Day 1
1. In right triangle $ABC$ with $\angle{A}= \textdegree{90}$, point $P$ is chosen. $D \in BC$ such that $PD \perp BC$. Let the intersection of $PD$ with $AB$ and $AC$ be $E$ and $F$ respectively. Denote by $X$ and $Y$ as the intersection of $(APE)$ and $(APF)$ with $BP$ and $CP$ respectively. Prove that $CX,BY,PD$ are concurrent.
2. Prove that for any $2\le n \in \mathbb{N}$ there exists positive integers $a_1,a_2,\cdots,a_n$ such that $\forall i\neq j: \text{gcd}(a_i,a_j) = 1$ and $\forall i: a_i \ge 1402$ and the given relation holds. $$[\frac{a_1}{a_2}]+[\frac{a_2}{a_3}]+\cdots+[\frac{a_n}{a_1}] = [\frac{a_2}{a_1}]+[\frac{a_3}{a_2}]+\cdots+[\frac{a_1}{a_n}]$$
3. We have a $n \times n$ board. We color the unit square $(i,j)$ black if $i=j$, red if $i<j$ and green if $i>j$. Let $a_{i,j}$ be the color of the unit square $(i,j)$. In each move we switch two rows and write down the $n$-tuple $(a_{1,1},a_{2,2},\cdots,a_{n,n})$. How many $n$-tuples can we obtain by repeating this process? (note that the order of the numbers are important)
Day 2
4. A positive integer n is given.Find the smallest $k$ such that we can fill a $3*k$ gird with non-negative integers such that: $\newline$ $i$) Sum of the numbers in each column is $n$. $ii$) Each of the numbers $0,1,\dots,n$ appears at least once in each row.
5. We call $(P_n)_{n\in \mathbb{N}}$ an arithmetic sequence with common difference $Q(x)$ if $\forall n: P_{n+1} = P_n + Q$ $\newline$ We have an arithmetic sequence with a common difference $Q(x)$ and the first term $P(x)$ such that $P,Q$ are monic polynomials with integer coefficients and don't share an integer root. Each term of the sequence has at least one integer root. Prove that: $\newline$ a) $P(x)$ is divisible by $Q(x)$ $\newline$ b) $\text{deg}(\frac{P(x)}{Q(x)}) = 1$
6. Circles $W_{1}$ and $W_{2}$ with equal radii are given. Let $P$,$Q$ be the intersection of the circles. points $B$ and $C$ are on $W_{1}$ and $W_{2}$ such that they are inside $W_{2}$ and $W_{1}$ respectively. Points $X$,$Y$ $\neq$ $P$ are on $W_{1}$ and $W_{2}$ respectively, such that $\angle{BPQ}=\angle{BYQ}$ and $\angle{CPQ}=\angle{CXQ}$.Denote by $S$ as the other intersection of $(YPB)$ and $(XPC)$. Prove that $QS,BC,XY$ are concurrent.