IR-TST 2016

Let $a,b,c,d$ be positive real numbers such that $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}+\frac{1}{d+1}=2$ . Prove that $$\sum_{cyc} \sqrt{\frac{a^2+1}{2}} \geq (3.\sum_{cyc} \sqrt{a}) -8$$

Let $ABC$ be an arbitrary triangle and $O$ is the circumcenter of $\triangle {ABC}$ .Points $X,Y$ lie on $AB,AC$ ,respectively such that the reflection of $BC$ WRT $XY$ is tangent to circumcircle of $\triangle {AXY}$ .Prove that the circumcircle of triangle $AXY$ is tangent to circumcircle of triangle $BOC$ .

Let $p \neq 13$ be a prime number of the form $8k+5$ such that $39$ is a quadratic non-residue modulo $p$ . Prove that the equation $$x_1^4+x_2^4+x_3^4+x_4^4 \equiv 0 \pmod p$$ has a solution in integers such that $p\nmid x_1x_2x_3x_4$ .

Let $AD,BF,CE$ be altitudes of triangle $ABC$ . $Q$ is a point on $EF$ such that $QF=DE$ and $F$ is between $E,Q$ . $P$ is a point on $EF$ such that $EP=DF$ and $E$ is between $P,F$ .Perpendicular bisector of $DQ$ intersect with $AB$ at $X$ and perpendicular bisector of $DP$ intersect with $AC$ at $Y$ .Prove that midpoint of $BC$ lies on $XY$ .