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International Mathematical Olympiad Volume 3: 1991?2004 available in Hardcover, Paperback

- 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

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## Product Details

ISBN-13: | 9781843312048 |
---|---|

Publisher: | Anthem Press |

Publication date: | 09/05/2005 |

Series: | Anthem Learning Series |

Edition description: | First Edition, 1 |

Pages: | 186 |

Product dimensions: | 156.00(w) x 234.00(h) x (d) |

## About the Author

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.

## Read an Excerpt

#### International Mathematical Olympiad Volume III 1991â"2004

**By Istvan Reiman**

**Wimbledon Publishing Company**

**Copyright © 2005 Typotex Ltd.**

All rights reserved.

ISBN: 978-1-84331-204-8

All rights reserved.

ISBN: 978-1-84331-204-8

CHAPTER 1

**International Mathematical Olympiad**

**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 *a*1, *a*2, ..., *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* = *v*0, *v*1, ..., *vm = y* such that each pair *vi, v*i+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 *x*0, *x*11, ..., such that

(1) |xi - xj} x |i - j|a ≥ 1

for every pair of distinct *i, j.* [An infinite sequence *x*0, *x*1, ..., 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), n*2 can be written as the sum of *k* positive squares.

(a) Prove that *S(n) ≤ n*2 - 114 for each *n* ≥ 4.

(b) Find an integer *n* such that *S(n) = n*2 - 14.

(c) Prove that there are infinitely many integers *n* such that *S(n) = n*2 - 14.

**1993**

**1993/1**. Let *f(x) = xn* + 5*xn*-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 *n*n 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 *L*0, *L*1, ..., *Ln*-1 in a circle. We use *Ln+k* to mean *Lk*. A lamp is at all times either on or off. Perform steps *s*0, *s*1, ... 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* = 2*k*, then we can take *M(n) = n*2 - 1.

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

**1994**

**1994/1**. Let *m* and *n* be positive integers. Let *a*i, *a*2, ..., *am* a be distinct elements of {1, 2, ..., *n}* such that whenever *ai + aj* ≤ *n* 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, ..., 2*k*} 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 *A*1, *A*2, ..., *An* in the plane, no three collinear and real numbers *r*1*r*2, ..., *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 *x*0 for which there exists a sequence *x*0, *x*1, ..., *x*1995 of positive reals with *x*0 = *x*1995 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 15*a* + 16*b* and 16*a* – 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 *x*0, *x*1, ..., *xn* be integers such that *x*0 = *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 *S*1 be the total area of the black part of the triangle, and *S*2 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 *x*1, *x*2, ..., *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 *i*th row and the *i*th 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 (*ab*2 + *b* + 7) divides (*a*2*b* + 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 *x*1, ..., *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* ≤ 2*p* and (*p* - 1)*n* + 1 is divisible by *np*-1.

**1999/5**. The circles *C*1 and *C*2 lie inside the circle *C*, and are tangent to it at *M* and *N*, respectively. *C*1 passes through the centre of *C*2. The common chord of *C*1 and *C*2 when extended, meets *C* at *A* and *B*. The lines *MA* and *MB* meet *C*1 again at *E* and *F*. Prove that the line *EE* is tangent to *C*2.

*(Continues...)*

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

All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.

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## Table of Contents

Preface; Problems 1991-2004; A Glossary of Theorems