IR-TST 2008

Find all functions $f: \mathbb R\longrightarrow \mathbb R$ such that for each $x,y\in\mathbb R$ : $$f(xf(y)) + y + f(x) = f(x + f(y)) + yf(x)$$

Suppose that $I$ is incenter of triangle $ABC$ and $l'$ is a line tangent to the incircle. Let $l$ be another line such that intersects $AB,AC,BC$ respectively at $C',B',A'$ . We draw a tangent from $A'$ to the incircle other than $BC$ , and this line intersects with $l'$ at $A_1$ . $B_1,C_1$ are similarly defined. Prove that $AA_1,BB_1,CC_1$ are concurrent.

Suppose that $T$ is a tree with $k$ edges. Prove that the $k$ -dimensional cube can be partitioned to graphs isomorphic to $T$ .

Let $P_1,P_2,P_3,P_4$ be points on the unit sphere. Prove that $\sum_{i\neq j}\frac1{|P_i-P_j|}$ takes its minimum value if and only if these four points are vertices of a regular pyramid.

Let $a,b,c > 0$ and $ab+bc+ca = 1$ . Prove that: $$\sqrt {a^3 + a} + \sqrt {b^3 + b} + \sqrt {c^3 + c}\geq2\sqrt {a + b + c}.$$

Prove that in a tournament with 799 teams, there exist 14 teams, that can be partitioned into groups in a way that all of the teams in the first group have won all of the teams in the second group.

Let $S$ be a set with $n$ elements, and $F$ be a family of subsets of $S$ with $2^{n-1}$ elements, such that for each $A,B,C\in F$ , $A\cap B\cap C$ is not empty. Prove that the intersection of all of the elements of $F$ is not empty.

Find all polynomials $p$ of one variable with integer coefficients such that if $a$ and $b$ are natural numbers such that $a + b$ is a perfect square, then $p\left(a\right) + p\left(b\right)$ is also a perfect square.

$I_a$ is the excenter of the triangle $ABC$ with respect to $A$ , and $AI_a$ intersects the circumcircle of $ABC$ at $T$ . Let $X$ be a point on $TI_a$ such that $XI_a^2=XA.XT$ . Draw a perpendicular line from $X$ to $BC$ so that it intersects $BC$ in $A'$ . Define $B'$ and $C'$ in the same way. Prove that $AA'$ , $BB'$ and $CC'$ are concurrent.

In the triangle $ABC$ , $\angle B$ is greater than $\angle C$ . $T$ is the midpoint of the arc $BAC$ from the circumcircle of $ABC$ and $I$ is the incenter of $ABC$ . $E$ is a point such that $\angle AEI=90^\circ$ and $AE\parallel BC$ . $TE$ intersects the circumcircle of $ABC$ for the second time in $P$ . If $\angle B=\angle IPB$ , find the angle $\angle A$ .

$k$ is a given natural number. Find all functions $f: \mathbb{N}\rightarrow\mathbb{N}$ such that for each $m,n\in\mathbb{N}$ the following holds: $$f(m)+f(n)\mid (m+n)^k$$

In the acute-angled triangle $ABC$ , $D$ is the intersection of the altitude passing through $A$ with $BC$ and $I_a$ is the excenter of the triangle with respect to $A$ . $K$ is a point on the extension of $AB$ from $B$ , for which $\angle AKI_a=90^\circ+\frac 34\angle C$ . $I_aK$ intersects the extension of $AD$ at $L$ . Prove that $DI_a$ bisects the angle $\angle AI_aB$ iff $AL=2R$ . ( $R$ is the circumradius of $ABC$ )