IMO 2010

Problem 1. Determine all functions $f:\mathbb{R}\to \mathbb{R}$ such that the equality

$f(\lfloor x\rfloor y)=f(x)\lfloor f(y)\rfloor$

holds for all $x,y\in \mathbb{R}$. (Here $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z$.)

Problem 2. 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 $\Gamma$.

Problem 3. Let $\mathbb{N}$ be the set of positive integers. Determine all functions $g:\mathbb{N}\to \mathbb{N}$ such that

$(g(m)+n)(m+g(n))$

is a perfect square for all $m,n\in \mathbb{N}$.

Problem 4. 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 $MK=ML$.

Problem 5. 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 $1\le j\le 5$. Remove one coin from $B_j$ and add two coins to $B_{j+1}$.
• Type 2: Choose a nonempty box $B_k$ with $1\le k\le 4$. 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 $a^{b^c}=a^{(b^c)}$.)

Problem 6. Let $a_1,a_2,a_3,\dots$ be a sequence of positive real numbers. Suppose that for some positive integer $s$, we have

$a_n=\max \{a_k+a_{n-k}\mid 1\le k\le n-1\}$

for all $n>s$. Prove that there exist positive integers $\ell$ and $N$, with $\ell \le s$ and such that $a_n=a_{\ell }+a_{n-\ell }$ for all $n\ge N$.