Problem descriptions in my math book titled The Hard Mathematical Olympiad Problems and Their Solutions

Solutions for these following Mathematical Olympiad problems, many of them not available in the web, are in the book titled

The hard Mathematical Olympiad problems and their solutions

by Steve Dinh, a.k.a. Vo Duc Dien published by AuthorHouse.

This book is available for purchase at Amazon.com

http://www.amazon.com/Mathematical-Olympiad-Problems-Their-Solutions/dp/1463444907

and other online outlets around the world. Many of the author’s solutions are available at the world’s largest mathematical webpage http://www.cut-the-knot.org. Just search for Steve Dinh.

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 2 of the Korean Mathematical Olympiad 2007

ABCD is a convex quadrilateral, and AB ≠ CD. Show that there exists a point M such that  AB/CD = MA/MD = MB/MC.

Problem 1 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 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 4 of the Vietnamese Mathematical Olympiad 1963

The tetrahedron SABC has the faces SBC and ABC perpendicular. The three angles at S are all 60° and SB = SC = 1. Find the volume of the tetrahedron.

Problem 3 of the Ibero-American Mathematical Olympiad 1988

Show that between all triangles such that the distances from their vertices to a given point P are 3, 5 and 7, the one with the greatest perimeter has P as incenter.

Problem 1 of International Mathematical Talent Search Round 10

Find x² + y² + z² if x, y and z are positive integers such that 7x² – 3y² + 4z² = 8 and 16x² – 7y² + 9z² = -3.

Problem 4 of Austrian 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 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 1 of Tournament of Towns 1984

On the island of Camelot live 13 gray, 15 brown and 17 crimson chameleons. If two chameleons of different colors meet they both simultaneously change color to the third color (e.g. if a gray and a brown chameleon meet each other they both change to crimson). Is it possible that they will eventually all be the same color?

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 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 1 of Russia Sharygin Geometry Olympiad 2010

Does there exist a triangle, whose side is equal to some of its altitudes, another side is equal to some of its bisectors, and the third is equal to some of its medians?

Problem 1 of India Mathematical Olympiad 1986

A person who left home between 4 p.m. and 5 p.m. returned between 5 p.m. and 6 p.m. and found that the hands of his watch had exactly exchanged place, when did he go out ?

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 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 4 of Austrian Mathematical Olympiad 1987

Find all triples (x, y, z) of natural numbers satisfying 2xz = y² and x + z = 1987.

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 the Austrian Mathematical Olympiad 2000

Determine all real solutions of the equation ||||||| x² – x –1 | – 3 | – 5 | – 7 | – 9 | – 11 | – 13 | =  x² – 2x – 48

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 Austrian 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 1 of the Canadian Mathematical Olympiad 1977

If f(x) = x² + x, prove that the equation 4f(a) = f(b) has no solutions in positive integers a and b.

Problem 3 of the Austrian Mathematical Olympiad 2001

We are given a triangle ABC and its circumcircle with midpoint U and radius r. Let K be the circle with midpoint U 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 1 of the Irish Mathematical Olympiad 1997

Find, with proof, all pairs of integers (x, y) satisfying the equation 1 + 1996x + 1998y = xy.

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 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 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 themid-pointofXC. Let AB be extended to D so that |AB| = |BD|. Prove that DX is perpendicular to AY.

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 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 17 of Russia Sharygin Geometry Olympiad 2010

Construct a triangle, if the lengths of the bisectrix (bisector) and of the altitude from one vertex, and of the median from another vertex are given.

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 Austrian 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 6 of Austrian 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 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 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 θ with DA where θ= ∠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 1 of Japan’s Kyoto University 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 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 1 of British Mathematical Olympiad 1974

The curves A and B and C are related in such a way that B “bisects” the area between A and C, that is, the area of the region U is equal to the area of the region V at all points of the curve B. Find the equation of the curve B given that the equation of curve A is y = x² and that the equation of curve C is y = x².

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 the 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 diameterOBbe 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 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 13 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 ofABC (not containing A) at D. Prove that DL/IL = DK/IK.

Problem 4 of the British Mathematical Olympiad 1997

Let ABCD be a convex quadrilateral. The midpoints of AB, BC, CD and DA are P, Q, R and S, respectively. Given that the quadrilateral PQRS has area 1, prove that the area of the quadrilateral ABCD is 2.

Problem 1 of the Hong Kong Mathematical Olympiad 2000

Let O 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 3 of Austrian 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 the British Mathematical Olympiad 2008

Let ABPC be a parallelogram such that ABPis an acute triangle. The circumcircle of triangle ABC meets the line CP again at Q. Prove that PQ = AC if, and only if, ∠ACP= 60°. The circumcircle of a triangle is the circle which passes through its vertices.

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 = alpha.  Note: (Ω) denotes the area of shape Ω.

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 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 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 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 the Irish Mathematical Olympiad 2010

Find the least k for which the number 2010 can be expressed as the sum of the squares of k integers.

