Solutions for these following Mathematical Olympiad problems used to select the top mathematical students in the world and nations are in the book titled
How to Solve the World’s Mathematical Olympiad problems, Volume I
by Steve Dinh, a.k.a. Vo Duc Dien
Available at Amazon.com and other online outlets around the world.
At the time of this book’s publication, the solutions to many of these problems were not yet available.
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 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 = (a + b + c) = 2. 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 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 the Belarusian Mathematical Olympiad 2000
Find all pairs of integers (x, y) satisfying the equality
y(x² + 36) + x(y² − 36) + y²(y − 12) = 0.
Problem 1 of the British Mathematical Olympiad 2008
Find all solutions in non-negative integers a, b to sqrt(a) + sqrt(b) = sqrt(2009).
Problem 1 of the Canadian Mathematical Olympiad 1969
Show that if a1/b1 = a2/b2 = a3/b3 and p1, p2, p3 are not all zero, then (a1/b1)^n= (p1a1^n+ p2a2^n + p3a3^n)/(p1b1^n+ p2b2^n+ p3b3^n)
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 [(1.2.4 + 2.4.8 + . . . + n.2n.4n)/(1.3.9 + 2.6.18 + . . . + n.3n.9n)]^(1/3)
Problem 1 of the Canadian Mathematical Olympiad 1978
Let n be an integer. If the tens digit of n² is 7, what is the units digit of n²?
Problem 1 of the Canadian Mathematical Olympiad 1980
If a679b is a five digit number (in base 10) which is divisible by 72, determine a and b.
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, [pi] = 3 and [−5/2] = -3. Show that the equation
[x] + [2x] + [4x] + [8x] + [16x] + [32x] = 12345 has no real solution.
Problem 1 of Canadian Mathematical Olympiad 1982
In the diagram, OBi is parallel and equal in length to AiAi+1 for i = 1, 2, 3 and 4 (A5 = A1). Show that the area of B1B2B3B4 is twice that of A1A2A3A4.
Problem 1 of Canadian Mathematical Olympiad 1983
Find all positive integers w, x, y and z which satisfy w! = x! + y! + z!
Problem 1 of Canadian Mathematical Olympiad 1984
Prove that the sum of the squares of 1984 consecutive positive integers cannot be the square of an integer.
Problem 1 of the Canadian Mathematical Olympiad 1985
The lengths of the sides of a triangle are 6, 8 and 10 units. Prove that there is exactly one straight line which simultaneously bisects the area and perimeter of the triangle.
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 Canadian Mathematical Olympiad 1988
For what values of b do the equations 1988x² + bx + 8891 = 0 and 8891x² + bx + 1988 = 0 have a common root?
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 Canadian Mathematical Olympiad 1993
Determine a triangle for which the three sides and an altitude are four consecutive integers and for which this altitude partitions the triangle into two right triangles with integer sides. Show that there is only one such triangle.
Problem 1 of China Mathematical Olympiad 2010
Circle Г1 and Г2 intersect at two points A and B. A line through B intersects Г1 and Г2 at points C and D, respectively. Another line through B intersects Г1 and Г2 at points E and F, respectively. Line CF intersects Г1 and Г2 at points P and Q, respectively. Let M and N be the midpoints of arcs PB and QB, respectively. Prove that if CD = EF, then C, F, M and N are concyclic.
Problem 1 of the Belarusian Mathematical Olympiad 2000
Find all pairs of integers (x, y) satisfying 3xy – x – 2y = 8.
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 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 1 of Irish Mathematical Olympiad 1994
Let x, y be positive integers with y > 3 and x² + y^4 = 2 [(x − 6)² + (y + 1)²]. Prove that x² + y^4= 1994.
Problem 1 of 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 [(r-1)/(r + 1)][(s-1)/(s + 1)][(t-1)/(t + 1)].
