JBMO-SL 2009

Determine all integers $a, b, c$ satisfying the identities:

$$\begin{gathered} a+b+c=15 \\ (a-3)^{3}+(b-5)^{3}+(c-7)^{3}=540 \end{gathered}$$

Find the maximum value of $z+x$, if $(x, y, z, t)$ satisfies the conditions:

$$\left\{\begin{array}{l} x^{2}+y^{2}=4 \\ z^{2}+t^{2}=9 \\ x t+y z \geq 6 \end{array}\right.$$

Find all values of the real parameter $a$, for which the system

$$\left\{\begin{array}{c} (|x|+|y|-2)^{2}=1 \\ y=a x+5 \end{array}\right.$$

has exactly three solutions.

Real numbers $x, y, z$ satisfy

$$0<x, y, z<1$$

and

$$x y z=(1-x)(1-y)(1-z) .$$

Show that

$$\frac{1}{4} \leq \max \{(1-x) y,(1-y) z,(1-z) x\}$$

Let $x, y, z$ be positive real numbers. Prove that:

$$\left(x^{2}+y+1\right)\left(x^{2}+z+1\right)\left(y^{2}+z+1\right)\left(y^{2}+x+1\right)\left(z^{2}+x+1\right)\left(z^{2}+y+1\right) \geq(x+y+z)^{6}$$

Each one of 2009 distinct points in the plane is coloured in blue or red, so that on every blue-centered unit circle there are exactly two red points. Find the gratest possible number of blue points.

Five players $(A, B, C, D, E)$ take part in a bridge tournament. Every two players must play (as partners) against every other two players. Any two given players can be partners not more than once per day. What is the least number of days needed for this tournament?

a) In how many ways can we read the word SARAJEVO from the table below, if it is allowed to jump from cell to an adjacent cell (by vertex or a side) cell?

b) After the letter in one cell was deleted, only 525 ways to read the word SARAJEVO remained. Find all possible positions of that cell.

Determine all pairs $(m, n)$ for which it is possible to tile the table $m \times n$ with "corners" as in the figure below, with the condition that in the tiling there is no rectangle (except for the $m \times n$ one) regularly covered with corners.

Let $A B C D$ be a parallelogram with $A C>B D$, and let $O$ be the point of intersection of $A C$ and $B D$. The circle with center at $O$ and radius $O A$ intersects the extensions of $A D$ and $A B$ at points $G$ and $L$, respectively. Let $Z$ be intersection point of lines $B D$ and $G L$. Prove that $\angle Z C A=90^{\circ}$.

In a right trapezoid $A B C D(A B \| C D)$ the angle at vertex $B$ measures $75^{\circ}$. Point $H$ is the foot of the perpendicular from point $A$ to the line $B C$. If $B H=D C$ and $A D+A H=8$, find the area of $A B C D$.

A parallelogram $A B C D$ with obtuse angle $\angle A B C$ is given. After rotating the triangle $A C D$ around the vertex $C$, we get a triangle $C D^{\prime} A^{\prime}$, such that points $B, C$ and $D^{\prime}$ are collinear. The extension of the median of triangle $C D^{\prime} A^{\prime}$ that passes through $D^{\prime}$ intersects the straight line $B D$ at point $P$. Prove that $P C$ is the bisector of the angle $\angle B P D^{\prime}$.

Let $A B C D E$ be a convex pentagon such that $A B+C D=B C+D E$ and let $k$ be a semicircle with center on side $A E$ that touches the sides $A B, B C, C D$ and $D E$ of the pentagon, respectively, at points $P, Q, R$ and $S$ (different from the vertices of the pentagon). Prove that $P S \| A E$.

Let $A, B, C$ and $O$ be four points in the plane, such that $\angle A B C>90^{\circ}$ and $O A=$ $O B=O C$. Define the point $D \in A B$ and the line $\ell$ such that $D \in \ell, A C \perp D C$ and $\ell \perp A O$. Line $\ell$ cuts $A C$ at $E$ and the circumcircle of $\triangle A B C$ at $F$. Prove that the circumcircles of triangles $B E F$ and $C F D$ are tangent at $F$.