 of the Irish Mathematical Olympiad 1990
Let ABC be a right-angled triangle with right-angle at A. Let X be the foot of the perpendicular from A to BC, and Y the mid-point of XC. Let AB be extended to D so that |AB| = |BD|. Prove that DX is perpendicular to AY.
Problem 7 of the British Mathematical Olympiad 1998
A triangle ABC has ∠BAC > ∠BCA. A line AP is drawn so that ∠PAC = ∠BCA where P is inside the triangle. A point Q outside the triangle is constructed so that PQ is parallel to AB, and BQ is parallel to AC. R is the point on BC (separated from Q by the line AP) such that ∠PRQ = ∠BCA.
Prove that the circumcircle of ΔABC touches the circumcircle
of ΔPQR.
Problem 3 of Austria Mathematical Olympiad 2000
Determine all real solutions of the equation
||||||| x² – x –1 | – 3 | – 5 | – 7 | – 9 | – 11 | – 13 | = x² – 2x – 48.
Problem 2 of Belarus Mathematical Olympiad 1997 Category D
Points D and E are taken on side CB of triangle ABC, with D between C and E, such that ∠BAE = ∠CAD. If AC < AB, prove
Problem 6 of Belarus Mathematical Olympiad 2004
Circles S and S meet at points A and B. A line through A is parallel to the line through the centers of S and S and meets S again at C and S again at D. The circle S with diameter CD meets S and S again at P and Q, respectively. Prove that lines CP, DQ, and AB are concurrent.
Problem 4 of the Vietnamese Mathematical Olympiad 1962
Let be given a tetrahedron ABCD such that triangle BCD equilateral and AB = AC = AD. The height is h and the angle between two planes ABC and BCD is α. The point X is taken on AB such that the plane XCD is perpendicular to AB. Find the volume of the tetrahedron XBCD.
Problem 4 of the Vietnamese Mathematical Olympiad 1986
Let ABCD be a square of side 2a. An equilateral triangle AMB is constructed in the plane through AB perpendicular to the plane of the square. A point S moves on AB such that SB = x. Let P be the projection of M on SC and E, O be the midpoints of AB and CM, respectively.
a) Find the locus of P as S moves on AB.
b) Find the maximum and minimum lengths of SO.
Problem 4 of the Irish Mathematical Olympiad 2006
Given a positive integer n, let b(n) denote the number of positive integers whose binary representations occur as blocks of consecutive integers in the binary expansion of n. For example, b(13) = 6 because 13 = 1101, which contains as consecutive blocks the binary representations of 13= 1101, 6 = 110, 5 = 101, 3 = 11, 2 = 10 and 1 = 1.
Show that if n ≤ 2500, then b(n) ≤ 39, and determine the values of n for which equality holds.
Problem 1 of the Canadian Mathematical Olympiad 1992
Prove that the product of the first n natural numbers is divisible by the sum of the first n natural numbers if and only if n + 1 is not an odd prime.
Problem 1 of the Ibero-American Mathematical Olympiad 1988
The measures of the angles of a triangle is an arithmetic progression and its altitudes is also another arithmetic progression. Prove that the triangle is equilateral.
Problem 2 of the Ibero-American Mathematical Olympiad 1997
In a triangle ABC draw a circumcircle with its center I being the incicle of the triangle to intersect twice each of the sides of the triangle: the segment BC on D and P (where D is nearer two B), the segment CA on E and Q (where E is nearer to C) and the segment AB on F and R (where F is nearer to A). Let S be the intersection of the diagonals of the quadrilateral EQFR, T be the intersection of the diagonals of the quadrilateral FRDP and U be the intersection of the diagonals of the quadrilateral DPEQ. Show that the circumcircles of the triangles FRT, DPU and EQS have a unique point in common.
Problem 1 of Tournament of Towns 1987
A machine gives out five pennies for each nickel inserted into it. The machine also gives out five nickels for each penny. Can Peter, who starts out with one penny, use the machine in such a way as to end up with an equal number of nickels and pennies?
Problem 1 of the Canadian Mathematical Olympiad 1981
For any real number t, denote by [t] the greatest integer which is less than or equal to t. For example: [8] = 8, [p] = 3 and [-] = -3. Show that the equation
[x] + [2x] + [4x] + [8x] + [16x] + [32x] = 12345 has no solution.
Problem 1 of Asian Pacific Mathematical Olympiad 1993
Let ABCD be a quadrilateral such that all sides have equal length and angle ABC is 60°. Let l be a line passing through D and not intersecting the quadrilateral (except at D). Let E and F be the points of intersection of l with AB and BC respectively. Let M be the point of intersection of CE and AF. Prove that CA² = CM×CE.
Problem 1 of the Asian Pacific Mathematical Olympiad 2010
Let ABC be a triangle with ∠BAC ≠ 90°. Let Obe the circum-center of the triangle ABCand let Г be the circumcircle of the triangle BOC. Suppose that Г intersects the line segment ABat Pdifferent from B, and the line segment ACat Qdifferent from C. Let ON be a diameter of the circle Г. Prove that the quadrilateral APNQis a parallelogram.
Problem 1 of Spain Mathematical Olympiad 1998
A unit square ABCD with center O is rotated about O by an angle a. Compute the common area of the two squares.
Problem 4 of the British Mathematical Olympiad 1987
The triangle ABC has orthocenter H. The feet of the perpendicu-lars from H to the internal and external bisectors of angle BAC (which is not a right angle) are P and Q. Prove that PQ passes
Problem 5 of India postal Coaching 2010
A point P lies on the internal angle bisector of ∠BAC of a triangle ABC. Point D is the midpoint of BC and PD meets the external angle bisector of ∠BAC at point E. If F is the point such that PAEF is a rectangle then prove that PF bisects ∠BFC internally or externally.
Problem 1 of the International Mathematical Olympiad 1998
In the convex quadrilateral ABCD, the diagonals AC and BD are perpendicular and the opposite sides AB and DC are not parallel. Suppose that the point P, where the perpendicular bisectors of AB and DC meet, is inside ABCD. Prove that ABCD is a cyclic quadrilateral if and only if the triangles ABP and CDP have equal areas.
Problem 2 of Austria Mathematical Olympiad 2005
For how many integer values a with |a| ≤ 2005 does the system of equations
x² = y + a
y² = x + a
have integer solutions?
Problem 4 of Indonesia MO Team Selection Test 2010
Let ABC be a non-obtuse triangle with CH and CM are the altitude and median, respectively. The angle bisector of ∠BAC intersects CH and CM at P and Q, respectively. Assume that ∠ABP = ∠PBQ = ∠QBC.
a) Prove that ABC is a right-angled triangle, and
b) Calculate .
Problem 5 of Spain Mathematical Olympiad 1987
In a triangle ABC, D lies on AB, E lies on AC and ∠ABE = 30°, ∠EBC = 50°, ∠ACD = 20°, ∠DCB = 60°. Find ∠EDC.
Problem 2 Asian Pacific Mathematical Olympiad 1992
In a circle C with center O and radius r, let C1, C2 be two circles with centers O1, O2 and radii r1, r2 respectively, so that each circle Ci is internally tangent to C at Ai and so that C1, C2 are externally tangent to each other at A. Prove that the three lines OA, O1A2, and O2A1 are concurrent.
Problem 6 of Russia Sharygin Geometry Olympiad 2008
In a plane, given two concentric circles with the center A. Let B be an arbitrary point on some of these circles, and C on the other one. For every triangle ABC, consider two equal circles mutually tangent at the point D, such that one of these circles is tangent to the line AB at point B and the other one is tangent to the line AC at point C. Determine the locus of points D.
Problem 2 of the Canadian Mathematical Olympiad 1977
Let O be the center of a circle and A a fixed interior point of the circle different from O. Determine all points P on the circum-ference of the circle such that the angle OPA is a maximum.
Problem 2 of the Canadian Mathematical Olympiad 1978
Find all pairs a, b of positive integers satisfying the equation 2a² = 3b³.
Problem 2 of the United States Mathematical Olympiad 1976
If A and B are fixed points on a given circle and XY is a variable diameter of the same circle, determine the locus of the point of intersection of lines AX and BY. You may assume that AB is not a diameter.
Problem 2 of the United States Mathematical Olympiad 1993
Let ABCD be a convex quadrilateral such that diagonals AC and BD intersect at right angles, and let E be their intersection. Prove that the reflections of E across AB, BC, CD, DA are concyclic.
Problem 3 of Austria Mathematical Olympiad 2005
In an acute-angled triangle ABC two circles C1 and C2 are drawn whose diameters are the sides AC and BC. Let E be the foot of the altitude hb on AC and let F be the foot of the altitude ha on BC. Let L and N be the intersections of the line BE with the circle C1 (L on the line BE) and let K and M be the intersections of the line AF with the circle C2 (K on the line AF). Show that KLMN is a cyclic quadrilateral.