Problem 1 of the British Mathematical Olympiad 2006
Triangle ABC has integer-length sides, and AC = 2007. The internal bisector of ∠BAC meets BC at D. Given that AB = CD, determine AB and BC.
Problem 1 of the British Mathematical Olympiad 2007
Find the minimum value of x² + y² + z² where x, y, z are real numbers such that x³ + y³ + z³ − 3xyz = 1.
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 1 of USA 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 USA Mathematical Olympiad 1984
The product of two of the four roots of the quartic equation x^4 – 18x^3 + kx^3 + 200x − 1984 = 0 is -32. Determine the value of k.
Problem 1 of the USA 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 William Lowell Putnam Competition 1986
Find, with explanation, the maximum value of f(x) = x³− 3x on the set of all real numbers x satisfying x^4 + 36 ≤ 13x².
Problem 2 of the Austrian 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 2 Asian Pacific Mathematical Olympiad 1992
In a circle C with center Oand 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 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 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 circumference 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 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 Canadian Mathematical Olympiad 1988
A house is in the shape of a triangle, perimeter P meters and area A square meters. The garden consists of all the land within5 metersof the house. How much land do the garden and house together occupy?
Problem 2 of Canadian Mathematical Olympiad 1989
Let ABC be a right angled triangle of area 1. Let A’B’C’ be the points obtained by reflecting A, B, C respectively, in their opposite sides. Find the area of triangle A’B’C’.
Problem 2 of Canadian Mathematical Olympiad 1992
For x, y, z ≥ 0, establish the inequality x(x − z)² + y(y − z)² ≥ (x − z)(y − z)(x + y − z) and determine when equality holds.
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 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 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 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 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 circleBOHlies on the line AB.
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 Asian Pacific Mathematical Olympiad 1989
Let A1, A2, A3 be three points in the plane, and for convenience, let A4 = A1, A5 = A2. For n = 1, 2, and 3, suppose that Bn is the midpoint of AnAn+1, and suppose that Cn is the midpoint of AnBn. Suppose that AnCn+1 and BnAn+2 meet at Dn, and that AnBn+1 and CnAn+2 meet at En. Calculate the ratio of the area of triangle D1D2D3 to the area of triangle E1E2E3.
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 3 of Asian Pacific Mathematical Olympiad 2000
Let ABC be a triangle. Let M and N be the points in which the median and the angle bisector, respectively, at A meet the side BC. Let Q and P be the points in which the perpendicular at N to NA meets MA and BA, respectively, and O the point in which the perpendicular at P to BA meets AN produced. Prove that QO is perpendicular to BC.
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 triangleABPand 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 sqrt(d/3).
Problem 3 of Belarusian Mathematical Olympiad 1997
Points D, M, N are chosen on the sides AC, AB, BC of a triangle ABC respectively, so that the intersection point P of AN and CM lies on BD. Prove that BD is a median of the triangle if and only if AP/PN= CP/PM.
Problem 3 of Canadian Mathematical Olympiad 1986
A chord ST of constant length slides around a semicircle with diameter AB. M is themid-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 3 of Canadian Mathematical Olympiad 1991
Let C be a circle and P a given point in the plane. Each line through P which intersects C determines a chord of C. Show that the midpoints of these chords lie on a circle.
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 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 tanα = 4nh/[(n² – 1)a].
Problem 3 of the Irish Mathematical Olympiad 2006
Prove that a square of side 2.1 units can be completely covered by seven squares of side 1 unit.
Problem 4 of the Austrian 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 4 of the Austrian 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 1969
Let ABC be an equilateral triangle, and P be an arbitrary point within the triangle. Perpendiculars PD, PE and PF are drawn to the three sides of the triangle. Show that, no mater where P is chosen. (PD + PE + PF)/(AB + BC + CA) = 1/2.
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 1/25 of the original integer.
b) Show that there is no integer such that deletion of the first digit produces a result which is 1/35 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 4 of Canadian Mathematical Olympiad 1972
Describe a construction of a quadrilateral ABCD given
a) the lengths of all four sides,
b) that AB and CD are parallel,
c) that BC and DA do not intersect.
