# [Solutions] International Zhautykov Mathematical Olympiad 2010

1. Find all primes $p,q$ such that $$p^3-q^7=p-q.$$
2. In a cyclic quadrilateral $ABCD$ with $AB=AD$ points $M$,$N$ lie on the sides $BC$ and $CD$ respectively so that $MN=BM+DN$. Lines $AM$ and $AN$ meet the circumcircle of $ABCD$ again at points $P$ and $Q$ respectively. Prove that the orthocenter of the triangle $APQ$ lies on the segment $MN$.
3. A rectangle formed by the lines of checkered paper is divided into figures of three kinds: isosceles right triangles with base of two units, squares with unit side, and parallelograms formed by two sides and two diagonals of unit squares (figures may be oriented in any way). Prove that the number of figures of the third kind is even.
4. Positive integers $1,2,...,n$ are written on а blackboard $(n >2)$. Every minute two numbers are erased and the least prime divisor of their sum is written. In the end only the number $97$ remains. Find the least $n$ for which it is possible.
5. In every vertex of a regular $n$-gon exactly one chip is placed. At each $step$ one can exchange any two neighbouring chips. Find the least number of steps necessary to reach the arrangement where every chip is moved by $\left[\dfrac{n}{2}\right]$ positions clockwise from its initial position.
6. Let $ABC$ arbitrary triangle ($AB \neq BC \neq AC \neq AB$) And O,I,H it's circum-center, incenter and ortocenter (point of intersection altitudes). Prove that
a) $\angle OIH > 90^\circ$.
b) $\angle OIH >135^\circ$.
 MOlympiad.NET là dự án thu thập và phát hành các đề thi tuyển sinh và học sinh giỏi toán. Quý bạn đọc muốn giúp chúng tôi chỉnh sửa đề thi này, xin hãy để lại bình luận facebook (có thể đính kèm hình ảnh) hoặc google (có thể sử dụng $\LaTeX$) bên dưới. BBT rất mong bạn đọc ủng hộ UPLOAD đề thi và đáp án mới hoặc liên hệbbt.molympiad@gmail.comChúng tôi nhận tất cả các định dạng của tài liệu: $\TeX$, PDF, WORD, IMG,...