Problem 6 of the Irish Mathematical Olympiad 1990

The sum of two consecutive squares can be a square: for instance, 3² + 4² = 5².

a)  Prove that the sum of m consecutive squares cannot be a square for the cases m = 3, 4, 5, 6.

b)  Find an example of eleven consecutive squares whose sum is a square.

Problem 8 of the Auckland Mathematical Olympiad 2009

What is the smallest positive integer n, such that there exist positive integers a and b, with b obtained from a by a rearrangement of its digits, so that a – b = 11…1 (n digits of 1’s)?

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 2 of the Hong Kong Mathematical Olympiad 2007

Points X, Y, Z are marked on the sides AB, BC, CD of the rhombus ABCD, respectively, so that XY || AZ. Prove that XZ, AY and BD are concurrent.

Problem 1 of the 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 the 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 = (tanθ)x on the coordinate plane with the origin, where p > 1, 0 < θ < pi/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 tanθ  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 < θ < pi/2) the line passing through the origin and is perpendicular to the line l is y =[tan(θ/3)]x.

Problem 3 of the Belgium Flanders Mathematical Olympiad 1995

Points A, B, C, D are on a circle with radius R. |AC| = |AB| = 500, while the ratio between |DC|, |DA|, |DB| is 1, 5, 7. Find R.

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 the 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 to 1999?

Problem 6 of Uruguay Mathematical Olympiad 2009

Is the sum 1^2009 + 2^2009 + 3^2009 + … + 2008^2009  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 [1.008^8×100].

Problem 6 of Hong Kong Mathematical Olympiad 2007

If R is the remainder of 1^6 + 2^6  + 3^6  + 4^6  + 5^6  + 6^6  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^n + 15^n + 8^n + 6^n 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 = [sqrt(3) – sqrt(2)]^2009 + 16, find the value of a.

Problem 3 of Hong Kong Mathematical Olympiad 2007

208208 = 8^5*a + 8^4*b + 8^3*c + 8²*d + 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 7 of Hong Kong Mathematical Olympiad 2007

If 14! is divisible by 6^k, where k is an integer, find the largest possible value of k.

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 3 of Uruguay Mathematical Olympiad 2010

In the triangle ABC, angle BAC is 120°, and the point P is on the side BC is such that angle PAC is right. Knowing that AC = PB = 1, calculate the length of side AB.

Problem 1 Belarusian 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 the 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 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 the 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 3^32 –2^32.

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 sqrt(1 + 12n^2) is an integer, then it is a perfect square.

Problem 3 of the Korean Mathematical Olympiad 2005

In a trapezoid ABCD with AD || BC, O1, O2, O3, O4 denote the circles with diameters AB, BC, CD, DA, respectively. Show that there exists a circle with center inside the trapezoid which is tangent to all the four circles O1, O2, O3, O4 if and only if ABCD is a parallelogram.

Silicon Valley Typical Engineering Job Interview Problem

Find the area and the perimeter of the circle C1 with diameter of not included in circle C2 with the diameter of 2. Both ends of this area are connected to form the diameter of C1.

Problem 1 of the British Mathematical Olympiad 2007

Find the value of  (1^4 + 2007^4 + 2008^4)/(1² + 2007² + 2008²).

Problem 2 of Pan African Mathematical Competition 2004

Is 4 sqrt(4 – 2 sqrt(3)) + sqrt(97 – 56 sqrt(3))  an integer?

Problem 8 of the British Mathematical Olympiad 1996

Two circles S1 and S2 touch each other externally at K; they also touch a circle S internally at A1 and A2, respectively. Let P be one point of intersection of S with the common tangent to S1 and S2 at K. The line PA1 meets S1 again at B1, and PA2 meets S2 again at B2. Prove that B1B2 is a common tangent to S1 and S2.

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 Brazilian Mathematical Olympiad 1995

A regular tetrahedron has side l. What is the smallest x such that the tetrahedron can be passed through a loop of twice of length x?

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,   (ax² – 24 x + b)/(x² – 1) = x has two solutions and the sum of them equals 12.

Problem 1 of the British Mathematical Olympiad 2001

Find all positive integers m, n, where n is odd, that satisfy 1/m + 4/n = 1/12.

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 5 of Belarus Mathematical Olympiad 1997

In a trapezoid ABCD with AB || CD it holds that ∠ADB + ∠DBC = 180°. Prove that AB×BC = AD×DC.

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^ n + 2×17^ n 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 2 of Austrian Mathematical Olympiad 2004

Solve the equation  sqrt{4 – x sqrt[4 – (x – 2) sqrt (1 + (x – 5)(x – 7))]} (all the square roots are non-negative)

Problem 3 of Vietnam 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 2 of the New Zealand MO Camp Selection 2010

AB is a chord of length 6 in a circle of radius 5 and center O. A square is inscribed in the sector OAB with two vertices on the circumference and two sides parallel to AB. Find the area of the square.

