# [Solutions] International Zhautykov Mathematical Olympiad 2013

1. Given a trapezoid $ABCD$ ($AD \parallel BC$) with $\angle ABC > 90^\circ$ . Point $M$ is chosen on the lateral side $AB$. Let $O_1$ and $O_2$ be the circumcenters of the triangles $MAD$ and $MBC$, respectively. The circumcircles of the triangles $MO_1D$ and $MO_2C$ meet again at the point $N$. Prove that the line $O_1O_2$ passes through the point $N$.
2. Find all odd positive integers $n>1$ such that there is a permutation $a_1, a_2, a_3, \ldots, a_n$ of the numbers $1, 2,3, \ldots, n$ where $n$ divides one of the numbers $a_k^2 - a_{k+1} - 1$ and $a_k^2 - a_{k+1} + 1$ for each $k$, $1 \leq k \leq n$ (we assume $a_{n+1}=a_1$).
3. Let $a, b, c$, and $d$ be positive real numbers such that $abcd = 1$. Prove that $\frac{(a-1)(c+1)}{1+bc+c} + \frac{(b-1)(d+1)}{1+cd+d} + \frac{(c-1)(a+1)}{1+da+a} + \frac{(d-1)(b+1)}{1+ab+b} \geq 0.$
4. A quadratic trinomial $p(x)$ with real coefficients is given. Prove that there is a positive integer $n$ such that the equation $p(x) = \frac{1}{n}$ has no rational roots.
5. Given convex hexagon $ABCDEF$ with $AB \parallel DE$, $BC \parallel EF$, and $CD \parallel FA$. The distance between the lines $AB$ and $DE$ is equal to the distance between the lines $BC$ and $EF$ and to the distance between the lines $CD$ and $FA$. Prove that the sum $AD+BE+CF$ does not exceed the perimeter of hexagon $ABCDEF$.
6. A $10 \times 10$ table consists of $100$ unit cells. A block is a $2 \times 2$ square consisting of $4$ unit cells of the table. A set $C$ of $n$ blocks covers the table (i.e. each cell of the table is covered by some block of $C$) but no $n -1$ blocks of $C$ cover the table. Find the largest possible value of $n$.
 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,...