The princess wishes to have a bracelet with $r$ rubies and $s$ emeralds arranged in such order that there exist two jewels on the bracelet such that starting with these and enumerating the jewels in the same direction she would obtain identical sequences of jewels. Prove that it is possible to fulfill the princess’s wish if and only if $r$ and $s$ have a common divisor.

Determine the smallest real number $C$ such that the inequality $$\frac x{\sqrt{yz}}\cdot\frac1{x+1}+\frac y{\sqrt{zx}}\cdot\frac1{y+1}+\frac z{\sqrt{xy}}\cdot\frac1{x+1}\le C$$holds for all positive real numbers $x,y$ and $z$ with $\frac1{x+1}+\frac1{y+1}+\frac1{z+1}=1$.

Let there be $2n$ positive reals $a_1,a_2,...,a_{2n}$. Let $s = a_1 + a_3 +...+ a_{2n-1}$, $t = a_2 + a_4 + ... + a_{2n}$, and $x_k = a_k + a_{k+1} + ... + a_{k+n-1}$ (indices are taken modulo $2n$). Prove that $$\frac{s}{x_1}+\frac{t}{x_2}+\frac{s}{x_3}+\frac{t}{x_4}+...+\frac{s}{x_{2n-1}}+\frac{t}{x_{2n}}>\frac{2n^2}{n+1}$$

Let $ABCD$ be a convex quadrilateral with positive integer side lengths, $\angle{A} = \angle{B} = 120^{\circ}, |AD - BC| = 42,$ and $CD = 98$. Find the maximum possible value of $AB$.

The formula expressing the relationship between $ x$ and $ y$ in the table is: \[ \begin{tabular}{|c|c|c|c|c|c|} \hline x & 2 & 3 & 4 & 5 & 6 \\ \hline y & 0 & 2 & 6 & 12 & 20 \\ \hline \end{tabular} \] $ \textbf{(A)}\ y \equal{} 2x \minus{} 4 \qquad\textbf{(B)}\ y \equal{} x^2 \minus{} 3x \plus{} 2 \qquad\textbf{(C)}\ y \equal{} x^3 \minus{} 3x^2 \plus{} 2x$ $ \textbf{(D)}\ y \equal{} x^2 \minus{} 4x \qquad\textbf{(E)}\ y \equal{} x^2 \minus{} 4$

Let $ABC$ be a triangle such that $AB = 7$, $BC = 8$, and $CA = 9$. There exists a unique point $X$ such that $XB = XC$ and $XA$ is tangent to the circumcircle of $ABC$. If $XA = \tfrac ab$, where $a$ and $b$ are coprime positive integers, find $a + b$. [i]Proposed by Alexander Wang[/i]

Alan, Barb, Cory, and Doug are on the golf team, Doug, Emma, Fran, and Greg are on the swim team, and Greg, Hope, Inga, and Alan are on the tennis team. These nine people sit in a circle in random order. The probability that no two people from the same team sit next to each other is $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

For each positive integer $\,n,\;S(n)\,$ is defined to be the greatest integer such that, for every positive integer $\,k\leq S(n),\;n^{2}\,$ can be written as the sum of $\,k\,$ positive squares. [b]a.)[/b] Prove that $\,S(n)\leq n^{2}-14\,$ for each $\,n\geq 4$. [b]b.)[/b] Find an integer $\,n\,$ such that $\,S(n)=n^{2}-14$. [b]c.)[/b] Prove that there are infintely many integers $\,n\,$ such that $S(n)=n^{2}-14.$

Does there exist a convex polyhedron in which each internal angle of each of its faces is either a right angle or an obtuse angle, and which has exactly $100$ edges? Justify your answer.

We say a finite set $S$ of points with $|S|\ge3$ is [i]good[/i] if for any three distinct elements of $S$, they are non-collinear and the orthocenter of them is also in $S$. Find all good sets.

a)Find a matrix $A\in \mathcal{M}_3(\mathbb{C})$ such that $A^2\neq O_3$ and $A^3=O_3$. b)Let $n,p\in\{2,3\}$. Prove that if there is bijective function $f:\mathcal{M}_n(\mathbb{C})\rightarrow \mathcal{M}_p(\mathbb{C})$ such that $f(XY)=f(X)f(Y),\ \forall X,Y\in \mathcal{M}_n(\mathbb{C})$, then $n=p$. [i]Ion Savu[/i]