Problem 3 of the Canadian Mathematical Olympiad 1973
Prove that if p and p + 2 are both prime integers greater than 3, then 6 is a factor of p + 1.
Problem 3 of the Canadian Mathematical Olympiad 1978
Determine the largest real number z such that
x + y + z = 5
xy + yz + xz = 3
and x, y are also real.
Problem 4 of the Ibero-American Mathematical Olympiad 2002
In a triangle ABC with all its sides of different length, D is on the side AC, such that BD is the angle bisector of ∠ABC. Let E and F, respectively, be the feet of the perpendicular drawn from A and C onto the line BD and let M be the point on BC such that DM is perpendicular to BC. Show that ∠EMD = ∠DMF.
Problem 3 of the Canadian Mathematical Olympiad 1980
Among all triangles having (i) a fixed angle A and (ii) an inscribed circle of fixed radius r, determine which triangle has the least perimeter.
Problem 3 of Canadian Mathematical Olympiad 1983
The area of a triangle is determined by the lengths of its sides. Is the volume of a tetrahedron determined by the areas of its faces?
Problem 3 of the Irish Mathematical Olympiad 2007
The point P is a fixed point on a circle and Q is a fixed point on a line. The point R is a variable point on the circle such that P, Q and R are not collinear. The circle through P, Q and R meets the line again at V. Show that the line VR passes through a fixed point.
Problem 3 of the British Mathematical Olympiad 2005
Let ABC be a triangle with AC > AB. The point X lies on the side BA extended through A, and the point Y lies on the side CA in such a way that BX = CA and CY = BA. The line XY meets the perpendicular bisector of side BC at P. Show that ∠BPC + ∠BAC = 180°.
Problem 3 of the British Mathematical Olympiad 2006
Let ABC be an acute-angled triangle with AB > AC and ∠BAC = 60°. Denote the circumcenter by O and the orthocenter by H and let OH meet AB at P and AC at Q. Prove that PO = HQ.
Problem 3 of Romanian Mathematical Olympiad 2006
In the acute-angle triangle ABC we have ∠ACB = 45°. The points Aand B are the feet of the altitudes from A and B, respectively. H is the orthocenter of the triangle. We consider the points D and E on the segments AA and BC such that AD = AE = AB. Prove that
a) AB = .
b) CH = DE.
Problem 4 of the Canadian Mathematical Olympiad 1976
Let AB be a diameter of a circle, C be any fixed point between A and B on this diameter, and Q be a variable point on the circumference of the circle. Let P be the point on the line determined by Q and C for which = . Describe, with proof, the locus of the point P.
Problem 4 of the Ibero-American Mathematical Olympiad 1997
In an acute triangle ABC, let AE and BF be its altitudes, and H the orthocenter. The symmetric line of AE with respect to the angle bisector of angle A and the symmetric line of BF with respect to the angle bisector of angle B intersect each other on the point O. The lines AE and AO intersect again the circumscribed circumference to ABC on the points M and N respectively. Let P be the intersection of BC with HN; R the intersection of BC with OM; and S the intersection of HR with OP. Show that AHSO is a parallelogram.
Problem 3 of the Canadian Mathematical Olympiad 1977
N is an integer whose representation in base b is 777. Find the smallest positive integer b for which N is the fourth power of an integer.
Problem 3 of Belarus Mathematical Olympiad 2004
Find all pairs of integers (x, y) satisfying the equation y²(x²+ y² − 2xy − x − y) = (x + y)²( x − y).
Problem 2 of the Vietnamese Regional Competition 1977
Compare . . . . with .
Problem 3 of Asian Pacific Mathematical Olympiad 2002
Let ABC be an equilateral triangle. Let P be a point on the side AC and Q be a point on the side AB so that both triangles ABP and ACQ are acute. Let R be the orthocenter of triangle ABP and S be the orthocenter of triangle ACQ. Let T be the point common to the segments BP and CQ. Find all possible values of
∠CBP and ∠BCQ such that triangle TRS is equilateral.
Problem 3 of the Balkan Mathematical Olympiad 1988
Let ABCD be a tetrahedron and let d be the sum of squares of its edges’ lengths. Prove that the tetrahedron can be included in a region bounded by two parallel planes, the distances between the planes being at most ½.
Problem 3 of the Canadian Mathematical Olympiad 1992
In the diagram, ABCD is a square, with U and V interior points of the sides AB and CD respectively. Determine all the possible ways of selecting U and V so as to maximize the area of the quadrilateral PUQV.
Problem 4 of the International Mathematical Olympia 1960
Construct triangle ABCgiven ha, hb(the altitudes from Aand B) and ma, the median from vertex A.
Problem 5 of the Ibero-American Mathematical Olympiad 1999
An acute triangle 4ABC is inscribed in a circumference of center O. The highs of the triangle are AD; BE and CF. The line EF cut the circumference on P and Q.
a) Show that OA is perpendicular to PQ.
b) If M is the midpoint of BC, show that AP² = 2AD×OM.
Problem 6 of the Canadian Mathematical Olympiad 1971
Show that, for all integers n, n² + 2n + 12 is not a multiple of 121.
Problem 6 of the Ibero-American Mathematical Olympiad 1987
Let ABCD be a plain convex quadrilateral. P, Q are points of AD and BC respectively such that = = .
Show that the angles that are formed by the lines PQ with AB and CD are equal.
Problem 6 of the United States Mathematical Olympiad 1999
Let ABCD be an isosceles trapezoid with AB║CD. The inscribed circle w of triangle BCD meets CD at E. Let F be a point on the (internal) angle bisector of ∠DAC such that EF⊥CD. Let the circumscribed circle of triangle ACF meet line CD at C and G. Prove that the triangle AFG is isosceles.
Problem 7 of Belarus Mathematical Olympiad 2004
Let be given two similar triangles such that the altitudes of the first triangle are equal to the sides of the other. Find the largest possible value of the similarity ratio of the triangles.
Problem 7 of the Canadian Mathematical Olympiad 1969
Show that there are no integers a, b, c for which a² + b² − 8c = 6.
Problem 1 of Austria Mathematical Olympiad 2004
Determine all integers a and b such that (a³ + b)(a + b³) = (a + b).
Problem 2 of the Irish Mathematical Olympiad 1994
Let A, B, C be three collinear points with B between A and C. Equilateral triangles ABD, BCE, CAF are constructed with D, E on one side of the line AC and F on the opposite side. Prove that the centroids of the triangles are the vertices of an equilateral triangle. Prove that the centroid of this triangle lies on the line AC.
Problem 2 of Poland Mathematical Olympiad 2001
ABC is a given triangle. ABDE and ACFG are the squares drawn outside of the triangle. The points M and N are the midpoints of DG and EF, respectively. Find all the values of the ratio MN : BC.
Problem 3 of Balkan Mathematical Olympiad 1993
Circles C1 and C2 with centers O1 and O2, respectively, are externally tangent at point C. A circle C3 with center O touches C1 at A and C2 at B so that the centers O1, O2 lie inside C3. The common tangent to C1 and C2 at C intersects the circle C3 at K and L. If D is the midpoint of the segment KL, show that ∠ADB = ∠O1OO2.
Problem 5 of the Canadian Mathematical Olympiad 1972
Prove that the equation x³ +11³ = y³ has no solution in positive integers x and y.
Problem 5 of the Canadian Mathematical Olympiad 1969
Let ABC be a triangle with sides of lengths a, b and c. Let the bisector of the angle C cut AB in D. Prove that the length of CD is .
Problem 2 of the Ibero-American Mathematical Olympiad 1985
Let P be a point in the interior of the equilateral triangle ABC such that PA = 5, PB = 7, PC = 8. Find the length of the side of the triangle ABC.
Problem 3 of the Ibero-American Mathematical Olympiad 1992
In a equilateral triangle of length 2, it is inscribed a circumference C.
a) Show that for all point P of C the sum of the squares of the distance of the vertices A, B and C is 5.
b) Show that for all point P of C it is possible to construct a triangle such that its sides has the length of the segments AP, BP and CP, and its area is ¼.
Problem 3 of the Ibero-American Mathematical Olympiad 2002
Let P be a point in the interior of the equilateral triangle ABC such that ∠APC = 120°. Let M be the intersection of CP with AB, and N the intersection of AP and BC. Find the locus of the circumcenter of the triangle MBN when P varies.
