31st International Mathematical Olympiad 1990 Problems & Solutions

A1.  Chords AB and CD of a circle intersect at a point E inside the circle. Let M be an interior point of the segment EB. The tangent at E to the circle through D, E and M intersects the lines BC and AC at F and G respectively. Find EF/EG in terms of t = AM/AB. A2.  Take n ≥ 3 and consider a set E of 2n-1 distinct points on a circle. Suppose that exactly k of these points are

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30th International Mathematical Olympiad 1989 Problems & Solutions

A1.  Prove that the set {1, 2, ... , 1989} can be expressed as the disjoint union of subsets A1, A2, ... , A117 in such a way that each Ai contains 17 elements and the sum of the elements in each Ai is the same. A2.  In an acute-angled triangle ABC, the internal bisector of angle A meets the circumcircle again at A1. Points B1 and C1 are defined similarly. Let A0 be the point of

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29th International Mathematical Olympiad 1988 Problems & Solutions

A1.  Consider two coplanar circles of radii R > r with the same center. Let P be a fixed point on the smaller circle and B a variable point on the larger circle. The line BP meets the larger circle again at C. The perpendicular to BP at P meets the smaller circle again at A (if it is tangent to the circle at P, then A = P). (i)  Find the set of values of AB2 + BC2 + CA2. (ii)  Find the

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28th International Mathematical Olympiad 1987 Problems & Solutions

A1.  Let pn(k) be the number of permutations of the set {1, 2, 3, ... , n} which have exactly k fixed points. Prove that the sum from k = 0 to n of (k pn(k) ) is n!. [A permutation f of a set S is a one-to-one mapping of S onto itself. An element i of S is called a fixed point if f(i) = i.] A2.  In an acute-angled triangle ABC the interior bisector of angle A meets BC at L

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27th International Mathematical Olympiad 1986 Problems & Solutions

A1.  Let d be any positive integer not equal to 2, 5 or 13. Show that one can find distinct a, b in the set {2, 5, 13, d} such that ab - 1 is not a perfect square. A2.  Given a point P0 in the plane of the triangle A1A2A3. Define As = As-3 for all s >= 4. Construct a set of points P1, P2, P3, ... such that Pk+1 is the image of Pk under a rotation center Ak+1 through an angle

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26th International Mathematical Olympiad 1985 Problems & Solutions

A1.  A circle has center on the side AB of the cyclic quadrilateral ABCD. The other three sides are tangent to the circle. Prove that AD + BC = AB. A2.  Let n and k be relatively prime positive integers with k < n. Each number in the set M = {1, 2, 3, ... , n-1} is colored either blue or white. For each i in M, both i and n-i have the same color. For each i in M not equal to

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25th International Mathematical Olympiad 1984 Problems & Solutions

A1.  Prove that 0 ≤ yz + zx + xy - 2xyz ≤ 7/27, where x, y and z are non-negative real numbers satisfying x + y + z = 1. A2.  Find one pair of positive integers a, b such that ab(a+b) is not divisible by 7, but (a+b)7 - a7 - b7 is divisible by 77. A3.  Given points O and A in the plane. Every point in the plane is colored with one of a finite number of colors.

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24th International Mathematical Olympiad 1983 Problems & Solutions

A1.  Find all functions f defined on the set of positive reals which take positive real values and satisfy:   f(x(f(y)) = yf(x) for all x, y; and f(x) → 0 as x → ∞. A2.  Let A be one of the two distinct points of intersection of two unequal coplanar circles C1 and C2 with centers O1 and O2 respectively. One of the common tangents to the circles touches C1 at P1 and C2 at P2,

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23rd International Mathematical Olympiad 1982 Problems & Solutions

A1.  The function f(n) is defined on the positive integers and takes non-negative integer values. f(2) = 0, f(3) > 0, f(9999) = 3333 and for all m, n:       f(m+n) - f(m) - f(n) = 0 or 1. Determine f(1982). A2.  A non-isosceles triangle A1A2A3 has sides a1, a2, a3 with ai opposite Ai. Mi is the midpoint of side ai and Ti is the point where the incircle touches side ai. Denote

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22nd International Mathematical Olympiad 1981 Problems & Solutions

A1.  P is a point inside the triangle ABC. D, E, F are the feet of the perpendiculars from P to the lines BC, CA, AB respectively. Find all P which minimise:         BC/PD + CA/PE + AB/PF. A2.  Take r such that 1 ≤ r ≤ n, and consider all subsets of r elements of the set {1, 2, ... , n}. Each subset has a smallest element. Let F(n,r) be the arithmetic mean of these smallest

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21st International Mathematical Olympiad 1979 Problems & Solutions

