International Competitions 2010

Determine all functions $f\colon \mathbb{R}\to \mathbb{R}$ such that the equality $$f \big (\lfloor x \rfloor y \big) = f (x) \big \lfloor f (y) \rfloor$$ holds for all $x, y \in \mathbb{R}$. (Here $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$.)

Let $I$ be the incentre of triangle $ABC$ and let $\Gamma$ be its circumcircle. Let the line $AI$ intersect $\Gamma$ again at $D$. Let $E$ be a point on the arc $\widehat{BDC}$ and $F$ a point on the side $BC$ such that $$\angle BAF = \angle CAE < \frac{1}{2}\angle BAC.$$ Finally, let $G$ be the midpoint of the segment $IF$. Prove that the lines $DG$ and $EI$ intersect on $\Gamma$.

Let $\mathbb{N}$ be the set of positive integers. Determine all functions $g\colon \mathbb{N}\to \mathbb{N}$ such that $$\left(g (m) + n\right) \left(m + g (n)\right)$$ is a perfect square for all $m, n \in \mathbb{N}$.

Let $P$ be a point inside the triangle $ABC$. The lines $AP$, $BP$ and $CP$ intersect the circumcircle $\Gamma$ of triangle $ABC$ again at the points $K$, $L$ and $M$ respectively. The tangent to $\Gamma$ at $C$ intersects the line $AB$ at $S$. Suppose that $SC = SP$. Prove that $MK = ML$.

In each of six boxes $B_{1}, B_{2}, B_{3}, B_{4}, B_{5}, B_{6}$ there is initially one coin. There are two types of operation allowed:

Type 1: Choose a nonempty box $B_{j}$ with $1 \leq j \leq 5$. Remove one coin from $B_{j}$ and add two coins to $B_{j+1}$.

Type 2: Choose a nonempty box $B_{k}$ with $1 \leq k \leq 4$. Remove one coin from $B_{k}$ and exchange the contents of (possibly empty) boxes $B_{k+1}$ and $B_{k+2}$.

Determine whether there is a finite sequence of such operations that results in boxes $B_{1}, B_{2}, B_{3}, B_{4}, B_{5}$ being empty and box $B_{6}$ containing exactly $2010^{2010^{2010}}$ coins. (Note that $a^{b^{c}} = a^{(b^{c})}$.)

Let $a_1, a_2, a_3, \ldots$ be a sequence of positive real numbers. Suppose that for some positive integer $s$, we have $$a _ {n} = \max \left\{a _ {k} + a _ {n - k} \mid 1 \leq k \leq n - 1 \right\}$$ for all $n > s$. Prove that there exist positive integers $\ell$ and $N$, with $\ell \leq s$ and such that $a_{n} = a_{\ell} + a_{n - \ell}$ for all $n \geq N$.

Find all functions $f\colon \mathbb{R}\to \mathbb{R}$ such that for all $x,y\in \mathbb{R}$, we have $$f(x + y) + f(x)f(y) = f(xy) + (y + 1)f(x) + (x + 1)f(y).$$

All positive divisors of a positive integer $N$ are written on a blackboard. Two players $A$ and $B$ play the following game taking alternate moves. In the first move, the player $A$ erases $N$. If the last erased number is $d$, then the next player erases either a divisor of $d$ or a multiple of $d$. The player who cannot make a move loses. Determine all numbers $N$ for which $A$ can win independently of the moves of $B$.

We are given a cyclic quadrilateral $ABCD$ with a point $E$ on the diagonal $AC$ such that $AD = AE$ and $CB = CE$. Let $M$ be the center of the circumcircle $k$ of the triangle $BDE$. The circle $k$ intersects the line $AC$ in the points $E$ and $F$. Prove that the lines $FM$, $AD$, and $BC$ meet at one point.

Find all positive integers $n$ which satisfy the following two conditions:

(i) $n$ has at least four different positive divisors;

(ii) for any divisors $a$ and $b$ of $n$ satisfying $1 < a < b < n$, the number $b - a$ divides $n$.

Three strictly increasing sequences $$a_1, a_2, a_3, \ldots, \qquad b_1, b_2, b_3, \ldots, \qquad c_1, c_2, c_3, \ldots$$ of positive integers are given. Every positive integer belongs to exactly one of the three sequences. For every positive integer $n$, the following conditions hold:

(i) $c_{a_n} = b_n + 1$;

(ii) $a_{n+1} > b_n$;

(iii) the number $c_{n+1}c_n - (n+1)c_{n+1} - nc_n$ is even.

Find $a_{2010}$, $b_{2010}$, and $c_{2010}$.

For each integer $n \geq 2$, determine the largest real constant $C_n$ such that for all positive real numbers $a_1, \ldots, a_n$, we have $$\frac{a_1^2 + \cdots + a_n^2}{n} \geq \left(\frac{a_1 + \cdots + a_n}{n}\right)^2 + C_n \cdot (a_1 - a_n)^2.$$

In each vertex of a regular $n$-gon there is a fortress. At the same moment each fortress shoots at one of the two nearest fortresses and hits it. The result of the shooting is the set of the hit fortresses; we do not distinguish whether a fortress was hit once or twice. Let $P(n)$ be the number of possible results of the shooting. Prove that for every positive integer $k \geq 3$, $P(k)$ and $P(k + 1)$ are relatively prime.

Let $n$ be a positive integer. A square $ABCD$ is partitioned into $n^2$ unit squares. Each of them is divided into two triangles by the diagonal parallel to $BD$. Some of the vertices of the unit squares are colored red in such a way that each of these $2n^2$ triangles contains at least one red vertex. Find the least number of red vertices.

The incircle of the triangle $ABC$ touches the sides $BC$, $CA$, and $AB$ in the points $D$, $E$, and $F$, respectively. Let $K$ be the point symmetric to $D$ with respect to the incenter. The lines $DE$ and $FK$ intersect at $S$. Prove that $AS$ is parallel to $BC$.

Let $A$, $B$, $C$, $D$, $E$ be points such that $ABCD$ is a cyclic quadrilateral and $ABDE$ is a parallelogram. The diagonals $AC$ and $BD$ intersect at $S$ and the rays $AB$ and $DC$ intersect at $F$. Prove that $\angle AFS = \angle ECD$.

For a nonnegative integer $n$, define $a_n$ to be the positive integer with decimal representation $$1\underbrace{0\ldots 0}_{n}2\underbrace{0\ldots 0}_{n}2\underbrace{0\ldots 0}_{n}1.$$

Prove that $a_n/3$ is always the sum of two positive perfect cubes but never the sum of two perfect squares.

We are given a positive integer $n$ which is not a power of $2$. Show that there exists a positive integer $m$ with the following two properties:

(i) $m$ is the product of two consecutive positive integers;

(ii) the decimal representation of $m$ consists of two identical blocks of $n$ digits.