Problem 3 of the International Mathematical Olympiad 1960
In a given right triangle ABC; the hypotenuse BC, of length a, is divided into n equal parts (n an odd integer). Let α be the acute angle subtending, from A, the segment which contains the midpoint of the hypotenuse. Let h be the length of the altitude to the hypotenuse of the triangle. Prove that tanα = .
Problem 1 of Tournament of Towns 1993
Point M and N are taken on the hypotenuse of a right triangle ABC so that BC = BM and AC = AN. Prove that the angle MCN is equal to 45 degrees.
Problem 2 of the Canadian Mathematical Olympiad 1981
Given a circle of radius r and a tangent line l to the circle through a given point P on the circle. From a variable point R on the circle, a perpendicular RQ is drawn to l with Q on l. Determine the maximum of the area of triangle PQR.
Problem 2 of Canadian Mathematical Olympiad 1985
Prove or disprove that there exists an integer which is doubled when the initial digit is transferred to the end.
Problem 2 of Canadian Mathematical Olympiad 1987
The number 1987 can be written as a three digit number xyz in some base b. If x + y + z = 1 + 9 + 8 + 7, determine all possible values of x, y, z, b.
Problem 2 of the International Mathematical Olympiad 2007
Consider five points A, B, C, D and E such that ABCD is a parallelogram and BCED is a cyclic quadrilateral. Let l be a line passing through A. Suppose that l intersects the interior of the segment DC at F and intersects line BC at G. Suppose also that EF = EG = EC. Prove that l is the bisector of angle DAB.
Problem 4 of Austria Mathematical Olympiad 2009
Let D, E and F be the midpoints of the sides of the triangle ABC (D on BC, E on CA and F on AB). Further let HaHbHc be the triangle formed by the base points of the altitudes of the triangle ABC. Let P, Q and R be the midpoints of the sides of the triangle HaHbHc (P on HbHc, Q on HcHa and R on HaHb).
Show that the lines PD, QE and RF share a common point.
Problem 4 of Asian Pacific Mathematical Olympiad 1995
Let C be a circle with radius R and center O, and S a fixed point in the interior of C. Let AA’ and BB’ be perpendicular chords through S. Consider the rectangles SAMB, SBN’A’, SA’M’B’, and SB’NA. Find the set of all points M, N’, M’, and N when A moves around the whole circle.
Problem 4 of Asian Pacific Mathematical Olympiad 1998
Let ABC be a triangle and D the foot of the altitude from A. Let E and F be on a line through D such that AE is perpendicular to BE, AF is perpendicular to CF, and E and F are different from D. Let M and N be the midpoints of the line segments BC and EF, respectively. Prove that AN is perpendicular to NM.
Problem 4 of the Canadian Mathematical Olympiad 1970
a) Find all positive integers with initial digit 6 such that the integer formed by deleting this 6 is of the original integer.
b) Show that there is no integer such that deletion of the first digit produces a result which is of the original integer.
Problem 4 of Canadian Mathematical Olympiad 1971
Determine all real numbers a such that the two polynomials x² + ax + 1 and x² + x + a have at least one root in common.
Problem 6 of TokyoUniversity Entrance Exam 2010
Given a tetrahedron with four congruent faces such that OA = 3, OB = , AB = 2. Denote by L a plane which contains three points O, A and B.
a) Let H be the foot of the perpendicular drawn from the point C to the plane L. Express vector OH in terms of vectors OA and OB.
b) For a real number t with 0 < t < 1, let Pt, Qt be the points which divide internally the line segments OA, OB into t : 1 – t, respectively. Denote by M a plane which is perpendicular to the plane L. Find the sectional area S(t) of the tetrahedron OABC cut by the plane M.
c) When t moves in the range of 0 < t < 1, find the maximum value of S(t).
Problem 22 of Tournament of Towns 2008
A 30-gon A1A2 …A30 is inscribed in a circle of radius of 2. Prove that one can choose a point Bk on the arc AkAk+1 for 1 ≤ k ≤ 29 and a point B30 on the arc A30A1, such that the numerical value of the area of the 60-gon A1B1A2B2…A30B30 is equal to the numerical value of the perimeter of the original 30-gon.
Problem 2 of British Mathematical Olympiad 1988
Points P, Q lie on the sides AB, AC respectively of triangle ABC and are distinct from A. The lengths AP, AQ are denoted by x, y respectively, with the convention that x > 0 if P is on the same side of A as B, and x < 0 on the opposite side; similarly for y. Show that PQ passes through the centroid of the triangle if and only if
3xy = bx + cy where b = AC, c = AB.
Problem 2 of Austria Mathematical Olympiad 1989
Find all triples (a, b, c) of integers with abc = 1989 and a + b – c = 89.
Problem 3 of Canadian Mathematical Olympiad 1975
For a positive number such as 3.27, 3 is referred to as the integral part of the number and .27 as the decimal part. Find a positive number such that its decimal part, its integral part, and the number itself form a geometric progression.
Problem 1 of International Mathematical Talent Search Round 16
Prove that if a + b + c = 0, then a³+ b³+ c³ = 3abc.
Problem 3 of British Mathematical Olympiad 1991
ABCD is a quadrilateral inscribed in a circle of radius r. The diagonal AC, BD meet at E. Prove that if AC is perpendicular to BD then EA² + EB² + EC² + ED² = 4r² (*). Is it true that if (*) holds then AC is perpendicular to BD? Give a reason for your answer.
Problem 3 of the British Mathematical Olympiad 2000
Triangle ABC has a right angle at A. Among all points P on the perimeter of the triangle, find the position of P such that AP + BP + CP is minimized.
Problem 9 of Canadian Mathematical Olympiad 1970
Let f(n) be the sum of the first n terms of the sequence 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, . . .
a) Give a formula for f(n).
b) Prove that f(s + t) − f(s − t) = st where s and t are positive integers and s > t.
Problem 1 of British Mathematical Olympiad 1988
Find all integers a, b, c for which
(x – a)(x – 10) + 1 = (x + b)(x + c) for all x.
Problem 1 of Canadian Mathematical Olympiad 1973
a) Solve the simultaneous inequalities, x < and x < 0; i.e, find a single inequality equivalent to the two given simultaneous inequalities.
b) What is the greatest integer which satisfies both inequalities 4x + 13 < 0 and x² + 3x > 16?
c) Give a rational number between 11/24 and 6/13.
d) Express 100000 as a product of two integers neither of which is an integral multiple of 10.
e) Without the use of logarithm tables evaluate 1/log36 + 1/log36.
Problem 4 of the British Mathematical Olympiad 1995
ABC is a triangle, right-angled at C. The internal bisectors of angles BAC and ABC meet BC and CA at P and Q, respectively. M and N are the feet of the perpendiculars from P and Q to AB. Find angle MCN.
Problem 1 of Hong Kong Mathematical Olympiad 2009 (Event 3)
Find the smallest prime factor of 101 + 303.
Problem 1 of Hong Kong Mathematical Olympiad 2009 (Event 2)
p = 2 – 2 – 2 – 2 – . . . – 2 – 2 + 2, find the value of p.
Problem 2 of the British Mathematical Olympiad 1994
In triangle ABC the point X lies on BC.
a) Suppose that ∠BAC = 90°, that X is the midpoint of BC, and that ∠BAX is one third of ∠BAC. What can you say and prove about triangle ACX?
b) Suppose that ∠BAC = 60°, that X lies one third of the way from B to C, and that AX bisects ∠BAC. What can you say and prove about triangle ACX?
Problem 3 of Hong Kong Mathematics Olympiad 2009
In the figure below, if ∠A = 60°, ∠B = ∠D = 90°, BC = 2, CD = 3 and AB = x. Find the value of x.
Problem 7 of Canadian Mathematical Olympiad 1975
A function f(x) is periodic if there is a positive number p such that
f (x + p) = f(x) for all x. For example, sinx is periodic with period 2p. Is the function sin(x²) periodic? Prove your assertion.
Problem 3 of Austria Mathematical Olympiad 1985
A line meets the lines containing sides BC, CA, AB of a triangle ABC at E, P, I, respectively. The points F, Q, J are symmetric to E, P, I with respect to the midpoints of BC, CA, AB, respectively. Prove that F, Q and J are collinear.
Problem 3 of British Mathematical Olympiad 2010
Let ABC be a triangle with ∠CAB a right-angle. The point L lies on the side BC between B and C. The circumcircle of triangle ABL meets the line AC again at M and the circle of triangle CAL meets the line AB again at N. Prove that L, M and N lie on a straight line.
Problem 1 of the Uzbekistan Mathematical Olympiad 2008
Let triangle ABC with AB = c, AC = b, BC = a, R the circum-radius, p the half perimeter of triangle ABC.
If = then find the value of cosA.
