IR-MO-R2 2018

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.