Let $ABC$ be a triangle with $AB < AC$. As shown below, $T$ is the point on $\overline{BC}$ such that $\overline{AT}$ is tangent to the circumcircle of $\triangle{}ABC$. Additionally, $H$ and $O$ are the orthocenter and circumcenter of $\triangle{}ABC$, respectively. Suppose that $\overline{CH}$ passes through the midpoint of $\overline{AT}$. Prove that $\overline{AO}$ bisects $\overline{CH}$. [asy] size(8cm); pair A = dir(132.5); pair B = dir(200); pair C = dir(340); draw(A--B--C--cycle, black); draw(circumcircle(A, B, C), black); pair O = circumcenter(A, B, C); pair U = 2*C*A/(C+A); pair V = 2*A*B/(A+B);pair T = extension(U, V, B, C); draw(A--T); draw(T--B);pair X = (T+A)/2; pair H = (A+B+C); draw(A--H); pair Y = (H+C)/2; draw(H--X, dashed); draw(C--H); draw(O--Y, dashed); draw(A--O); dot("$A$", A, dir(A)); dot("$B$", B, SW); dot("$C$", C, dir(C)); dot("$O$", O, NE); dot("$T$", T, dir(T)); dot("$H$", H, SW); dot("$X$", X, NW); dot("$Y$", Y, SW); [/asy]

Given an acute triangle $ PBC$ with $ PB\neq PC.$ Points $ A,D$ lie on $ PB,PC,$ respectively. $ AC$ intersects $ BD$ at point $ O.$ Let $ E,F$ be the feet of perpendiculars from $ O$ to $ AB,CD,$ respectively. Denote by $ M,N$ the midpoints of $ BC,AD.$ $ (1)$: If four points $ A,B,C,D$ lie on one circle, then $ EM\cdot FN \equal{} EN\cdot FM.$ $ (2)$: Determine whether the converse of $ (1)$ is true or not, justify your answer.

A sphere $S$ is tangent to the edges $AB,BC,CD,DA$ of a tetrahedron $ABCD$ at the points $E,F,G,H$ respectively. The points $E,F,G,H$ are the vertices of a square. Prove that if the sphere is tangent to the edge $AC$, then it is also tangent to the edge $BD.$

Let $ A\equal{}\{1,2,\ldots, 2008\}$. We will say that set $ X$ is an $ r$-set if $ \emptyset \neq X \subset A$, and $ \sum_{x\in X} x \equiv r \pmod 3$. Let $ X_r$, $ r\in\{0,1,2\}$ be the set of $ r$-sets. Find which one of $ X_r$ has the most elements.

The twelve-sided figure shown has been drawn on $1 \text{ cm}\times 1 \text{ cm}$ graph paper. What is the area of the figure in $\text{cm}^2$? [asy] unitsize(8mm); for (int i=0; i<7; ++i) { draw((i,0)--(i,7),gray); draw((0,i+1)--(7,i+1),gray); } draw((1,3)--(2,4)--(2,5)--(3,6)--(4,5)--(5,5)--(6,4)--(5,3)--(5,2)--(4,1)--(3,2)--(2,2)--cycle,black+2bp); [/asy] $\textbf{(A) } 12 \qquad \textbf{(B) } 12.5 \qquad \textbf{(C) } 13 \qquad \textbf{(D) } 13.5 \qquad \textbf{(E) } 14$

Let $\mathcal{C}$ be a circle in the $xy$ plane with radius $1$ and center $(0, 0, 0)$, and let $P$ be a point in space with coordinates $(3, 4, 8)$. Find the largest possible radius of a sphere that is contained entirely in the slanted cone with base $\mathcal{C}$ and vertex $P$.

Solve the equation $7^x -3^y =4$ in positive integers.

Let $x, y, z$ be real numbers greater than $1$ such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2$. Prove that \[\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}\leq \sqrt{x+y+z}.\]

Show that we cannot tile a $10 x 10$ board with $25$ pieces of type $A$, or with $25$ pieces of type $B$, or with $25$ pieces of type $C$.

The function $\psi : \mathbb{N}\rightarrow\mathbb{N}$ is defined by $\psi (n)=\sum_{k=1}^n\gcd (k,n)$. $(a)$ Prove that $\psi (mn)=\psi (m)\psi (n)$ for every two coprime $m,n \in \mathbb{N}$. $(b)$ Prove that for each $a\in\mathbb{N}$ the equation $\psi (x)=ax$ has a solution.

Find integer $ n$ with $ 8001 < n < 8200$ such that $ 2^n \minus{} 1$ divides $ 2^{k(n \minus{} 1)! \plus{} k^n} \minus{} 1$ for all integers $ k > n$.

The incircle of a right-angled triangle $ABC$ touches its catheti $AC$ and $BC$ at points $B_1$ and $A_1$, the hypotenuse touches the incircle at point $C_1$. Lines $C_1A_1$ and $C_1B_1$ meet $CA$ and $CB$ respectively at points $B_0$ and $A_0$. Prove that $AB_0 = BA_0$. (J. Zajtseva, D. Shvetsov )

Find all function $ f: \mathbb{R} \rightarrow \mathbb{R}$ such that \[ f(x \plus{} y)(f(x) \minus{} y) \equal{} xf(x) \minus{} yf(y) \] for all $ x,y \in \mathbb{R}$.

Prove that the inequality \[ \frac{ x+y}{x^2-xy+y^2 } \leq \frac{ 2\sqrt 2 }{\sqrt{ x^2 +y^2 } } \] holds for all real numbers $x$ and $y$, not both equal to 0.