International Competitions 2009

Let $n$ be a positive integer and let $a_1, \ldots, a_k$ ($k \geq 2$) be distinct integers in the set $\{1, \ldots, n\}$ such that $n$ divides $a_i(a_{i+1} - 1)$ for $i = 1, \ldots, k-1$. Prove that $n$ does not divide $a_k(a_1 - 1)$.

Let $ABC$ be a triangle with circumcentre $O$. 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 $\Gamma$ be the circle passing through $K$, $L$ and $M$. Suppose that the line $PQ$ is tangent to the circle $\Gamma$. Prove that $OP = OQ$.

Suppose that $s_1, s_2, s_3, \ldots$ is a strictly increasing sequence of positive integers such that the subsequences

$$s_{s_1}, s_{s_2}, s_{s_3}, \ldots \quad \text{and} \quad s_{s_1 + 1}, s_{s_2 + 1}, s_{s_3 + 1}, \ldots$$

are both arithmetic progressions. Prove that the sequence $s_1, s_2, s_3, \ldots$ is itself an arithmetic progression.

Let $ABC$ be a triangle with $AB = AC$. The angle bisectors of $\angle CAB$ and $\angle ABC$ meet the sides $BC$ and $CA$ at $D$ and $E$, respectively. Let $K$ be the incentre of triangle $ADC$. Suppose that $\angle BEK = 45^{\circ}$. Find all possible values of $\angle CAB$.

Determine all functions $f$ from the set of positive integers to the set of positive integers such that, for all positive integers $a$ and $b$, there exists a non-degenerate triangle with sides of lengths

$$a, f(b) \text{ and } f(b + f(a) - 1).$$

(A triangle is non-degenerate if its vertices are not collinear.)

Let $a_1, a_2, \ldots, a_n$ be distinct positive integers and let $M$ be a set of $n - 1$ positive integers not containing $s = a_1 + a_2 + \cdots + a_n$. A grasshopper is to jump along the real axis, starting at the point $0$ and making $n$ jumps to the right with lengths $a_1, a_2, \ldots, a_n$ in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in $M$.

Find all functions $f\colon \mathbb{R}\to \mathbb{R}$ such that

$$f (x f (y)) + f (f (x) + f (y)) = y f (x) + f (x + f (y))$$

for all $x,y\in \mathbb{R}$, where $\mathbb{R}$ denotes the set of real numbers.

Suppose that we have $n \geqslant 3$ distinct colours. Let $f(n)$ be the greatest integer with the property that every side and every diagonal of a convex polygon with $f(n)$ vertices can be coloured with one of $n$ colours in the following way:

• at least two distinct colours are used, and

• any three vertices of the polygon determine either three segments of the same colour or of three different colours.

Show that $f(n) \leqslant (n - 1)^2$ with equality for infinitely many values of $n$.

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 $F$. The line $EF$ meets segments $AB$ and $CD$ at $G$ and $H$, respectively. Show that $\measuredangle AGH = \measuredangle DHG$.

Determine all integers $k \geqslant 2$ such that for all pairs $(m, n)$ of different positive integers not greater than $k$, the number $n^{n-1} - m^{m-1}$ is not divisible by $k$.

Let $x, y, z$ be real numbers satisfying $x^2 + y^2 + z^2 + 9 = 4(x + y + z)$. Prove that $$x^4 + y^4 + z^4 + 16(x^2 + y^2 + z^2) \geqslant 8(x^3 + y^3 + z^3) + 27$$ and determine when equality holds.

Let $a, b, c$ be real numbers such that for every two of the equations $$x^2 + ax + b = 0, \quad x^2 + bx + c = 0, \quad x^2 + cx + a = 0$$ there is exactly one real number satisfying both of them. Determine all the possible values of $a^2 + b^2 + c^2$.

The numbers $0, 1, 2, \ldots, n$ ($n \geqslant 2$) are written on a blackboard. In each step we erase an integer which is the arithmetic mean of two different numbers which are still left on the blackboard. We make such steps until no further integer can be erased. Let $g(n)$ be the smallest possible number of integers left on the blackboard at the end. Find $g(n)$ for every $n$.

We colour every square of the $2009 \times 2009$ board with one of $n$ colours (we do not have to use every colour). A colour is called connected if either there is only one square of that colour or any two squares of the colour can be reached from one another by a sequence of moves of a chess queen without intermediate stops at squares having another colour (a chess queen moves horizontally, vertically or diagonally). Find the maximum $n$, such that for every colouring of the board at least one colour present at the board is connected.

Let $ABCD$ be a parallelogram with $\measuredangle BAD = 60°$ and denote by $E$ the intersection of its diagonals. The circumcircle of the triangle $ACD$ meets the line $BA$ at $K \neq A$, the line $BD$ at $P \neq D$ and the line $BC$ at $L \neq C$. The line $EP$ intersects the circumcircle of the triangle $CEL$ at points $E$ and $M$. Prove that the triangles $KLM$ and $CAP$ are congruent.

Suppose that $ABCD$ is a cyclic quadrilateral and $CD = DA$. Points $E$ and $F$ belong to the segments $AB$ and $BC$ respectively, and $\measuredangle ADC = 2\measuredangle EDF$. Segments $DK$ and $DM$ are height and median of the triangle $DEF$, respectively. $L$ is the point symmetric to $K$ with respect to $M$. Prove that the lines $DM$ and $BL$ are parallel.

Find all pairs $(m, n)$ of integers which satisfy the equation $$(m + n)^4 = m^2n^2 + m^2 + n^2 + 6mn.$$

Find all non-negative integer solutions of the equation $$2^x + 2009 = 3^y5^z.$$