Find all pairs of nonzero rational numbers $x, y$, such that every positive rational number can be written as $\frac{\{rx\}} {\{ry\}}$ for some positive rational $r$.

Solve the system of equations:$$\left\{ \begin{array}{l} ({x^2} + y)\sqrt {y - 2x} - 4 = 2{x^2} + 2x + y\\ {x^3} - {x^2} - y + 6 = 4\sqrt {x + 1} + 2\sqrt {y - 1} \end{array} \right.(x,y \in \mathbb{R}).$$

Call a polynomial $f$ with positive integer coefficients \textit{triangle-compatible} if any three coefficients of $f$ satisfy the triangle inequality. For instance, $3x^3 + 4x^2 + 6x + 5$ is triangle-compatible, but $3x^3 + 3x^2 + 6x + 5$ is not. Given that $f$ is a degree $20$ triangle-compatible polynomial with $-20$ as a root, what is the least possible value of $f(1)$? [i] Note: this problem is also Premier #3 [/i]

Two circles $\Gamma_1$ and $\Gamma_2$are given with centres $O_1$ and $O_2$ and common exterior tangents $\ell_1$ and $\ell_2$. The line $\ell_1$ intersects $\Gamma_1$ in $A$ and $\Gamma_2$ in $B$. Let $X$ be a point on segment $O_1O_2$, not lying on $\Gamma_1$ or $\Gamma_2$. The segment $AX$ intersects $\Gamma_1$ in $Y \ne A$ and the segment $BX$ intersects $\Gamma_2$ in $Z \ne B$. Prove that the line through $Y$ tangent to $\Gamma_1$ and the line through $Z$ tangent to $\Gamma_2$ intersect each other on $\ell_2$.

A acute triangle $\triangle ABC$ has circumcenter $O$. The circumcircle of $OAB$, called $O_1$, and the circumcircle of $OAC$, called $O_2$, meets $BC$ again at $D ( \not=B )$ and $E ( \not= C )$ respectively. The perpendicular bisector of $BC$ hits $AC$ again at $F$. Prove that the circumcenter of $\triangle ADE$ lies on $AC$ if and only if the centers of $O_1, O_2$ and $F$ are colinear.

For how many numbers $n$ ranging from $1$ to $10$, inclusive, is $5n + 1$ a prime number?

Hui is an avid reader. She bought a copy of the best seller [i]Math is Beautiful[/i]. On the first day, she read $1/5$ of the pages plus $12$ more, and on the second day she read $1/4$ of the remaining pages plus $15$ more. On the third day she read $1/3$ of the remaining pages plus $18$ more. She then realizes she has $62$ pages left, which she finishes the next day. How many pages are in this book? $ \textbf{(A)}\ 120 \qquad\textbf{(B)}\ 180\qquad\textbf{(C)}\ 240\qquad\textbf{(D)}\ 300\qquad\textbf{(E)}\ 360 $

[color=darkred]Let $ m$ and $ n$ be two nonzero natural numbers. In every cell $ 1 \times 1$ of the rectangular table $ 2m \times 2n$ are put signs $ \plus{}$ or $ \minus{}$. We call [i]cross[/i] an union of all cells which are situated in a line and in a column of the table. Cell, which is situated at the intersection of these line and column is called [i]center of the cross[/i]. A transformation is defined in the following way: firstly we mark all points with the sign $ \minus{}$. Then consecutively, for every marked cell we change the signs in the cross, whose center is the choosen cell. We call a table [i]accesible[/i] if it can be obtained from another table after one transformation. Find the number of all [i]accesible[/i] tables.[/color]

How many pairs $(a,b)$ of non-zero real numbers satisfy the equation \[ \frac{1}{a} + \frac{1}{b} = \frac{1}{a+b}? \] $\text{(A)} \ \text{none} \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ \text{one pair for each} ~b \neq 0$ $\text{(E)} \ \text{two pairs for each} ~b \neq 0$

Let $k$ be a circle with center $M.$ There is another circle $k_1$ whose center $M_1$ lies on $k,$ and we denote the line through $M$ and $M_1$ by $g.$ Let $T$ be a point on $k_1$ and inside $k.$ The tangent $t$ to $k_1$ at $T$ intersects $k$ in two points $A$ and $B.$ Denote the tangents (diifferent from $t$) to $k_1$ passing through $A$ and $B$ by $a$ and $b$, respectively. Prove that the lines $a,b,$ and $g$ are either concurrent or parallel.

Let $ABC$ be an isosceles triangle with $AB = AC$, $P$ be the midpoint of the minor arc $AB$ of its circumcircle, and $Q$ be the midpoint of $AC$. A circumcircle of triangle $APQ$ centered at $O$ meets $AB$ for the second time at point $K$. Prove that lines $PO$ and $KQ$ meet on the bisector of angle $ABC$.

