ISBN-10:
1843312042
ISBN-13:
9781843312048
Pub. Date:
09/05/2005
Publisher:
Anthem Press # International Mathematical Olympiad Volume 3: 1991?2004

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## Overview

The famed International Mathematical Olympiad has been challenging students worldwide for over 40 years. The first competition was held in Romania in 1959 with seven countries participating. It has since expanded to attract competitors from over 80 countries, representing all five continents. This third volume features every question set from 1991-2004, along with comprehensive solutions and multiple answers where applicable. A fantastic selection of mathematical puzzles, this fully updated three volume series will be of interest to serious mathematicians and enthusiasts alike. István Reiman's compilation of logic puzzles and questions will tease the intellect of all those with a mathematical mind.

## Product Details

ISBN-13: 9781843312048 Anthem Press 09/05/2005 Anthem Learning Series First Edition, 1 186 156.00(w) x 234.00(h) x (d)

István Reiman was formerly Leader of the Chair of Geometry at the Budapest University of Technology. He has guided the Youth Mathematical Circle of the J Bolyai Mathematical Society and directed the preparation of Hungarian students for the annual International Maths Olympiad for 40 years. He was awarded the Paul Erdos Award by the World Federation of Mathematics Competitions in early 2000.

By Istvan Reiman

#### Wimbledon Publishing Company

ISBN: 978-1-84331-204-8

CHAPTER 1

Problems

1991

1991/1. Given a triangle ABC, let I be the incentre. The internal bisectors of angles A, B and C meet the opposite sides at A', B', C' respectively. Prove that

(1) 1/4 < AI x B x CI/ AA' x BB' x CC' ≤ 8/27.

1991/2. Let n > 6 be an integer and let a1, a2, ..., ak be all the positive integers less than n and relatively prime to n. If

a2 - a1 = a3 - a2 = ... = ak - ak-1 > 0,

prove that n must be either a prime number or a power of 2.

1991/3. Let S = {1, 2, 3, ..., 280}. Find the smallest integer n such that each n-element subset of S contains five numbers which are pairwise relatively prime.

1991/4. Suppose G is a connected graph with k edges. Prove that it is possible to label the edges 1, 2, 3, ..., k in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labelling those edges is 1.

[A graph is a set of points, called vertices, together with a set of edges joining certain pairs of distinct vertices. Each pair of points belongs to at most one edge. The graph is connected if for each pair of distinct vertices x, y there is some sequence of vertices x = v0, v1, ..., vm = y such that each pair vi, vi+1 (0 ≤ i< m) is joined by an edge.]

1991/5. Let ABC be a triangle and X an interior point of ABC. Show that at least one of the angles X AB, X BC, PCA is less than or equal to 30°.

1991/6. Given any real number a > 1 construct a bounded infinite sequence x0, x11, ..., such that

(1) |xi - xj} x |i - j|a ≥ 1

for every pair of distinct i, j. [An infinite sequence x0, x1, ..., of real numbers is bounded if there is a constant C such that |xi| for all i.]

1992/1. Find all integers a, b, c satisfying 1 < a < b < c such that (a - 1) x (b - 1) x (c - 1) is a divisor of abc - 1.

1992/2. Find all functions f defined on the set of all real numbers with real values, such that

(1) f (x2 + f(y)) = y + (f(x))2

for all x, y.

1992/3. Consider 9 points in space, no 4 coplanar. Each pair of points is joined by a line segment which is coloured either blue or red or left uncoloured. Find the smallest value of n such that whenever exactly n edges are coloured, the set of coloured edges necessarily contains a triangle all of whose edges have the same colour.

1992/4. L is tangent to the circle C and M is a point on L. Find the locus of all points P such that there exist points Q and R on L equidistant from M with C the incircle of the triangle PQR.

1992/5. Let S be a finite set of points in three-dimensional space. Let Sx, Sy, Sz be the sets consisting of the orthogonal projections of the points of S onto the yz-plane, xz-plane, xy-plane respectively. Prove that

|S|2|Sx Sy Sz|,

where |A| denotes the number of points in the set A.

1992/6. For each positive integer n, S(n) is defined as the greatest integer such that for every positive integer k ≤ S(n), n2 can be written as the sum of k positive squares.

(a) Prove that S(n) ≤ n2 - 114 for each n ≥ 4.

(b) Find an integer n such that S(n) = n2 - 14.

(c) Prove that there are infinitely many integers n such that S(n) = n2 - 14.

1993

1993/1. Let f(x) = xn + 5xn-1 + 3, where n > 1 is an integer. Prove that f(x) cannot be expressed as the product of two non-constant polynomials with integer coefficients.

1993/2. Let D be a point inside the acute-angled triangle ABC such that [angle]ABD = 90° + [angle]ACB and AC x BD = AD x BC.

(a) Calculate the ratio AB x CD/AC x BD.

(b) Prove that the tangents at C to the circumcircles of ACD and BCD are perpendicular.

1993/3. On an infinite chessboard a game is played as follows. At the start nn pieces are arranged in an n x n block of adjoining squares, one piece on each square. A move in the game is a jump in a horizontal or vertical direction over an adjacent occupied square to an unoccupied square immediately beyond. The piece which has been jumped over is removed. Find those values of n for which the game can end with only one piece remaining on the board.