Problem 4 of Hong KongMO 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 P3 Tournament of Towns 2008

Acute triangle A1A2A3 is inscribed in a circle of radius 2. Prove that one can choose points B1, B2, B3 on the arcs A1A2, A2A3 and A3A1, respectively, such that the numerical value of the area of the hexagon A1B1A2B2A3B3 is equal to the numerical value of the perimeter of the triangle A1A2A3.

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 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 AN/ND = BM/MC. The lines AM and BN intersect at P, while the lines CN and DM intersect at Q. Prove that if S(ABP) + S(CDQ) = S(MNPQ), 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 trianglesBAB1, 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 quadrangular 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 Institute of Technology Entrance Exam 2011

On a plane, given a square D with side length 1 and a line which intersects with D. For the solid obtained by a rotation of D about the line as the axis, answer the following questions:
a) Suppose that the line l on a plane the same with D isn’t parallel to any edges. Prove that the line by which the volume of the solid is maximized has only intersection point with D. Note that the line as axis of rotation is parallel to l.
b) Find the possible maximum volume for which all solid formed by the rotation axis as line intersecting with D.

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 2 of the Irish Mathematical Olympiad 1988

A, B, C, D are the vertices of a square, and P is a point on the arc CD of its circumcircle. Prove that

|PA|² – |PB|² = |PB|×|PD| – |PA|×|PC|.

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 4 of the British Mathematical Olympiad 1993

Two circles touch internally at M. A straight line touches the inner circle at P and cuts the outer circle at Q and R. Prove that ∠QMP = ∠RMP.

Problem 7 of the British Mathematical Olympiad 1997

In the acute-angled triangle ABC, CF is an altitude, with F on AB, and BM is a median, with M on CA. Given that BM = CF and ∠MBC = ∠FCA, prove that the triangle ABC is equilateral.

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 2 of the British Mathematical Olympiad 1995

Let ABC be a triangle, and D, E, F be the midpoints of BC, CA, AB, respectively. Prove that ∠DAC = ∠ABE if, and only if, ∠AFC = ∠ADB.

Problem 6 of the British Mathematical Olympiad 2009

Points A, B, C, D and E lie, in that order, on a circle and the lines AB and ED are parallel. Prove that ∠ABC = 90° if, and only if, AC² = BD² + CE².

Problem 1 of India Postal Coaching 2010

Let g, Γ be two concentric circles with radii r, R with r < R. Let ABCD be a cyclic quadrilateral inscribed in g. If vector AB denotes the ray starting from A and extending indefinitely in B′s direction then let vectors AB, BC, CD, DA meet Γ at the points C1, D1, A1 and B1, respectively. Prove that ≥   where (.) denotes area.

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 2 of Cono Sur Olympiad 1994

Consider a circle C with diameter AB = 1. A point P0 is chosen on C, P0 ≠ A, and starting in P0 a sequence of points P1, P2, … , Pn, …  is constructed on C, in the following way: Qn is the symmetrical point of A with respect to Pn and the straight line that joins B and Qn cuts C at B and Pn+1 (not necessary different). Prove that it is possible to choose P0 such that:
i) ∠P0AB < 1.
ii) In the sequence that starts with P0 there are 2 points, Pk and Pj, such that triangle APkPj is equilateral.

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+ sqrt(2^2 – 1)) (2 – sqrt(2^2 – 1)). Find the value of  Q.

Problem 9 of Hong Kong Mathematical Olympiad 2008

Let F = 1 + 2 + 2^2 + 2^3 + … +2^s and T = sqrt(log(1 + F)/log2).  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 the Austrian 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 India Postal Coaching 2005

Let <Γ> be a sequence of concentric circles such that the sequence <R>, where R denotes the radius of Γ, is increasing and Γ→ ∞ as j → ∞. Let ABC be a triangle inscribed in Γ. Extend the rays A B, BC, CA to meet Γ at A, B and C, respectively to form the triangle ABC. Continue this process. Show that the sequence of triangles <ABC> tends to an equilateral triangle as n→ ∞.

Problem 3 of the Irish Mathematical Olympiad 1988

ABC is a triangle inscribed in a circle, and E is the mid-point of the arc subtended by BC on the side remote from A. If through E a diameter ED is drawn, show that the measure of the angle DEA is half the magnitude of the difference of the measures of the angles at B and C.

Problem at Art of the Problem Solving web page 2011

There is a point P inside a rectangle ABCD such that ∠APD= 110°, ∠PBC = 70°, ∠PCB = 30°. Find ∠PAD.

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 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 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 (x + 1)yz = 12, (y + 1)zx = 4 and (z + 1)xy = 4.

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 5 of International Mathematical Talent Search Round 13

Armed with just a compass – no straightedge – draw two circles that intersect at right angle; that is, construct overlapping circles in the same plane, having perpendicular tangents at the two points where they meet.

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 50^th 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 1 of Tournament of Towns 1995

Prove that the number 40..09 (with at least one zero) is not a perfect square.