JBMO-SL 2004

Prove that

$$(1+a b c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right) \geq 3+a+b+c$$

for any real numbers $a, b, c \geq 1$.

Prove that, for all real numbers $x, y, z$ :

$$\frac{x^{2}-y^{2}}{2 x^{2}+1}+\frac{y^{2}-z^{2}}{2 y^{2}+1}+\frac{z^{2}-x^{2}}{2 z^{2}+1} \leq(x+y+z)^{2}$$

When the equality holds?

Prove that for all real $x, y$

$$\frac{x+y}{x^{2}-x y+y^{2}} \leq \frac{2 \sqrt{2}}{\sqrt{x^{2}+y^{2}}}$$

Prove that if $0<\frac{a}{b}<b<2 a$ then

$$\frac{2 a b-a^{2}}{7 a b-3 b^{2}-2 a^{2}}+\frac{2 a b-b^{2}}{7 a b-3 a^{2}-2 b^{2}} \geq 1+\frac{1}{4}\left(\frac{a}{b}-\frac{b}{a}\right)^{2}$$

A polygon having $n$ sides is arbitrarily decomposed in triangles having all the vertices among the vertices of the polygon. We paint in black the triangles that have two sides that are also sides of the polygon, in red if only one side of the triangle is side of the polygon and white those triangles that have in common with the poligon only vertices.

Prove that there are 2 more black triangles than white ones.

Given $m \times n$ table, each cell signed with "-". The following operations are

(i) to change all the signs in entire row to the opposite, i. e. every "-" to "+", and every "+" to "-";

(ii) to change all the signs in entire column to the opposite, i. e. every "-" to "+" and every "+" to " -".

(a) Prove that if $m=n=100$, using the above operations one can not obtain 2004 signs "t".

(b) If $m=1004$, find the least $n>100$ for which 2004 signs " + " can be obtained.

Two circles $k_{1}$ and $k_{2}$ intersect a.t points $A$ and $B$. A circle $k_{3}$ centered at $A$ meet $k_{1}$ at $M$ and $P$ and $k_{2}$ at $N$ and $Q$, such that $N$ and $Q$ are on different sides of $M P$ and $A B>A M$.

Prove rhat the angles $\angle M B Q$ and $\angle N B P$ are equal.

Let $A B C$ be scalene triangle inscribed in the circle $k$. Circles $\alpha, \beta, \gamma$ are internally tangent to $k$ at points $A_{1}, B_{1}, C_{1}$ respectively, and tangent to the sides $B C, C A, A B$ at points $A_{2}, B_{2}, C_{2}$ respectively, so that $A$ and $A_{1}$ are on opposite sides of $B C^{\prime}, B$ and $B_{1}$ mre on opposite sides of $C A$, and $C$ and $C_{1}$ are on opposite sides of $A B$. Lines $A_{1} A_{2}, B_{1} B_{2}$ and $C_{1} C_{2}$ meet again the circle $k$ in the points $A^{\prime}, B^{\prime}, C^{\prime \prime}$ respectively. is right.

Prove that if $M$ is the intersection point of the lines $B B^{\prime}$ and $C^{\prime} C^{\prime \prime}$, the angle $\angle M A .4^{\prime}$

Let $A B C$ be a triangle with $\angle C=90^{\circ}$ and $D \in C .4, E \in C^{\prime} B$, and $k_{1}, k_{2}, k_{3}, k_{4}$ semicircles with diameters $C A, C B, C D, C E$ respectively, which have common part with the triangle $A B C$. Let also,

$$k_{1} \cap k_{2}=\{C, K\}, k_{3} \cap k_{4}=\{C, M\}, k_{2} \cap k_{3}=\{C, L\}, k_{1} \cap k_{4}=\{C, N\}$$

Prove that $K, L, M$ and $N$ are cocyclic points.