Problem 3 of the Irish Mathematical Olympiad 2001
Prove that if an odd prime number p can be expressed in the form x – y, for some integers x, y, then = for some odd integer v.
Problem 3 of Poland Mathematical Olympiad 2008
Disjoint circles G1 and G2, with O1, O2 as their respective centers, are tangent to the line k at A1, A2. The point C on the segment
O1O2 satisfies ∠A1CA2 = 90°. Let B1 be the second intersection point of A1C with G1 and B2 be the second intersection point of A2C with G2. Prove that B1B2 is tangent to the circles G1, G2.
Problem 1 of British Mathematical Olympiad 2009
Find all integers x, y and z such that x² + y² + z² = 2(yz + 1) and x + y + z = 4018.
Problem 2 of Spain Mathematical Olympiad 1996
Let G be the centroid of a triangle ABC. Prove that if AB + GC = AC + GB, then the triangle is isosceles.
Problem 6 of Spain Mathematical Olympiad 1996
A regular pentagon is constructed externally on each side of a regular pentagon of side 1. The figure is then folded and the two edges of the external pentagons meeting at each vertex of the original pentagon are glued together. Find the volume of water that can be poured into the obtained container.
Problem 2 of Junior Balkan Mathematical Olympiad 1998
Let ABCDE be a convex pentagon such that AB = AE = CD = 1, ∠ABC = ∠DEA = 90° and BC + DE = 1. Compute the area of the pentagon.
Problem 4 of International Mathematical Talent Search Round 19
Suppose that f satisfies the functional equation
2f (x) + 3f ( ) = 100x + 80.
Find f (3).
Problem 2 of the Central America Mathematical Olympiad 2011
In a scalene triangle ABC, D is the foot of the altitude through A,
E is the intersection of AC with the bisector of ∠ABC and F is a point on AB. Let O be the circumcenter of ABC and X = AD ∩ BE, Y = BE ∩ CF, Z = CF ∩ AD. If XYZ is an equilateral triangle, prove that one of the triangles OXY, OYZ, OZX must be equilateral.
Problem 1 of the Irish Mathematical Olympiad 2001
Find, with proof, all solutions of the equation 2 = a! + b! + c! in positive integers a, b, c and n. (Here, ! means “factorial”.)
Problem 2 of the Irish Mathematical Olympiad 2001
Let ABC be a triangle with sides BC, CA, AB of lengths a, b, c, respectively. Let D, E be the midpoints of the sides AC, AB, respectively. Prove that BD is perpendicular to CE if, and only if, b² + c² = 5a².
Problem 2 of the Canadian Mathematical Olympiad 1979
It is known in Euclidean geometry that the sum of the angles of a triangle is constant. Prove, however, that the sum of the dihedral angles of a tetrahedron is not constant.
Problem 5 of Malaysia National Olympiad 2010 Muda Category
Find the number of triples of nonnegative integers (x, y, z) such that x² + 2xy + y² – z² = 9.
Problem 2 of the Iranian Mathematical Olympiad 1993
In the figure below, areas of triangles AOD, DOC, and AOB are given. Find the area of triangle OEF in terms of areas of these three triangles.
Problem 6 of the Irish Mathematical Olympiad 1993
The real numbers x, y satisfy the equations
x³ – 3x² + 5x – 17 = 0 (i)
y³ – 3y² + 5y + 11 = 0 (ii)
Find x + y.
Problem 1 of Mediterranean Mathematics Olympiad 2008
Let ABCDEF be a convex hexagon such that all of its vertices are on a circle. Prove that AD, BE, and CF are concurrent if and only if ×× = 1.
Problem 1 of International Mathematical Talent Search Round 2
What is the smallest integer multiple of 9997, other than 9997 itself, which contains only odd digits?
Problem 6 of Canadian MO Qualification Repechage 2011
In the diagram, ABDF is a trapezoid with AF parallel to BD and AB perpendicular to BD. The circle with center B and radius AB meets BD at C and is tangent to DF at E. Suppose that x is equal to the area of the region inside quadrilateral ABEF but outside the circle, that y is equal to the area of the region inside ΔEBD but outside the circle, and that a = ∠EBC. Prove that there is exactly one measure a, with 0° ≤ a ≤ 90°, for which x = y and that this value of a satisfies < sina < .
Problem 7 of Australia Mathematical Olympiad 2010
On the edges of a triangle ABC are drawn three similar isosceles triangles APB (with AP = PB), AQC (with AQ = QC) and BRC (with BR = RC). The triangles APB and AQC lie outside the triangle ABC and the triangle BRC is lying on the same side of the line BC as the triangle ABC. Prove that the quadrilateral PAQR is a parallelogram.
Problem 5 of Turkey Mathematical Olympiad 2007
Let ABC be a triangle with ∠B = 90°. The incircle of ABC touches the side BC at D. The incenters of triangles ABD and ADC are X and Z, respectively. The lines XZ and AD are intersecting at the point K. XZ and circumcircle of ABC are intersecting at U and V. Let M be the midpoint of line segment [UV]. AD intersects the circumcircle of ABC at Y other than A. Prove that |CY| = 2|MK|.
Problem 2 of Turkey MO Team Selection Test 1996
In a parallelogram ABCD with ∠A < 90°, the circle with diameter AC intersects the lines CB and CD again at E and F, and the tangent to this circle at A meets the line BD at P. Prove that the points P, E, F are collinear.
Problem 6 of Pan African 2009
Points C, E, D and F lie on a circle with center O. Two chords CD and EF intersect at a point N. The tangents at C and D intersect at A, and the tangents at E and F intersect at B. Prove that ON ⊥ AB.
Problem 7 of Belarus Mathematical Olympiad 1997
If ABCD is a convex quadrilateral with ∠ADC = 30° and BD = AB + BC + CA, prove that BD bisects ∠ABC.
Problem 2 of the Vietnamese Mathematical Olympiad 1986
Let R and r be the respective circumradius and inradius of a regular 1986-gonal pyramid. Prove that ≥ 1 + and find the total area of the surface of the pyramid when equality occurs.
Problem 5 of British Mathematical Olympiad 1990
The diagonal of a convex quadrilateral ABCD intersect at O. The centroids of triangles AOD and BOC are P and Q; the orthocenters of triangles AOB and COD are R and S, respectively. Prove that PQ is perpendicular to RS.
Problem 9 of Russia Sharygin Geometry Olympiad 2010
A point inside a triangle is called “good” if three cevians passing through it are equal. Assume for an isosceles triangle ABC with AB = BC the total number of “good” points is odd. Find all possible values of this number.
Sample Mathematical Olympiad Problem
Given triangle ABC, its orthocenter H and its altitude AD, BE and CF such that the perimeters of the triangles AHB, AHC and BHC are the same. Prove that triangle ABC is equilateral. (This problem was proposed but has never been selected for any competition.)
Problem 10 of Russia Sharygin Geometry Olympiad 2010
Let three lines forming a triangle ABC be given. Using a two-sided ruler and drawing at most eight lines construct a point D on the side AB such that = .
Problem 1 of the Russian Mathematical Olympiad 2008
Do there exist 14 positive integers such that, upon increasing each of them by 1, their product increases exactly 2008 times?
Problem 6 Tournament of Towns 2008
Let ABC be a non-isosceles triangle. Two isosceles triangles AB’C with base AC and CA’B with base BC are constructed outside of triangle ABC. Both triangles have the same base angle j. Let C1 be a point of intersection of the perpendicular from C to A’B’ and the perpendicular bisector of the segment AB. Determine the value of ∠AC1B.
Problem 4 of Bulgaria Mathematical Olympiad 2011
Point O is inside triangle ABC. The feet of perpendicular from O to AB, BC, CA are D, E, F, respectively. Perpendiculars from A and B, respectively to DF and ED meet at P. Let H be the foot of perpendicular from P to AB. Prove that D, E, F, H are
Problem 4 of Hong Kong Mathematical Olympiad 2004
In the figure below, BEC is a semicircle and F is a point on the diameter BC. Given that BF:FC = 3:1, AB = 8 and AE = 4. Find
Problem 2 of Hong Kong Mathematical Olympiad 2009
Let n be the integral part of ; find the value of n.
Problem 3 of Hong Kong Mathematical Olympiad 2009 (Event 2)
Given that x is a positive real number and x·3 = 3. If k is a positive integer and k < x < k + 1, find the value of k.
Problem 6 of Mongolian Mathematical Olympiad 2000
In a triangle ABC, the angle bisectors at A, B, C meet the opposite sides at A, B, C, respectively. Prove that if the quadrilateral BABC is cyclic, then = + .
Problem 3 of Spain Mathematical Olympiad 2003
The altitudes of triangles ABC meet at H. We know that AB = CH. Determine the angle BCA.
