In decimal representation $$\text {34!=295232799039a041408476186096435b0000000}.$$Find the numbers $a$ and $b$.

Prove that if for a real number $a $ , $a+\frac {1}{a} $is integer then $a^n+\frac {1}{a^n} $ is also integer where $n$ is positive integer.

$65$ distinct natural numbers not exceeding $2016$ are given. Prove that among these numbers we can find four $a,b,c,d$ such that $a+b-c-d$ is divisible by $2016.$

Let $O$ be the circumcenter of $\triangle ABC.$ The circle $k$ passing through $A$ and $B$ cuts $AC$ and $BC$ at $P$ and $Q,$ respectively. Prove that $PQ$ and $OC$ are perpendicular.

Positive integers $(p,a,b,c)$ called good quadruple if a) $p $ is odd prime, b) $a,b,c $ are distinct , c) $ab+1,bc+1$ and $ca+1$ are divisible by $p $. Prove that for all good quadruple $p+2\le \frac {a+b+c}{3} $, and show the equality case.

For all reals $x,y,z$ prove that $$\sqrt {x^2+\frac {1}{y^2}}+ \sqrt {y^2+\frac {1}{z^2}}+ \sqrt {z^2+\frac {1}{x^2}}\geq 3\sqrt {2}. $$

In $\triangle ABC$ the median $AM$ is drawn. The foot of perpendicular from $B$ to the angle bisector of $\angle BMA$ is $B_1$ and the foot of perpendicular from $C$ to the angle bisector of $\angle AMC$ is $C_1.$ Let $MA$ and $B_1C_1$ intersect at $A_1.$ Find $\frac{B_1A_1}{A_1C_1}.$