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 triangleADC. Suppose that ∠BEK = 45°. Find all possible values of ∠CAB.
Problem 4 of USA Stanford Mathematical Tournament 2006
Simplify a³/[(a – b)(b – c)] + b³/[(b – a)(b – c)] + c³/[(c – a)(c – b)].
Problem 4 of USA 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 USA 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 (1/FP) + (1/PE) is a maximum.
Problem 4 of the USA 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 4 of the Vietnam 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 Vietnam 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 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 5 of USA 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, Q lie on a common circle.
Problem 5 of USA 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 6 of Austrian 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 the Australian Mathematical Olympiad 2010
Prove that [6 + 845^(1/3) + 325^(1/3)]^(1/3) + [6 + 847^(1/3) + 539^(1/3)]^(1/3) = [4 + 245^(1/3) + 175^(1/3)]^(1/3) + [8 + 1859^(1/3) + 1573^(1/3)]^(1/3)
Problem 6 of Belarusian Mathematical Olympiad 2000
The equilateral trianglesABFandCAGare 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 6 of the Belarusian 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 6 of the Canadian Mathematical Olympiad 1970
Given three non-collinear points A, B, C, construct a circle with center C such that the tangents from A and B to the circle are parallel.
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 n/m is an integer.
Problem 7 of Irish Mathematical Olympiad 1994
Let p, q, r be distinct real numbers which satisfy the equations
q = p(4 − p)
r = q(4− q)
p = r(4 − r)
Find all possible values of p + q + r.
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 8 of the Canadian Mathematical Olympiad 1971
A regular pentagon is inscribed in a circle of radius r. P is any point inside the pentagon. Perpendiculars are dropped from P to the sides, or the sides produced, of the pentagon.
a) Prove that the sum of the lengths of these perpendiculars is constant.
b) Express this constant in terms of the radius r.
Problem 9 of the Auckland Mathematical Olympiad 2009
Through the incenter I of triangle ABC a straight line is drawn intersecting AB and BC at points M and N, respectively, in such a way that the triangle BMN is acute-angled. On the side AC the points K and L are chosen such that ∠ILA = ∠IMB and ∠IKC = ∠INB. Prove that AC = AM + KL + CN.
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 9 of 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 Auckland Mathematical Olympiad 2009
Is it possible to write the number 1² + 2² + 3² + . . . + 12² as a sum of 11 distinct squares?
Problem 2 of the Australian 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 CD/CE = CF/CD.
Problem 2 of Belarusian 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 that AC×AE < AB×AD.
Problem 2 of the Canadian Mathematical Olympiad 1969
Determine which of the two numbers sqrt(c + 1) − sqrt(c), sqrt(c) − sqrt(c – 1) is greater for any c ≥ 1.
Problem 2 of the Canadian Mathematical Olympiad 1970
Given a triangle ABC with angle A obtuse and with altitudes of length h and k as shown in the diagram, prove that a + h ≥ b + k. Find under what conditions a + h = b + k.
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/x)(1 + 1/y) ≥ 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 1974
Let ABCD be a rectangle with BC = 3AB. Show that if P, Q are the points on side BC with BP = PQ = QC, then ∠DBC + ∠DPC = ∠DQC.
Problem 2 of the Canadian Mathematical Olympiad 2010
Let A, B, P be three points on a circle. Prove that if a and b are the distances from P to the tangents at A and B and c is the distance from P to the chord AB, then c² = ab.
