SMO 2009

let $O$ be the center of the circle inscribed in a rhombus ABCD. points E,F,G,H are chosen on sides AB, BC, CD, DA respectively so that EF and GH are tangent to inscribed circle. show that EH and FG are parallel.

a palindromic number is a number which is unchanged when order of its digits is reversed. prove that the arithmetic progression 18, 37,.. contains infinitely many palindromic numbers.

for $k\in\mathbb{N}$ , define $A_n$ for $n=1,2,...$ by $A_{n+1} = \frac{ nA_n+2(n+1)^{2k} }{n+2} , A_1=1$ Prove $A_n$ is integer for all $n\geq 1$ , and $A_n$ is odd if and only if $n\equiv$ 1 or 2(mod 4)

find largest constant C st $\sum_{i=1}^{4} (x_i+1/x_i)^3 \geq C$ for all positive real numbers $x_1,..,x_4$ st $x_1^3+x_3^3+3x_1x_3=x_2+x_4=1$

Find all integers x,y,z with $2\leq x\leq y\leq z$ st $xy \equiv 1$ (mod z) $xz\equiv 1$ (mod y) $yz \equiv 1$ (mod x)