Problem 3 of Spain Mathematical Olympiad 2006
ABC is an isosceles triangle with AB = AC. Let H be a point on the circle tangent to the sides AB at B and AC at C. We call a, b, and c the distances from H to the sides BC, AB and AC, respectively. Prove that a² = bc.
Problem 3 of Spain Mathematical Olympiad 2004
ABCD is a quadrilateral with P and Q the midpoints of the diagonals BD and AC, respectively. The line through P and is parallel to AC meets the line through Q and is parallel to BD at O; X, Y, Z and T are the midpoints of AB, BC, CD and AD, respectively. Prove that the four quadrilaterals OXBY, OYCZ, OZDT and OTAX have the same area.
Problem 2 of Spain Mathematical Olympiad 2002
In a triangle ABC, A’ is the foot of the vertex A onto BC and H the orthocenter.
a) Given a positive real number k such that = k, find the relationship between the angles B and C as a function of k.
b) If B and C are fixed, find the locus of the vertex A for each value of k.
Problem 1 of British Mathematical Olympiad 1985
Two circles S1 and S2 each touch a straight line p at the same point P. All points of S2, except P, are in the interior of S1. A straight line q (i) is perpendicular to p; (ii) touches S2 at R; (iii) cuts p at L; and (iv) cuts S1 at N and M, where M is between L and R.
a) Prove that RP bisects angle MPN.
b) If MP bisects angle RPL, find, with proof, the ratio of the areas of S1 and S2.
Problem 5 of British Mathematical Olympiad 2010
Circles S1 and S2 meet at L and M. Let P be a point on S2. Let PL and PM meet S1 again at Q and R, respectively. The lines QM and RL meet at K. Show that, as P varies on S2, K lies on a fixed circle.
Problem 4 of the Vietnamese Mathematical Olympiad 1989
Are there integers x, y, not both divisible by 5, such that x + 19y = 198×10?
Problem 2 of Tournament of Towns 2008
Twenty-five of the numbers 1, 2, … , 50 are chosen. Twenty-five of the numbers 51, 52, … , 100 are also chosen. No two chosen numbers differ by 0 or 50. Find the sum of all 50 chosen numbers.
Problem 4 of Turkey MO Team Selection Test 1997
A convex ABCDE is inscribed in a unit circle, AE being its diameter. If AB = a, BC = b, CD = c, DE = d and ab = cd = , compute AC + CE in terms of a, b, c, d.
Problem 1 of Spain Mathematical Olympiad 1999
The lines t and t’ are tangent to the parabola of equation y = x² at points A and B intersect at point C. The median of the triangle ABC corresponds to the vertex C has length m. Determine the area of triangle ABC in terms of m.
Problem 2 of the Irish Mathematical Olympiad 2006
P and Q are points on the equal sides AB and AC respectively of an isosceles triangle ABC such that AP = CQ. Moreover, neither P nor Q is a vertex of ABC. Prove that the circumcircle of the triangle APQ passes through the circumcenter of the triangle ABC.
Problem 2 of the Irish Mathematical Olympiad 2007
Prove that a triangle ABC is right-angled if and only if sin²A + sin²B + sin²C = 2.
Problem 2 of the British Mathematical Olympiad 2005
In triangle ABC, ∠BAC = 120°. Let the angle bisectors of angles A, B and C meet the opposite sides in D, E and F, respectively. Prove that the circle on diameter EF passes through D.
Problem 3 of Asian Pacific Mathematical Olympiad 1989
Let A, A, A be three points in the plane, and for convenience, let A = A, A = A. For n = 1, 2, and 3, suppose that B is the midpoint of AA, and suppose that C is the midpoint of AB. Suppose that AC and BA meet at D, and that AB and CA meet at E. Calculate the ratio of the area of triangle DDD to the area of triangle EEE.
Problem 3 of Asian Pacific Mathematical Olympiad 1990
Consider all the triangles ABC which have a fixed base BC and whose altitude from A is a constant h. For which of these triangles is the product of its altitudes a maximum?
Problem 3 of Asian Pacific Mathematical Olympiad 1995
Let PQRS be a cyclic quadrilateral such that the segments PQ and RS are not parallel. Consider the set of circles through P and Q, and the set of circles through R and S. Determine the set I of points of tangency of circles in these two sets.
Problem 3 of Asian Pacific Mathematical Olympiad 1999
Let C1 and C2 be two circles intersecting at P and Q. The common tangent, closer to P, of C1 and C2 touches C1 at Aand C2 at B. The tangent of C1 at Pmeets C2 at C, which is different from P, and the extension of AP meets BC at R. Prove that the circumcircle of triangle PQR is tangent to BP and BR.
Problem 1 of Turkey MO Team Selection Test 1998
Squares BAXX’ and CAYY’ are drawn on the exterior of a triangle ABC with AB = AC. Let D be the midpoint of BC, and E and F be the feet of the perpendiculars from an arbitrary point K on the segment BC to BY and CX, respectively.
a) Prove that DE = DF.
b) Find the locus of the midpoint of EF.
Problem 2 of the Argentine MO Team Selection Test 2008
Triangle ABC is inscribed in a circumference Γ. A chord MN = 1 of Γ intersects the sides AB and AC at X and Y, respectively, with M, X, Y, N in that order in MN. Let UV be the diameter of Γ perpendicular to MN with U and A in the same semi-plane respect to MN. Lines AV, BU and CU cut MN in the ratios , and ,
respectively (start counting from M). Find XY.
Problem 4 of International Mathematical Talent Search Round 2
Let a, b, c, and d be the areas of the triangular faces of a tetrahedron, and let ha, hb, hc, and hd be the corresponding altitudes of the tetrahedron. If V denotes the volume of the tetrahedron, prove that
(a + b + c + d)(ha + hb + hc + hd ) ≥ 48V.
Problem 3 of International Mathematical Talent Search Round 3
Find k if P, Q, R and S are points on the sides of quadrilateral ABCD so that = = = = k, and the area of quadri-lateral PQRS is exactly 52% of the area of quadrilateral ABCD.
Problem 13 of the Iranian Mathematical Olympiad 2010
In a quadrilateral ABCD, E and F are on BC and AD, respectively such that each of the area of triangle AED or triangle BFC is of the area of ABCD. R is the intersection point of diagonals of ABCD. It’s also given that = , and = .
a) In what ratio does EF cut the diagonals?
b) Find .
Problem 1 of International Mathematical Talent Search Round 4
Use each of the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 exactly twice to form two distinct prime numbers whose sum is as small as possible. What must be this minimal sum be? (Note: The five smallest primes are 2, 3, 5, 7 and 11.)
Problem 4 of International Mathematical Talent Search Round 4
Let ΔABC be an arbitrary triangle, and construct P, Q, and R so that each of the angles marked is 30°. Prove that ΔPQR is an equilateral triangle.
Problem 3 of International Mathematical Talent Search Round 41
Suppose = for some angle x, 0 ≤ x ≤ . Determine for the same x.
Problem 1 of International Mathematical Talent Search Round 41
Determine the unique positive integers m and n for which the approximation = .2328767 is accurate to the seven decimals; i.e., 0.2328767 ≤ < 0.2328768.
Problem 4 of the Vietnamese Mathematical Olympiad 1964
The tetrahedron SABC has the faces SBC and ABC perpendicular to each other. The three angles at S are all 60° and SB = SC = 1. Find its volume.
Problem 5 of the Vietnamese Mathematical Olympiad 1964
The triangle ABC has perimeter p. Find the side length AB and the area S in terms of ∠A, ∠B and p.
Problem B6 of British Mathematical Olympiad 1974
X and Y are the feet of the perpendiculars from P to CA and CB respectively, where P is in the plane of triangle ABC and PX = PY. The straight line through P which is perpendicular to AB cuts XY at Z. Prove that CZ bisects AB.
Problem 3 of Austria Mathematical Olympiad 2001
In a convex pentagon, the areas of the triangles ABC, ABD, ACD and ADE are all equal to the same value F. What is the area of the triangle BCE?
Problem 4 of Spain Mathematical Olympiad 1994
In a triangle ABC with ∠A = 36° and AB = AC, the bisector of the angle at C meets the opposite side at D. Compute the angles of ΔBCD. Express the length a of side BC in terms of the length b of side AC without using trigonometric functions.
Problem 7 Baltic Way 1995
Prove that sin³18° + sin²18° = .
Problem 1 of the Vietnamese Mathematical Olympiad 1982
Determine a quadric polynomial with integral coefficients whose roots are cos72° and cos144°.
