JBMO-SL 2010

The real numbers $a, b, c, d$ satisfy simultaneously the equations

$$a b c-d=1, b c d-a=2, c d a-b=3, d a b-c=-6$$

Prove that $a+b+c+d \neq 0$.

Determine all four digit numbers $\overline{a b c d}$ such that

$$a(a+b+c+d)\left(a^{2}+b^{2}+c^{2}+d^{2}\right)\left(a^{6}+2 b^{6}+3 c^{6}+4 d^{6}\right)=\overline{a b c d}$$

Find all pairs $(x, y)$ of real numbers such that $|x|+|y|=1340$ and $x^{3}+y^{3}+2010 x y=670^{3}$.

Let $a, b, c$ be positive real numbers such that $a b c(a+b+c)=3$. Prove the inequality

$$(a+b)(b+c)(c+a) \geq 8$$

and determine all cases when equality holds.

The real positive numbers $x, y, z$ satisfy the relations $x \leq 2$, $y \leq 3, x+y+z=11$. Prove that $\sqrt{x y z} \leq 6$.

There are two piles of coins, each containing 2010 pieces. Two players A and B play a game taking turns (A plays first). At each turn, the player on play has to take one or more coins from one pile or exactly one coin from each pile. Whoever takes the last coin is the winner. Which player will win if they both play in the best possible way?

A $9 \times 7$ rectangle is tiled with pieces of two types, shown in the picture below.

Find the possible values of the number of the $2 \times 2$ pieces which can be used in such a tiling.

Consider a triangle $A B C$ with $\angle A C B=90^{\circ}$. Let $F$ be the foot of the altitude from $C$. Circle $\omega$ touches the line segment $F B$ at point $P$, the altitude $C F$ at point $Q$ and the circumcircle of $A B C$ at point $R$. Prove that points $A, Q, R$ are collinear and $A P=A C$.

Consider a triangle $A B C$ and let $M$ be the midpoint of the side $B C$. Suppose $\angle M A C=\angle A B C$ and $\angle B A M=105^{\circ}$. Find the measure of $\angle A B C$.

Let $A B C$ be an acute-angled triangle. A circle $\omega_{1}\left(O_{1}, R_{1}\right)$ passes through points $B$ and $C$ and meets the sides $A B$ and $A C$ at points $D$ and $E$, respectively. Let $\omega_{2}\left(O_{2}, R_{2}\right)$ be the circumcircle of the triangle $A D E$. Prove that $O_{1} O_{2}$ is equal to the circumradius of the triangle $A B C$.

Let $A L$ and $B K$ be angle bisectors in the non-isosceles triangle $A B C(L \in B C, K \in A C)$. The perpendicular bisector of $B K$ intersects the line $A L$ at point $M$. Point $N$ lies on the line $B K$ such that $L N \| M K$. Prove that $L N=N A$.

Find all positive integers $n$ such that $n 2^{n+1}+1$ is a perfect square.

Find all positive integers $n$ such that $36^{n}-6$ is a product of two or more consecutive positive integers.