ISL 2004

Does there exist a function $s\colon \mathbb{Q} \rightarrow \{-1,1\}$ such that if $x$ and $y$ are distinct rational numbers satisfying ${xy=1}$ or ${x+y\in \{0,1\}}$ , then ${s(x)s(y)=-1}$ ? Justify your answer.

Proposed by Dan Brown, Canada

Find all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying the equation $$f(x^2+y^2+2f(xy)) = (f(x+y))^2.$$ for all $x,y \in \mathbb{R}$ .

The following operation is allowed on a finite graph: Choose an arbitrary cycle of length 4 (if there is any), choose an arbitrary edge in that cycle, and delete it from the graph. For a fixed integer ${n\ge 4}$ , find the least number of edges of a graph that can be obtained by repeated applications of this operation from the complete graph on $n$ vertices (where each pair of vertices are joined by an edge).

Proposed by Norman Do, Australia

$A$ and $B$ play a game, given an integer $N$ , $A$ writes down $1$ first, then every player sees the last number written and if it is $n$ then in his turn he writes $n+1$ or $2n$ , but his number cannot be bigger than $N$ . The player who writes $N$ wins. For which values of $N$ does $B$ win?

Proposed by A. Slinko & S. Marshall, New Zealand

For a finite graph $G$ , let $f(G)$ be the number of triangles and $g(G)$ the number of tetrahedra formed by edges of $G$ . Find the least constant $c$ such that $$g(G)^3\le c\cdot f(G)^4$$ for every graph $G$ .

Proposed by Marcin Kuczma, Poland

Let $A_1A_2A_3\ldots A_n$ be a regular $n$ -gon. Let $B_1$ and $B_{n-1}$ be the midpoints of its sides $A_1A_2$ and $A_{n-1}A_n$ . Also, for every $i\in\left\{2,3,4,\ldots ,n-2\right\}$ . Let $S$ be the point of intersection of the lines $A_1A_{i+1}$ and $A_nA_i$ , and let $B_i$ be the point of intersection of the angle bisector bisector of the angle $\measuredangle A_iSA_{i+1}$ with the segment $A_iA_{i+1}$ .

Prove that $\sum_{i=1}^{n-1} \measuredangle A_1B_iA_n=180^{\circ}$ .

Proposed by Dusan Dukic, Serbia and Montenegro

For a given triangle $ABC$ , let $X$ be a variable point on the line $BC$ such that $C$ lies between $B$ and $X$ and the incircles of the triangles $ABX$ and $ACX$ intersect at two distinct points $P$ and $Q.$ Prove that the line $PQ$ passes through a point independent of $X$ .

Given a cyclic quadrilateral $ABCD$ , let $M$ be the midpoint of the side $CD$ , and let $N$ be a point on the circumcircle of triangle $ABM$ . Assume that the point $N$ is different from the point $M$ and satisfies $\frac{AN}{BN}=\frac{AM}{BM}$ . Prove that the points $E$ , $F$ , $N$ are collinear, where $E=AC\cap BD$ and $F=BC\cap DA$ .

Proposed by Dusan Dukic, Serbia and Montenegro

The function $f$ from the set $\mathbb{N}$ of positive integers into itself is defined by the equality $$f(n)=\sum_{k=1}^{n} \gcd(k,n), n\in \mathbb{N}.$$ a) Prove that $f(mn)=f(m)f(n)$ for every two relatively prime ${m,n\in\mathbb{N}}$ .

b) Prove that for each $a\in\mathbb{N}$ the equation $f(x)=ax$ has a solution.

c) Find all ${a\in\mathbb{N}}$ such that the equation $f(x)=ax$ has a unique solution.

Given an integer ${n>1}$ , denote by $P_{n}$ the product of all positive integers $x$ less than $n$ and such that $n$ divides ${x^2-1}$ . For each ${n>1}$ , find the remainder of $P_{n}$ on division by $n$ .

Proposed by John Murray, Ireland

Let $p$ be an odd prime and $n$ a positive integer. In the coordinate plane, eight distinct points with integer coordinates lie on a circle with diameter of length $p^{n}$ . Prove that there exists a triangle with vertices at three of the given points such that the squares of its side lengths are integers divisible by $p^{n+1}$ .

Proposed by Alexander Ivanov, Bulgaria