Note: You have 4 hours to solve as many problems as you can from the following list of 5 problems. Each solution should be written clearly and in detail on a separate sheet of paper. Each problem is worth $ 7$ points. Partial credit will be awarded for partial solutions.


Problem 1. Prove that there are no integers $ x$ and $ y$ satisfying the equation $ x^2-3y^2=17$.


Problem 2. $ \triangle ABC$ is inscribed in a circle $ k$ with diameter $ BD$. Let $ H$ be its orthocenter.


(a)
Prove that point $ D$ is symmetric to $ H$ with respect to the midpoint $ F$ of side $ AC$.


(b)
If $ AC=DH$, find the measure of $ \angle ABC$.


Problem 3. Let $ x_1,\,\,x_2,...,\,\,x_n$ be non-negative numbers whose sum is $ 1/2$. Prove that

$\displaystyle \frac{1-x_1}{1+x_1}\cdot \frac{1-x_2}{1+x_2}\cdots \frac{1-x_n}{1+x_n}\geq \frac{1}{3}\cdot$

For what $ x_i$'s is equality achieved?


Problem 4. Let $ a_1,\,\,a_2,....,a_n,...$ be a sequence of positive numbers such that $ a_n^2\leq a_n-a_{n+1}$ for all $ n=1,2,...$ Prove that $ a_n\leq 1/n$ for all $ n$.


Problem 5. Let $ a_1,a_2,...,a_{mn+1}$ be a sequence of distinct numbers. Prove that we can find either an increasing subsequence of length greater than $ m$ or a decreasing subsequence of length greater than $ n$.