IMAC 2008

Find all prime numbers $p$ for which $1 + p\cdot 2^{p}$ is a perfect square.

In the $ABC$ triangle, the bisector of $A$ intersects the $[BC]$ at the point $A_ {1}$ , and the circle circumscribed to the triangle $ABC$ at the point $A_ {2}$ . Similarly are defined $B_ {1}$ and $B_ {2}$ , as well as $C_ {1}$ and $C_ {2}$ . Prove that $$\frac {A_{1}A_{2}}{BA_{2} + A_{2}C} + \frac {B_{1}B_{2}}{CB_{2} + B_{2}A} + \frac {C_{1}C_{2}}{AC_{2} + C_{2}B} \geq \frac {3}{4}$$

Let $0 \leq x \leq 2\pi$ . Prove the inequality $\sqrt {\frac {\sin^{2}x}{1 + \cos^{2}x}} + \sqrt {\frac {\cos^{2}x}{1 + \sin^{2}x}}\geq 1$