Let $P $ be the intersection of $AC $ and $BD $ in isosceles trapezoid  $ABCD $ ($AB\parallel CD$ , $BC=AD $) . The circumcircle of triangle $ABP $ inersects $BC $ for the second time at $X $. Point $Y $ lies on $AX $ such that $DY\parallel BC $. Prove that $\hat {YDA} =2.\hat {YCA} $.

Let $n$ be odd natural number and $x_1,x_2,\cdots,x_n$ be pairwise distinct numbers. Prove that someone can divide the difference of these number into two sets with equal sum. ( $X=\{\mid x_i-x_j \mid | i<j\}$ )

Let $a>k$ be natural numbers and $r_1<r_2<\dots r_n,s_1<s_2<\dots <s_n$ be sequences of natural numbers such that: $(a^{r_1}+k)(a^{r_2}+k)\dots (a^{r_n}+k)=(a^{s_1}+k)(a^{s_2}+k)\dots (a^{s_n}+k)$ Prove that these sequences are equal.

Find all functions $f:\Bbb {R} \rightarrow \Bbb {R} $ such that: $$f(x+y)f(x^2-xy+y^2)=x^3+y^3$$for all reals $x, y $.

Lamps of the hall switch by only five keys. Every key is connected to one or more lamp(s). By switching every key, all connected lamps will be switched too. We know that no two keys have same set of connected lamps with each other. At first all of the lamps are off. Prove that someone can switch just three keys to turn at least two lamps on.

Two circles $\omega_1,\omega_2$ intersect at $P,Q $. An arbitrary line passing through $P $ intersects $\omega_1 , \omega_2$ at $A,B $ respectively. Another line parallel to $AB $ intersects $\omega_1$ at $D,F $ and $\omega_2$ at $E,C $ such that $E,F $ lie between $C,D $.Let $X\equiv AD\cap BE $ and $Y\equiv BC\cap AF $. Let $R $ be the reflection of $P $ about $CD$. Prove that: a. $R $ lies on $XY $. b. PR is the bisector of $\hat {XPY}$.