UZ-MO 2013

Let real numbers $a,b$ such that $a\ge b\ge 0$ . Prove that $$\sqrt{a^2+b^2}+\sqrt[3]{a^3+b^3}+\sqrt[4]{a^4+b^4} \le 3a+b .$$

Let $x$ and $y$ are real numbers such that $x^2y^2+2yx^2+1=0.$ If $S=\frac{2}{x^2}+1+\frac{1}{x}+y(y+2+\frac{1}{x})$ , find (a)max $S$ and (b) min $S$ .

Find all functions $f:Q\rightarrow Q$ such that $$f(x+y)+f(y+z)+f(z+t)+f(t+x)+f(x+z)+f(y+t)\ge 6f(x-3y+5z+7t)$$ for all $x,y,z,t\in Q.$

Let circles $\Gamma$ and $\omega$ are circumcircle and incircle of the triangle $ABC$ , the incircle touches sides $BC,CA,AB$ at the points $A_1,B_1,C_1$ . Let $A_2$ and $B_2$ lies the lines $A_1I$ and $B_1I$ ( $A_1$ and $A_2$ lies different sides from $I$ , $B_1$ and $B_2$ lies different sides from $I$ ) such that $IA_2=IB_2=R$ . Prove that : (a) $AA_2=BB_2=IO$ ; (b) The lines $AA_2$ and $BB_2$ intersect on the circle $\Gamma ;$

Let $SABC$ is pyramid, such that $SA\le 4$ , $SB\ge 7$ , $SC\ge 9$ , $AB=5$ , $BC\le 6$ and $AC\le 8$ . Find max value capacity(volume) of the pyramid $SABC$ .