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 2 of the Ibero-American Mathematical Olympiad 1990
Let ABC a triangle, and let I be the center of the circumference inscribed and D, E, F its tangent points with BC, CA and AB respectively. Let P be the other point of intersection of the line AD with the circumference inscribed. If M is the mid point of EF, show that the four points P, I, M, D are either on the same circumference or they are collinear. (Exact wording from the exam text for easy Internet searching)
Problem 2 of the Ibero-American Mathematical Olympiad 1996
Let ABC be a triangle, D the midpoint of BC, and M be the midpoint of AD. The line BM intersects the side AC at point N. Show that AB is tangent to the circumcircle of the triangle NBC if and only if the following equality is true BM/MN= BC²/BN².
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 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 2 of USA Stanford Mathematical Tournament 2006
Find the minimum value of 2x² + 2y² + 5z² − 2xy – 4yz – 4x – 2z + 15 for real numbers x, y, z.
Problem 2 of USA Mathematical Olympiad 1976
If Aand 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 USA 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 2 of Vietnam Regional Competition 1977
Compare [(2³ + 1)/(2³ – 1)][(3³ + 1)/(3³ – 1)] . . . [(100³ + 1)/(100³ – 1)] with 3/2.
Problem 3 of the Austrian 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 3 of the Austrian Mathematical Olympiad 2004
We are given a convex quadrilateral ABCD with ∠ADC= ∠BCD > 90°. Let E be the intersection of the line AC with the line parallel to AD through B and F be the intersection of the line BD with the line parallel to BC through A. Show that EF is parallel to CD.
Problem 3 of the Austrian 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 Austrian 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 3 of Belarusian 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 3 of the Canadian Mathematical Olympiad 1971
ABCD is a quadrilateral with AD = BC. If ∠ADCis greater than ∠BCD, prove that AC > BD.
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 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 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 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 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 Middle European Mathematical Olympiad 2009
Let ABCD be a convex quadrilateral such that AB and CD are not parallel and AB = CD. The midpoints of the diagonals AC and BD are E and F, respectively. The line EF meets segments AB and CD at G and H, respectively. Show that ∠AGH= ∠DHG.
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) A1B1 = sqrt[(A1B² + A1C²)/2]
b) CH = DE.
Problem 3 of the Vietnam Mathematical Olympiad 1989
A square ABCD of side length 2 is given on a plane. The segment AB is moved continuously towards CD until A and B coincide with C and D, respectively. Let S be the area of the region formed by the segment AB while moving. Prove that AB can be moved in such a way that S < .
Problem 4 of the Austrian 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 4 of the Austrian Mathematical Olympiad 2003
In a parallelogram ABCD, let E be the midpoint of the side AB and F the midpoint of BC. Let P be the intersection point of the lines EC and FD. Show that the segments AP, BP, CP and DP divide the parallelogram into four triangles with areas in 1 : 2 : 3 : 4 ratio.
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 AC/CB = QC/CP. Describe, with proof, the locus of the point P.
Problem 4 of the Canadian Mathematical Olympiad 1978
The sides AD and BC of a convex quadrilateral ABCD are extended to meet at E. Let H and G be the midpoints of BD and AC, respectively. Find the ratio of the area of the triangle EHG to that of the quadrilateral ABCD.
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 4 of the Ibero-American Mathematical Olympiad 1990
Let C be a circumference, AB one of its diameters, t its tangent in B, and M a point on C distinct of A and B. It is constructed a circumference C tangent to C on M, and to the line t.
a) Find the point of tangency P to t and C, and the locus of the centers of the circumferences C when M varies.
b) Show that there exists a circle that is always orthogonal to Γ, regardless of the position of M. Note: Two circumferences are orthogonal to each other if they intersect and the respective tangents to the point of intersection are orthogonal.
Problem 4 of the Ibero-American Mathematical Olympiad 1993
Let ABC be an equilateral triangle and Γ its inscribed circle. If D and E are points in the sides AB and AC respectively, such that DE is tangent to Γ, show that Ad/DB + AE/EC = 1.
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 withOM; and S the intersection of HR with OP. Show that AHSO is a parallelogram.
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 4 of the International Mathematical Olympia 1960
Construct triangle ABC given ha, hb(the altitudes from A and B) and ma, the median from vertex A.