Problem 5 of the Vietnamese Mathematical Olympiad 1994
S is a sphere with center O. G and G’ are two perpendicular great circles on S. Take A, B, C on G and D on G’ such that the altitudes of the tetrahedron ABCD intersect at a point. Find the locus of the intersection.
Problem 2 of Tournament of Towns 1984
Prove that among 18 consecutive three digit numbers there must be at least one which is divisible by the sum of its digits.
Problem 4 of Canada Students Math Olympiad 2011
Circles Γ1 and Γ2 have centers O1 and O2 and intersect at P and Q. A line through P intersects Γ1 and Γ2 at A and B, respectively, such that AB is not perpendicular to PQ. Let X be the point on PQ such that XA = XB and let Y be the point within AO1O2B such that AYO1 and BYO2 are similar. Prove that 2∠O1AY = ∠AXB.
Problem 2 of the Russian Mathematical Olympiad 2001
Let the circle w1 be internally tangent to another circle w2 at N. Take a point K on w1 and draw a tangent AB which intersects w2 at A and B. Let M be the midpoint of the arc AB which is on the opposite side of N. Prove that the circumradius of the triangle KBM doesn’t depend on the choice of K.
Problem 2 of Austria Mathematical Olympiad 2001
Determine all real solutions of the equation
(x + 1) + (x + 1)(x – 2) + (x + 1)(x –2) + … + (x + 1)(x –2) + (x + 1)(x –2) + (x –2) = 0.
Problem 1 of Tournament of Towns 2007 Senior Level
A, B, C and D are points on the parabola y = x² such that AB and CD intersect on the y-axis. Determine the x-coordinate of D in terms of the x-coordinates of A, B and C, which are a, b and c, respectively.
Problem 1 Set 6 of India Postal Coaching 2011
Let ABCD be a quadrilateral with an inscribed circle, center O. Let AO = 5, BO = 6, CO = 7, DO = 8. If M and N are the midpoints of the diagonals AC and BD, determine .
Problem 5 of International Mathematical Talent Search Round 21
Assume that triangle ABC, shown below, is isosceles, with ∠ABC = ∠ACB = 78°. Let D and E be points on sides AB and AC, respectively, so that ∠BCD = 24° and ∠CBE = 51°. Determine, with proof, ∠BED.
Problem 4 of International Mathematical Talent Search Round 22
As shown below, a large wooden cube has one corner sawed off forming a tetrahedron ABCD. Determine the length of CD, if AD = 6, BD = 8 and area of triangle ABC = 74.
Problem 5 of International Mathematical Talent Search Round 27
Is it possible to construct in the plane the midpoint of a given segment using compasses alone (i.e., without using a straight edge, except for drawing the segment)?
Problem 1 of International Mathematical Talent Search Round 22
In 1996 nobody could claim that on their birthday their age was the sum of the digits of the year in which they were born. What was the last year prior to 1996 which had the same property?
Problem 3 of the Vietnamese Mathematical Olympiad 1982
Let be given a triangle ABC. Equilateral triangles BCA1 and BCA2 are drawn so that A and A1 are on one side of BC, whereas A2 is on the other side. Points B1, B2, C1, C2 are analogously defined. Prove that S(A2B2C2) = 5S(ABC) – S(A1B1C1).
Problem 1 of Canada Students Math Olympiad 2011
In triangle ABC, ∠BAC = 60° and the incircle of ABC touches AB and AC at P and Q, respectively. Lines PC and QB intersect at G. Let R be the circumradius of BGC. Find the minimum value of .
Problem 1 of British Mathematical Olympiad 2011
One number is removed from the set of integers from 1 to n. The
average of the remaining numbers is 40. Which integer was
Problem 4 of Morocco Mathematical Olympiad 2011 (Day 3)
Two circles C1 and C2 intersect at A and B. A line passing through B intersects C1 at C and C2 at D. Another line passing through B intersects C1 at E and C2 at F; CF intersects C1 and C2 at P and Q, respectively. Make sure that in your diagram, B, E, C, A, P ∈ C1 and B, D, F, A, Q ∈ C2, in this order. Let M and N be the midpoints of the arcs BP and BQ, respectively. Prove that if CD = EF, then the points C, F, M, N are concylic, in this order.
Problem 3 of Austria Mathematical Olympiad 2005
In an acute-angled triangle ABC two circles k1 and k2 are drawn whose diameters are the sides AC and BC. Let E be the foot of the altitude hb on AC and let F be the foot of the altitude ha on BC.
Let L and N be the intersections of the line BE with the circle k1 (L on the line BE) and let K and M be the intersections of the line AF with the circle k2 (K on the line AF). Show that KLMN is a cyclic quadrilateral.
Problem 3 of pre-Vietnamese Mathematical Olympiad 2011
Two circles Γ and Π intersect at A and B. Take two points P, Q on Γ and Π, respectively, such that AP = AQ. The line PQ intersects Γ and Π, respectively at M and N. Let E, F, respectively be the centers of the two arcs BP and BQ (which do not contain A). Prove that MNEF is a cyclic quadrilateral.
Problem 3 of International Mathematical Talent Search Round 4
Prove that a positive integer can be expressed in the form 3x² + y² if and only if it can also be expressed in the form u² + uv + v², where x, y, u and v are positive integers.
Problem 5 of International Mathematical Talent Search Round 13
Armed with just a compass – no straightedge – draw two circles that intersect at right angles; that is, construct overlapping circles in the same plane, having perpendicular tangents at the two points where they meet.
Draw arbitrary circles Γ with center O1 and Φ with Φ larger than Γ and overlapping Γ at two points A and B as shown. Next, draw circle D with center A and radius AO1. Continue by drawing circle.
Problem 10 of Austria Mathematical Olympiad 2006
Let A be a nonzero integer. Solve the following system in integers:
x + y + z = A (i)
+ + = (ii)
xyz = A (iii)
Problem 7 of Malaysia National Olympiad 2010 Sulung Category
A line segment of length 1 is given on the plane. Show that a line segment of length can be constructed using only a straight-edge and a compass.
Problem 4 Set 4 of India Postal Coaching 2011
Let C1, C2 be two circles in the plane intersecting at two distinct points. Let P be the midpoint of a variable chord AB of C2 with the property that the circle on AB as diameter meets C1 at a point T such that PT is tangent to C1. Find the locus of P.
Problem 8 of Malaysia National Olympiad 2010 Bongsu category
Find the last digit of 7 × 7 × 7 × … × 7 × 7.
Problem 9 of Malaysia National Olympiad 2010 Bongsu category
Show that there exist integers m and n such that = – .
Problem 2 of the Vietnamese MO Team Selection Test 1985
Let ABC be a triangle with AB = AC. A ray Ax is constructed in space such that the three planar angles of the trihedral angle ABCx at its vertex A are equal. If a point S moves on Ax, find the locus of the incenter of triangle SBC.
Problem 3 of Italian Mathematical Olympiad 2002
Let A and B are two points on a plane, M be the midpoint of AB, r be a line, R and S be the projections of A and B onto r. Assuming that A, M, and R are not collinear, prove that the circumcircle of triangle AMR has the same radius as the circumcircle of BSM.
Problem 3 of Italian Mathematical Olympiad 2003
Let a semicircle is given with diameter AB and center O and let C be an arbitrary point on the segment OB. Point D on the semi-circle is such that CD is perpendicular to AB. A circle with center P is tangent to the arc BD at F and to the segment CD and AB at E and G, respectively. Prove that the triangle ADG is isosceles.
Problem 4 of Italian Mathematical Olympiad 2002
Find all values of n for which all solutions of the equation x³ – 3x + n = 0 are integers.
Problem 6 of Austria Mathematical Olympiad 2003
Let ABC be an acute-angled triangle. The circle k with diameter AB intersects AC and BC again at P and Q, respectively. The tangents to k at A and Q meet at R, and the tangents at B and P meet at S. Show that C lies on the line RS.
Problem 2 of Australia Mathematical Olympiad 2010
Let the number of different divisors of the integer n be N(n); e.g. 24 has the divisors 1, 2, 3, 4, 6, 8, 12 and 24, so N(24) = 8. Determine whether the sum
N(1) + N(2) + . . . + N(1998) is odd or even.
Problem 2 of the Ibero-American Mathematical Olympiad 1987
In a triangle ABC, M and N are the midpoints of the sides AC and AB respectively, and P is the point of intersection of BM and CN. Show that if it is possible to inscribe a circumference in the quadrilateral ANPM, then the triangle ABC is isosceles.
Problem 4 of Mongolian Mathematical Olympiad 1999
Is it possible to place a triangle with area 1999 and perimeter 1999² in the interior of a triangle with area 2000 and perimeter 2000²?