Two circles $S_1$ and $S_2$ touch externally at $F$. their external common tangent touches $S_1$ at $A$ and $S_2$ at $B$. A line, parallel to $AB$ and tangent to $S_2$ at $C$, intersects $S_1$ at $D$ and $E$. Prove that points $A,F,C$ are collinear. (A. Kalinin)

Triangle $ABC$ has sides $a = |BC| > b = |AC|$. The points $K$ and $H$ on the segment $BC$ satisfy $|CH| = (a + b)/3$ and $|CK| = (a - b)/3$. If $G$ is the centroid of triangle $ABC$, prove that $\angle KGH = 90^o$.

Let $p$ be a prime number. Prove that from a $p^2\times p^2$ array of squares, we can select $p^3$ of the squares such that the centers of any four of the selected squares are not the vertices of a rectangle with sides parallel to the edges of the array.

Let $\{x_n\}$ be a sequence of natural numbers such that \[(a) 1 = x_1 < x_2 < x_3 < \ldots; \quad (b) x_{2n+1} \leq 2n \quad \forall n.\] Prove that, for every natural number $k$, there exist terms $x_r$ and $x_s$ such that $x_r - x_s = k.$

Let $A$ be a set of positive integers containing the number $1$ and at least one more element. Given that for any two different elements $m, n$ of A the number $ \frac{m+1 }{(m+1,n+1) }$ is also an element of $A$, prove that $A$ coincides with the set of positive integers.

Prove that if the angles $\alpha$ and $\beta$ satisfy $\sin(\alpha + \beta) = 2 \sin \alpha$, Then $$\alpha < \beta$$

You are given a checkered square, the side of which is $n – 1$ long and contains $n \geq 10$ nodes. A non-return path is a path along edges, the intersection of which with any horizontal or vertical line is a segment, point or empty set, and which does not pass along any edge more than once. What is the smallest number of non-return paths that can cover all the edges? (An edge is a unit segment between adjacent nodes.)

Find all natural numbers which can be written in the form $\frac{(a+b+c)^2}{abc}$ , where $a,b,c \in N$.

In the game of Yahtzee , players have to achieve various combinations of values with $5$ dice. In a round, a player can roll the dice three times. At the second and third rolls, he can choose which dice to re-roll and which to keep. What is the probability that a player achieves at least four $6$’s in a round, given that he plays with the optimal strategy to maximise this probability? Writing the answer as $p/q$ where $p$ and $q$ are coprime, you should submit the sum of all prime factors of $p$, counted with multiplicity. So for example if you obtained $\frac{p}{q} = \frac{3^4 \cdot 11}{ 2^5 \cdot 5}$ then the submitted answer should be $4 \cdot 3 + 11 = 23$.

Given triangle $ABC$ and its incircle $\omega$ prove you can use just a ruler and drawing at most 8 lines to construct points$A',B',C'$ on $\omega$ such that $A,B',C'$ and $B,C',A'$ and $C,A',B'$ are collinear.

For a 3-digit-number $n=\overline{abc}$, if $a,b,c$ can be three sides of an isosceles triangle (regular triangle included), then the number of such numbers is $\text{(A)}45\qquad\text{(B)}81\qquad\text{(C)}165\qquad\text{(D)}216$

Let $X = \{(x,y) \in \mathbb{R}^2 | y \geq 0, x^2+y^2 = 1\} \cup \{(x,0),-1\leq x\leq 1\} $ be the edge of the closed semicircle with radius 1. a) Let $n>1$ be an integer and $P_1,P_2,\dots,P_n \in X$. Show that there exists a permutation $\sigma \colon \{1,2,\dots,n\}\to \{1,2,\dots,n\}$ such that \[\sum_{j=1}^{n}|P_{\sigma(j+1)}-P_{\sigma(j)}|^2\leq 8\]. Where $\sigma(n+1) = \sigma(1)$. b) Find all sets $\{P_1,P_2,\dots,P_n \} \subset X$ such that for any permutation $\sigma \colon \{1,2,\dots,n\}\to \{1,2,\dots,n\}$, \[\sum_{j=1}^{n}|P_{\sigma(j+1)}-P_{\sigma(j)}|^2 \geq 8\]. Where $\sigma(n+1) = \sigma(1)$.

Let $ABCDE$ be a convex pentagon such that: $\angle ABC=90,\angle BCD=135,\angle DEA=60$ and $AB=BC=CD=DE$. Find angle $\angle DAE$.

In trapezoid $ABCD$, diagonal $AC$ is the bisector of angle $A$. Point $K$ is the midpoint of diagonal $AC$. It is known that $DC = DK$. Find the ratio of the bases $AD: BC$.