A1.  Let m and n be positive integers such that:       m/n = 1 - 1/2 + 1/3 - 1/4 + ... - 1/1318 + 1/1319. Prove that m is divisible by 1979. A2.  A prism with pentagons A1A2A3A4A5 and B1B2B3B4B5 as the top and bottom faces is given. Each side of the two pentagons and each of the 25 segments AiBj is colored red or green. Every triangle whose vertices are vertices of the prism

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20th International Mathematical Olympiad 1978 Problems & Solutions

A1.  m and n are positive integers with m < n. The last three decimal digits of 1978m are the same as the last three decimal digits of 1978n. Find m and n such that m + n has the least possible value. A2.  P is a point inside a sphere. Three mutually perpendicular rays from P intersect the sphere at points U, V and W. Q denotes the vertex diagonally opposite P in the

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19th International Mathematical Olympiad 1977 Problems & Solutions

A1.  Construct equilateral triangles ABK, BCL, CDM, DAN on the inside of the square ABCD. Show that the midpoints of KL, LM, MN, NK and the midpoints of AK, BK, BL, CL, CM, DM, DN, AN form a regular dodecahedron. A2.  In a finite sequence of real numbers the sum of any seven successive terms is negative, and the sum of any eleven successive terms is positive. Determine the

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18th International Mathematical Olympiad 1976 Problems & Solutions

A1.  A plane convex quadrilateral has area 32, and the sum of two opposite sides and a diagonal is 16. Determine all possible lengths for the other diagonal. A2.  Let P1(x) = x2 - 2, and Pi+1 = P1(Pi(x)) for i = 1, 2, 3, ... . Show that the roots of Pn(x) = x are real and distinct for all n. A3.  A rectangular box can be completely filled with unit cubes. If one

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17th International Mathematical Olympiad 1975 Problems & Solutions

A1.  Let x1 ≥ x2 ≥ ... ≥ xn, and y1 ≥ y2 ≥ ... ≥ yn be real numbers. Prove that if zi is any permutation of the yi, then:       ∑1≤i≤n (xi - yi)2 ≤ ∑1≤i≤n (xi - zi)2. A2.  Let a1 < a2 < a3 < ... be positive integers. Prove that for every i ≥ 1, there are infinitely many an that can be written in the form an = rai + saj, with r, s positive integers and j > i. A3. 

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16th International Mathematical Olympiad 1974 Problems & Solutions

A1.  Three players play the following game. There are three cards each with a different positive integer. In each round the cards are randomly dealt to the players and each receives the number of counters on his card. After two or more rounds, one player has received 20, another 10 and the third 9 counters. In the last round the player with 10 received the largest number of counters. Who

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15th International Mathematical Olympiad 1973 Problems & Solutions

A1.  OP1, OP2, ... , OP2n+1 are unit vectors in a plane. P1, P2, ... , P2n+1 all lie on the same side of a line through O. Prove that |OP1 + ... + OP2n+1| ≥ 1. A2.  Can we find a finite set of non-coplanar points, such that given any two points, A and B, there are two others, C and D, with the lines AB and CD parallel and distinct? A3.  a and b are real numbers

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14th International Mathematical Olympiad 1972 Problems & Solutions

A1.  Given any set of ten distinct numbers in the range 10, 11, ... , 99, prove that we can always find two disjoint subsets with the same sum. A2.  Given n > 4, prove that every cyclic quadrilateral can be dissected into n cyclic quadrilaterals. A3.  Prove that (2m)!(2n)! is a multiple of m!n!(m+n)! for any non-negative integers m and n. B1. 

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13th International Mathematical Olympiad 1971 Problems & Solutions

A1.  Let En = (a1 - a2)(a1 - a3) ... (a1 - an) + (a2 - a1)(a2 - a3) ... (a2 - an) + ... + (an - a1)(an - a2) ... (an - an-1). Let Sn be the proposition that En ≥ 0 for all real ai. Prove that Sn is true for n = 3 and 5, but for no other n > 2. A2.  Let P1 be a convex polyhedron with vertices A1, A2, ... , A9. Let Pi be the polyhedron obtained from P1 by a translation that moves

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12th International Mathematical Olympiad 1970 Problems & Solutions

A1.  M is any point on the side AB of the triangle ABC. r, r1, r2 are the radii of the circles inscribed in ABC, AMC, BMC. q is the radius of the circle on the opposite side of AB to C, touching the three sides of AB and the extensions of CA and CB. Similarly, q1 and q2. Prove that r1r2q = rq1q2. A2.  We have 0 ≤ xi < b for i = 0, 1, ... , n and xn > 0, xn-1 > 0. If a > b, and

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