Problem 4 of International Mathematical Talent Search Round 18
Let a, b, c, d be distinct real numbers such that a + b + c + d = 3 and a² + b² + c² + d² = 45. Find the value of the expression
+ + + .
Problem 3 of the Korean Mathematical Olympiad 2000
A rectangle ABCD is inscribed in a circle with center O. The exterior bisectors of ∠ABD and ∠ADB intersect at P; those of ∠DAB and ∠DBA intersect at Q; those of ∠ACD and ∠ADC intersect at R, and those of ∠DAC and ∠DCA intersect
Problem 1 of International Mathematical Talent Search Round 27
Are there integers M, N, K, such that M + N = K and
i) each of them contains each of the seven digits 1, 2, 3, … , 7 exactly once?
ii) each of them contains each of the nine digits 1, 2, 3, … , 9 exactly once?
Problem 1 of Italy Mathematical Olympiad 2003
Find all three digit integers n which are equal to the integer formed by three last digit of n².
Problem 3 of Spain Mathematical Olympiad 1988
Prove that if one of the number 25x + 3y, 3x + 7y (where x, y ∈ ℤ) is a multiple of 41, then so is the other.
Problem 4 of Germany Mathematical Olympiad 1998
Do there exist three consecutive odd integers whose sum of squares is a four-digit number having all its digits equal?
Problem 5 of International Mathematical Talent Search Round 25
As shown in the figure on the right, PABCD is a pyramid, whose base, ABCD, is a rhombus with ∠DAB = 60°. Assume that PC² = PB² + PD². Prove that PA = AB.
Problem 3 of Italy Mathematical Olympiad 2009
A natural number n is called nice if it enjoys the following properties:
• the expression is made up of 4 decimal digits;
• the first and third digits of n are equal;
• the second and fourth digits of n are equal;
• the product of the digits of n divides n².
Determine all nice numbers.
Problem 2 of Spain Mathematical Olympiad 1992
Given two circles of radii r and r’ exterior to each other, construct a line parallel to a given line and intersecting the two circles in chords with the sum of lengths l.
Problem 5 of Spain Mathematical Olympiad 1992
Given a triangle ABC, show how to construct the point P such that ∠PAB = ∠PBC = ∠PCA. Express this angle in terms of ∠A, ∠B, ∠C using trigonometric functions.
Problem 2 Asian Pacific Mathematical Olympiad 2003
Suppose ABCD is a square piece of cardboard with side length a. On a plane are two parallel lines l and l, which are also a units apart. The square ABCD is placed on the plane so that sides AB and AD intersect l at E and F respectively. Also, sides CB and CD intersect l at G and H respectively. Let the perimeters of triangle AEF and triangle CGH be m and m respectively. Prove that no matter how the square was placed, m + m remains constant.
Problem 2 of Asian Pacific Mathematical Olympia 2004
Let O be the circumcenter and H the orthocenter of an acute triangle ABC. Prove that the area of one of the triangles AOH, BOH and COH is equal to the sum of the areas of the other two.
Problem 4 of the Ibero-American Mathematical Olympiad 1989
The incircle of the triangle ABC, is tangential to both sides AC and BC at M and N, respectively. The angle bisectors of the angles A and B intersect MN at points P and Q, respectively. Let O be the incenter of the triangle ABC. Prove that MP×OA = BC×OQ.
Problem 6 of Austria Mathematical Olympiad 1990
A convex pentagon ABCDE is inscribed in a circle. The distances of A from the lines BC, CD, DE are a, b, c, respectively. Compute the distance of A from the line BE.
Problem 6 of the British Mathematical Olympiad 2000
Two intersecting circles C1 and C2 have a common tangent which touches C1 at P and C2 at Q. The two circles intersect at M and N, where N is nearer to PQ than M is. Prove that the triangles MNP and MNQ have equal areas.
Problem 3 of Austria Mathematical Olympiad 2001
We are given a triangle ABC and its circumcircle with center O and radius r. Let K be the circle with midpoint O and radius 2r, and let c’ be the tangent to K that is parallel to c = AB and has the property that C lies between c and c’. Analogously, the tangents a’ and b’ are determined. The resulting triangle with sides a’, b’, c’ is called triangle A’B’C’. Prove that the lines joining the midpoints of corresponding sides of the triangles ABC and A’B’C’ pass through a common point.
Problem at Art Of the Problem Solving website 2011
There is a point P inside a rectangle ABCD such that ∠APD = 110°, ∠PBC = 70°, ∠PCB = 30°. Find ∠PAD.
Problem 1 of the Asian Pacific Mathematical Olympiad 1991
Let G be the centroid of triangle ABC and M be the midpoint of BC. Let X be on AB and Y on AC such that the points X, Y, and G are collinear and XY and BC are parallel. Suppose that XC and GB intersect at Q and Y B and GC intersect at P. Show that triangle MPQ is similar to triangle ABC.
Problem 1 of the Asian Pacific Mathematical Olympiad 1992
A triangle with sides a, b, and c is given. Denote by s the semi-perimeter, that is s = . Construct a triangle with sides s − a, s − b, and s − c. This process is repeated until a triangle can no longer be constructed with the side lengths given. For which original triangles can this process be repeated indefinitely?
Problem 1 of the British Mathematical Olympiad 2008
Find all solutions in non-negative integers a, b to + = .
Problem 1 of the Canadian Mathematical Olympiad 1969
Show that if = = and p, p, p are not all zero, then ()= for every positive integer n.
Problem 1 of the Canadian Mathematical Olympiad 1971
DEB is a chord of a circle such that DE = 3 and EB = 5. Let O be the center of the circle. Join OE and extend OE to cut the circle at C. Given EC = 1, find the radius of the circle.
Problem 1 of the Canadian Mathematical Olympiad 1972
Given three distinct unit circles, each of which is tangent to the other two, find the radii of the circles which are tangent to all three circles.
Problem 1 of the Canadian Mathematical Olympiad 1975
Simplify
Problem 1 of Canadian Mathematical Olympiad 1982
In the diagram, OBi is parallel and equal in length to AA for i = 1, 2, 3 and 4 (A = A). Show that the area of BBBB is twice that of AAAA.
Problem 1 of the Canadian Mathematical Olympiad 1986
In the diagram line segments AB and CD are of length 1 while angles ABC and CBD are 90° and 30°, respectively. Find AC.
Problem 1 of the Irish Mathematical Olympiad 2007
Let r, s and t be the roots of the cubic polynomial p(x) = x³ − 2007x + 2002. Determine the value of . ..
Problem 1 of Romanian Mathematical Olympiad 2006
Let ABC be a triangle and the points M and N on the sides AB and BC, respectively, such that 2CN/BC = AM/AB. Let Pbe a point on the line AC. Prove that the lines MN and NP are perpendicular if and only if PN is the interior angle bisector of ∠MPC.
Problem 2 of the British Mathematical Olympiad 2007
Let triangle ABC have incenter I and circumcenter O. Suppose that ∠AIO = 90° and ∠CIO = 45°. Find the ratio AB : BC : CA.
Problem 2 of the British Mathematical Olympiad 2008
Let ABC be an acute-angled triangle with ∠B = ∠C. Let the circumcenter be O and the orthocenter be H. Prove that the center of the circle BOH lies on the line AB. The circumcenter of a triangle is the center of its circumcircle. The orthocenter of a triangle is the point where its three altitudes meet.
Problem 2 of the British Mathematical Olympiad 2009
In triangle ABC the centroid is G and D is the midpoint of CA. The line through G parallel to BC meets AB at E. Prove that ∠AEC = ∠DGC if, and only if, ∠ACB = 90°. The centroid of a triangle is the intersection of the three medians, the lines which join each vertex to the midpoint of the opposite side.
Problem 3 of Canadian Mathematical Olympiad 1986
A chord ST of constant length slides around a semicircle with diameter AB. M is the mid-point of ST and P is the foot of the perpendicular from S to AB. Prove that angle SPM is constant for all positions of ST.
Problem 4 of Austria Mathematical Olympiad 2008
In a triangle ABC let E be the midpoint of the sides AC and F the midpoint of the side BC. Furthermore let G be the foot of the altitude through C on the side AB (or its extension). Show that the triangle EFG is isosceles if and only if ABC is isosceles.
Problem 6 of Austria Mathematical Olympiad 2008
We are given a square ABCD. Let P be different from the vertices of the square and from its center M. For a point P for which the line PD intersects the line AC, let E be this intersection. For a point P for which the line PC intersects the line DB, let F be this intersection. All those points P for which E and F exist are called acceptable points. Determine the set of acceptable points for which the line EF is parallel to AD.
Problem 6 of Australia Mathematical Olympiad 2010
Prove that + = + .