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 2ab×cos(C/2)/(a + b).
Problem 3 of the Canadian Mathematical Olympiad 1969
Let c be the length of the hypotenuse of a right angle triangle whose other two sides have lengths a and b. Prove that a + b ≤ c. When does the equality hold?
Problem 5 of the Canadian Mathematical Olympiad 1970
A quadrilateral has one vertex on each side of a square of side-length 1. Show that the lengths a, b, c and d of the sides of the quadrilateral satisfy the inequalities 2 ≤ a² + b² + c² + d² ≤ 4.
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 1974
Given a circle with diameter AB and a point X on the circle different from A and B, let ta, tb and tx be the tangents to the circle at A, B and X respectively. Let Z be the point where line AX meets tb and Y the point where line BX meets ta. Show that the three lines YZ, tx and AB are either concurrent (i.e., all pass through the same point) or parallel.
Problem 5 of the Canadian Mathematical Olympiad 1975
A, B, C, D are four “consecutive” points on the circumference of a circle and P, Q, R, S are points on the circumference which are respectively the midpoints of the arcs AB, BC, CD, DA. Prove that PR is perpendicular to QS.
Problem 5 of the Ibero-American Mathematical Olympiad 1992
Let Γ be a circle and let h and m be positive numbers such that there exists a trapezoid ABCD inscribed in Γ of height h and such that the sum of the bases AB + CD is m. Construct the trapezoid ABCD.
Problem 5 of the Ibero-American Mathematical Olympiad 1999
An acute triangle ABC 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 5 of the International Mathematical Olympiad 1959
An arbitrary point M is selected in the interior of the segment AB. The squares AMCD and MBEF are constructed on the same side of AB, with the segments AM and MB as their respective bases. The circles circumscribed about these squares, with centers P and Q, intersect at M and also at another point N1. Let N denote the point of intersection of the straight lines AF and BC.
a) Prove that the points N1 and N coincide.
b) Prove that the straight lines MN pass through a fixed point S independent of the choice of M.
c) Find the locus of the midpoints of the segments PQ as M varies between A and B.
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 AP/PD = AB/DC = BQ/QC.
Show that the angles that are formed by the lines PQ with AB and CD are equal.
Problem 6 of the USA 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 triangleACFmeet line CD at C and G. Prove that the triangleAFGis isosceles.
Problem 6 of the Australian Mathematical Olympiad 2008
Let A1A2A3 and B1B2B3 be triangles. If p = A1A2 + A2A3 + A3A1 + B1B2 + B2B3 + B3B1 and q = A1B1 + A1B2 + A1B3 + A2B1 + A2B2 + A2B3 + A3B1 + A3B2 + A3B3 , prove that 3p ≤ 4q.
Problem 7 of the Australian Mathematical Olympiad 2009
Let I be the incenter of a triangle ABCin which AC ≠ BC. Let Γ be the circle passing through A, I and B. Suppose Γ intersects the line AC at A and X and intersects the line BC at B and Y. Show that AX = BY .
Problem 7 of Belarusian 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 7 of Belarusian 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.
A Russian Mathematical Problem
Given triangle ABC, construct three pair-wise orthogonal circles that each pass through a pair of vertices.
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 1 of Austrian Mathematical Olympiad 2004
Determine all integers a and b such that (a³ + b)(a + b³) = (a + b).
Problem 2 of Austrian Mathematical Olympiad 2000
The trapezoid ABCD (ABCD labeled counterclockwise, AB ≠ CD) is inscribed into a circle k. On the arc AB two points P and Q (P ≠ Q) are chosen (with APQB labeled in counterclockwise order). Let X be the intersection of the lines CP and AQ and Y be the intersection of the lines BP and DQ. Show that P, Q, X and Y lie on a circle.
Problem 2 of the Austrian 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 2 of 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.
Silicon Valley Math University 