We have a rectangle with it sides being a mirror.A light Ray enters from one of the corners of the rectangle and after being reflected several times enters to the opposite corner it started.Prove that at some time the light Ray passed the center of rectangle(Intersection of diagonals.)

$ABC$ is an isosceles triangle ($AB=AC$). Point $X$ is an arbitrary point on $BC$. $Z \in AC$ and $Y \in AB$ such that $\angle BXY = \angle ZXC$. A line parallel to $YZ$ passes through $B$ and cuts $XZ$ at $T$. Prove that $AT$ bisects $\angle A$.

$x_1,x_2,...,x_n>1$ are natural numbers and $n \geq 3$ Prove that : $(x_1x_2...x_n)^2 \ne x_1^3 + x_2^3 +...+x_n^3$

Consider a circle with diameter $AB$ and let $C,D$ be points on its circumcircle such that $C,D$ are not in the same side of $AB$.Consider the parallel line to $AC$ passing from $D$ and let it intersect $AB$ at $E$.Similarly consider the paralell line to $AD$ passing from $C$ and let it intersect $AB$ at $F$.The perpendicular line to $AB$ at $E$ intersects $BC$ at $X$ and the perpendicular line to $AB$ at $F$ intersects $DB$ at $Y$.Prove that the permiter of triangle $AXY$ is twice $CD$. Remark:This problem is proved to be wrong due to a typo in the exam papers you can find the correct version here.

Ali and Naqi are playing a game. At first, they have Polynomial $P(x) = 1+x^{1398}$. Naqi starts. In each turn one can choice natural number $k \in [0,1398]$ in his trun, and add $x^k$ to the polynomial. For example after 2 moves $P$ can be : $P(x) = x^{1398} + x^{300} + x^{100} +1$. If after Ali's turn, there exist $t \in R$ such that $P(t)<0$ then Ali loses the game. Prove that Ali can play forever somehow he never loses the game!

Consider lattice points of a $6*7$ grid.We start with two points $A,B$.We say two points $X,Y$ connected if one can reflect several times WRT points $A,B$ and reach from $X$ to $Y$.Over all choices of $A,B$ what is the minimum number of connected components?