Problem 6 of Belarus Mathematical Olympiad 2000
The equilateral triangles ABF and CAG are constructed in the exterior of a right-angled triangle ABC with ∠C = 90◦. Let M be the midpoint of BC. Given that MF = 11 and MG = 7, find the length of BC.
Problem 8 of the Canadian Mathematical Olympiad 1970
Consider all line segments of length 4 with one end-point on the line y = x and the other end-point on the line y = 2x. Find the equation of the locus of the midpoints of these line segments.
Problem 2 of the Canadian Mathematical Olympiad 1971
Let x and y be positive real numbers such that x + y = 1. Show that (1 + )(1 + ) ≥ 9.
Problem 2 of the Canadian Mathematical Olympiad 1973
Find all the real numbers which satisfy the equation |x + 3| – |x – 1| = x + 1. (Note: |a| = a if a ≥ 0; |a| = –a if a < 0.)
Problem 2 of the Canadian Mathematical Olympiad 1969
Determine which of the two numbers − , − is greater for any c ≥ 1.
Problem 2 of the Auckland Mathematical Olympiad 2009
Is it possible to write the number 1² + 2² + 3² + . . . + 12² as a sum of 11 distinct squares?
Problem 3 of Austria Mathematical Olympiad 2004
In a trapezoid ABCD with circumcircle K the diagonals AC and BD are perpendicular. Two circles Ka and Kc are drawn whose diameters are AB and CD respectively.
Calculate the circumference and the area of the region that lies within the circumcircle K, but outside of the circles Ka and Kc.
Problem 4 of Austria Mathematical Olympiad 2002
We are given three mutually distinct points A, C and P in the plane. A and C are opposite corners of a parallelogram ABCD, the point P lies on the bisector of the angle DAB, and the angle APD is a right angle. Construct all possible parallelograms ABCD that satisfy these conditions.
Problem 3 of the Canadian Mathematical Olympiad 1977
N is an integer whose representation in base b is 777. Find the smallest positive integer b for which N is the fourth power of an integer.
Problem 3 of Austria Mathematical Olympiad 2008
The line g is given, and on it lie the four points P, Q, R, and S (in this order from left to right).
Construct all squares ABCD with the following properties:
P lies on the line through A and D.
Q lies on the line through B and C.
R lies on the line through A and B.
S lies on the line through C and D.
Problem 7 of Belarus Mathematical Olympiad 2000
On the side AB of a triangle ABC with BC < AC < AB, points B and C are marked so that AC = AC and BB = BC. Points B on side AC and C on the extension of CB are marked so that CB = CB and CC = CA. Prove that the lines CC and BB are parallel.
Problem 2 of the Ibero-American Mathematical Olympiad 1991
Two perpendicular lines divide a square in four parts; three of them have area equal to 1. Show that the area of the full square is four.
Problem 2 of the British Mathematical Olympiad 1993
A square piece of toast ABCD of side length 1 and center O
is cut in half to form two equal pieces ABC and CDA. If the triangle ABC has to be cut into two parts of equal area, one would usually cut along the line of symmetry BO. However, there are other ways of doing this. Find, with justification, the length and location of the shortest straight cut which divides the triangle ABC into two parts of equal area.
Problem 9 of the Irish Mathematical Olympiad 1998
The year 1978 was “peculiar” in that the sum of the numbers formed with the first two digits and the last two digits is equal to the number formed with the middle two digits, i.e., 19 + 78 = 97. What was the last previous peculiar year, and when will the next one occur?
Problem 2 of Austria Mathematical Olympiad 2005
A semicircle h with diameter AB and center M is drawn. A second semicircle k with diameter MB is drawn on the same side of the line AB. Let X and Y be points on k such that the arc BX is one and a half times as long as the arc BY. The line MY intersects the line BX at C and the larger semicircle h at D. Show that Y is the midpoint of the line segment CD.
Problem 1 of Japan’s KyotoUniversity Entrance Exam 2010
Given a ΔABC such that AB = 2, AC = 1. A bisector of ∠BAC intersects BC at D. If AD = BD, then find the area of ΔABC.
Problem 4 of the Vietnamese Mathematical Olympiad 1990
A triangle ABC is given in the plane. Let M be a point inside the triangle and A’, B’, C’ be its projections on the sides BC, CA, AB, respectively. Find the locus of M for which MA×MA’ = MB×MB’ = MC×MC’.
Problem 3 of the British Mathematical Olympiad 2007
Let ABC be a triangle, with an obtuse angle at A. Let Q be a point
(other than A, B or C) on the circumcircle of the triangle, on the same side of chord BC as A, and let P be the other end of the diameter through Q. Let V and W be the feet of the perpendiculars from Q onto CA and AB, respectively. Prove that the triangles PBC and AWV are similar. Note: The circumcircle of the triangle ABC is the circle which passes through the vertices A, B and C.
Problem 1 of the Irish Mathematical Olympiad 2001
In a triangle ABC, AB = 20, AC = 21 and BC = 29. The points D and E lie on the line segment BC, with BD = 8 and EC = 9. Calculate the angle ∠DAE.
Problem 1 of the Irish Mathematical Olympiad 1997
Find, with proof, all pairs of integers (x, y) satisfying the equation 1 + 1996x + 1998y = xy.
Problem 1 of the Irish Mathematical Olympiad 1991
Three points Y, X and Z are given that are, respectively, the circumcenter of a triangle ABC, the midpoint of BC, and the foot of the altitude from B on AC. Show how to reconstruct the triangle ABC.
Problem 3 of the Canadian Mathematical Olympiad 1990
Let ABCD be a convex quadrilateral inscribed in a circle, and let diagonals AC and BD meet at X. The perpendiculars from X meet the sides AB, BC, CD, DA at A’, B’, C’, D’, respectively. Prove that |A’B’| + |C’D’| = |A’D’| + |B’C’|. (|A’B’| is the length of line segment A’B’, etc.)
Problem 1 of International Mathematical Talent Search Round 7
In trapezoid ABCD, the diagonals intersect at E. The area of triangle ABE is 72, and the area of triangle CDE is 50. What is the area of trapezoid ABCD?
Problem 2 of the British Mathematical Olympiad 2005
Let x and y be positive integers with no prime factors larger than 5. Find all such x and y which satisfy x² – y² = 2k for some non-negative integer k.
Problem 2 of Poland Mathematical Olympiad 2010
The orthogonal projections of the vertices A, B, C of the tetrahedron ABCD on the opposite faces are denoted by O, I, G, respectively. Suppose that point O is the circumcenter of the triangle BCD, point I is the incenter of the triangle ACD and G is the centroid of the triangle ABD. Prove that tetrahedron ABCD is regular.
Problem 2 of Italy Mathematical Olympiad 2004
Two parallel lines r, s and two points P ∈ r and Q ∈ s are given in a plane. Consider all pairs of circles (CP, CQ) in that plane such that CP touches r at P and CQ, touches s at Q and which touch each other externally at some point T. Find the locus of T.
Problem 4 of Germany Mathematical Olympiad 1996
A pupil wants to construct a triangle ABC, given the length c = AB, the altitude hc from C and the angle e = a – b. Here c and hc are arbitrary and satisfies 0 < e < 90.
a) Is there such a triangle for any c, hc and e?
b) Is this triangle unique up to the congruence?
c) Show how to construct one such triangle, if it exists.
Problem 3 of Germany Mathematical Olympiad 1997
In a convex quadrilateral ABCD we are given that ∠CBD= 10°, ∠CAD = 20°, ∠ABD = 40°, ∠BAC= 50°. Determine the angles ∠BCD and ∠ADC.
Problem 1 of Mongolia Teacher Level 1999
In a convex quadrilateral ABCD, ∠ABD = 65°, ∠CBD = 35°, ∠ADC = 130°, and BC = AB. Find the angles of ABCD.
Problem 3 of Germany Mathematical Olympiad 2001
Let be given a circle of radius 1 and points J, A, B on it. We denote by k the arc AB of the circle not containing J. For every point P on k, point P′ on the ray JP is such that JP×JP′ = 4. Describe and draw the locus of points P′.
Problem 2 of KyotoUniversity Entrance Exam 2012
Given a regular tetrahedron OABC and points P, Q, R on the sides OA, OB, OC, respectively. Note that P, Q, R are different from the vertices of the tetrahedron OABC. If triangle PQR is an equilateral triangle, then prove that three sides PQ, QR, RP are parallel to three sides AB, BC, CA, respectively.
Problem 3 of KyotoUniversity Entrance Exam 2012
When real numbers x, y moves in the constraint with x² + xy + y² = 6. Find the range of x²y + xy² – x² – 2xy – y² + x + y.
Silicon Valley Math University 