1993/4. For three points P, Q, R in the plane define m(PQR) as the minimum length of the three altitudes of the triangle PQR (or zero if the points are collinear). Prove that for any points A, B, C, X

m(ABC) ≤ m(ABX) + m(AXC) + m(XBC).

1993/5. Does there exist a function f from the positive integers to the positive integers such that f(1) = 2, f(f(n)) = f(n) + n for all n, and f(n) < f(n + 1) for all n?

1993/6. There are n > 1 lamps L0, L1, ..., Ln-1 in a circle. We use Ln+k to mean Lk. A lamp is at all times either on or off. Perform steps s0, s1, ... as follows: at step si, if Li-1 is lit, then switch Li from on to off or vice versa, otherwise do nothing. Show that:

(a) There is a positive integer M(n) such that after M(n) steps all the lamps are on again;

(b) If n = 2k, then we can take M(n) = n2 - 1.

(c) If n = 2k + 1 then we can take M(n) = n2 - n + 1.

1994

1994/1. Let m and n be positive integers. Let ai, a2, ..., am a be distinct elements of {1, 2, ..., n} such that whenever ai + ajn for some i, j (possibly the same) we have ai + aj = ak for some k. Prove that

a1 + a2 + ... + am/m ≥ n + 1/2.

1994/2. ABC is an isosceles triangle with AB = AC. M is the midpoint of BC and O is the point on the line AM such that OB is perpendicular to AB. Q is an arbitrary point on BC different from B and C. E lies on the line AB and F lies on the line AC such that E, Q and F are distinct and collinear. Prove that OQ is perpendicular to EF if and only if QE = QF.

1994/3. For any positive integer k, let f(k) be the number of elements in the set Ak = {k + 1, k + 2, ..., 2k} which have exactly three Is when written in base 2. Prove that for each positive integer m, there is at least one k with f(k) = m and determine all m for which there is exactly one k.

1994/4. Determine all ordered pairs (m, n) of positive integers for which

(1) n3 + 1/mn - 1

is an integer.

1994/5. Let S be the set of all real numbers greater than (-1). Find all functions f from S to S such that f (x + f(y) + xf(y)) = y + f(x) + yf(x) for all x and y, f(x)/x is strictly increasing on each of the intervals – 1 < x< 0 and 0 < x.

1994/6. Show that there exists a set A of positive integers with the following property: for any infinite set of primes, there exist two positive integers m in A and n not in A, each of which is a product of k distinct elements of S for some k ≥ 2.

1995

1995/1. Let A, B, C, D be four distinct points on a line, in that order. The circles with diameter AC and BD intersect at X and Y. Let P be a point on the line XY other than Z. The line CP intersects the circle with diameter AC at C and M, and the line BP intersects the circle with diameter BD at B and N. Prove that the lines AM, DN, XY are concurrent.

1995/2. Let a, b, c be positive real numbers with abc = 1. Prove that

1/a3(b + c) + 1/b3(c + a) + 1/c3(a + b) ≥ 3/2.

1995/3. Determine all integers n > 3 for which there exist n points A1, A2, ..., An in the plane, no three collinear and real numbers r1r2, ..., rn such that for any distinct i, j, k the area of the triangle AiAjAk is ri + rj + rk.

1995/4. Find the maximum value of x0 for which there exists a sequence x0, x1, ..., x1995 of positive reals with x0 = x1995 such that for i = 1, 2, ..., 1995

(i) x0 = x1995;

(ii) xi-1 + 2/xi-1 = 2xi + 1/xi

1995/5. Let ABCDEF be a convex hexagon with AB = BC = CD and DE = EF = FA such that [angle]BCD = [angle]EFA = 60°. Suppose that G and # are points in the interior of the hexagon such that [angle]AGB = [angle]DHE -120°. Prove that

(1) AG + GB + GH + DH + HE ≥ CF.

1995/6. Let p be an odd prime number. How many p-element subsets A of {1, 2, ..., 2p} are there, the sum of whose elements is divisible by p?

1996

1996/1. We are given a positive integer r and a rectangular board divided into 20 x 12 unit squares. The following moves are permitted on the board: one can move from one square to another only if the distance between the centres of the two squares is [square root of r]. The task is to find a sequence of moves leading between two adjacent corners of the board which lie on the long side.

(a) Show that the task cannot be done if r is divisible by 2 or 3.

(b) Prove that the task is possible for r = 73.

(c) Can the task be done for r = 97?

1996/2. Let P be a point inside the triangle ABC such that

(1) [angle]APP - [angle]ACB = [angle]APC - [angle]ABC.

Let D, E be the incentres of triangles APB, APC respectively. Show that AP, BD and CE meet at a point.

1996/3. Let S be the set of non-negative integers. Find all functions f from S to itself such that

(1) f(m + f(n)) = f(f(m)) + f(n)

for all m, n.

1996/4. The positive integers a and b are such that 15a + 16b and 16a – 156 are both squares of positive integers. What is the least possible value that can be taken by the smaller of these two squares?

1996/5. Let ABCDEF be a convex hexagon of perimeter p such that AB is parallel to DE, BC is parallel to EE and CD is parallel to FA. Let RA, RC, and RE denote the circumradii of triangles FAB, BCD, DEF respectively.

Prove that

(1) RA + RC + RE ≥ p/2.

1996/6. Let p, q, n be positive integers with p + q <n. Let x0, x1, ..., xn be integers such that x0 = xn = 0, and for each 1 ≤ i ≤ n, xi - xi-1 = p or -q. Show that there exist indices i < j with (i, j) ≠ (0, n) such that xi = xj

1997

1997/1. In the plane the points with integer coordinates are the vertices of unit squares. The squares are coloured alternately black and white as on a chess-board. For any pair of positive integers m and n, consider a right-angled triangle whose vertices have integer coordinates and whose legs, of lengths m and n, lie along the edges of the squares. Let S1 be the total area of the black part of the triangle, and S2 be the total area of the white part. Let

f(m, n) = |S1 - S2.

(a) Calculate f(m, n) for all positive integers which are either both even or both odd.

(b) Prove that f(m, n) ≤ 1/2 max(m, n) for all m, n.

(c) Show that there is no constant C such that f(m, n) C for all (m, n).

1997/2. The angle at A is the smallest angle in the triangle ABC. The points B and C divide the circumcircle of the triangle into two arcs. Let U be an interior point of the arc between B and C which does not contain A. The perpendicular bisectors of AB and AC meet the line AU at V and W, respectively. The lines BV and CW meet at T. Show that

AU = TB + TC.

1997/3. Let x1, x2, ..., xn be real numbers satisfying

|x1 + x2 + ... + xn| = 1

and

|xi| ≤ n + 1/2 (i = 11, 2, ... n).

Show that there exists a permutation yi of xi such that

|y1 + 2y2 + 3y3 + ... + nyn| ≤ n + 1/2.

1997/4. An n x n matrix whose entries come from the set S = {1, 2, ..., 2n - 1} is called silver matrix if, for each i = 1, 2, ..., n, the ith row and the ith column together contain all elements of S. Show that:

(a) there is no silver matrix for n = 1997;

(b) silver matrices exist for infinitely many values of n.

1997/5. Find all pairs (a, b) of positive integers that satisfy

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

1997/6. For each positive integer n, let f (n) denote the number of ways of representing n as a sum of powers of 2 with non-negative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For example, f(4) = 4, because 4 can be represented as

4; 2 + 2; 2 + 1 + 1; 1 + 1 + 1 + 1.

Prove that for any integer n ≥ 3

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

1998

1998/1. In the convex quadrilateral ABCD, the diagonals AC and BD are perpendicular and the opposite sides AB and CD are not parallel. The point P, where the perpendicular bisectors of AB and CD meet, is inside ABCD. Prove that ABCD is cyclic if and only if the triangles ABP and CDP have equal areas.

1998/2. In a competition there are a contestants and b judges, where b ≥ 3 is an odd integer. Each judge rates each contestant as either "pass" or "fail". Suppose k is a number such that for any two judges their ratings coincide for at most k contestants. Prove

k/a ≥ b - 1/2b.

1998/3. For any positive integer n, let d(n) denote the number of positive divisors of n (including 1 and n.) Determine all positive integers k such that

d(n2)/d(n) = k

for some n.

1998/4. Determine all pairs (a, b) of positive integers such that (ab2 + b + 7) divides (a2b + a + b).

1998/5. Let I be the incentre of the triangle ABC. Let the incircle of ABC touch the sides BC, CA, AB at K, L, M, respectively. The line through B parallel to MK meets the lines LM and LK at R and S, respectively. Prove that the angle RIS is acute.

1998/6. Consider all functions f from the set of all positive integers into itself satisfying

(1) f(t2f(s)) = s(f(t))2

for all s and t. Determine the least possible value of f (1998).

1999

1999/1. Find all finite sets S of at least three points in the plane such that for all distinct points A, B in S, the perpendicular bisector of AB is an axis of symmetry for S.

1999/2. Let n ≥ 2 be a fixed integer. Find the smallest constant C such that for all non-negative reals x1, ..., xn:

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Determine when equality occurs.

1999/3. Given an n x n square board with n even. Two distinct squares of the board are said to be adjacent if they share a common side, but a square is not adjacent to itself. Find the minimum number of squares that can be marked so that every square (marked or not) is adjacent to at least one marked square.

1999/4. Find all pairs (n, p) of positive integers, such that p is a prime, n ≤ 2p and (p - 1)n + 1 is divisible by np-1.

1999/5. The circles C1 and C2 lie inside the circle C, and are tangent to it at M and N, respectively. C1 passes through the centre of C2. The common chord of C1 and C2 when extended, meets C at A and B. The lines MA and MB meet C1 again at E and F. Prove that the line EE is tangent to C2.

(Continues...)

Excerpted from International Mathematical Olympiad Volume III 1991â"2004 by Istvan Reiman. Copyright © 2005 Typotex Ltd.. Excerpted by permission of Wimbledon Publishing Company.
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