Let $ f(x)\equal{}a_{2n}x^{2n}\plus{}a_{2n\minus{}1}x^{2n\minus{}1}\plus{}\cdots\plus{}a_1x\plus{}a_0$, with $ a_i\equal{}a_{2n\minus{}1}$ for all $ i\equal{}1,2,\ldots,n$ and $ a_{2n}\ne0$. Prove that there exists a polynomial $ g(x)$ of degree $ n$ such that $ g\left(x\plus{}\frac1x\right)x^n\equal{}f(x)$.

Let $r,g,b$ be non negative integers and $\Gamma$ be a connected graph with $r+g+b+1$ vertices. Its edges are colored in red green and blue. It turned out that $\Gamma $ contains A spanning tree with exactly $r$ red edges. A spanning tree with exactly $g$ green edges. A spanning tree with exactly $b$ blue edges. Prove that $\Gamma$ contains a spanning tree with exactly $r$ red edges, $g$ green edges and $b$ blue edges.

Pikachu, Charmander, and Vulpix are three of the four equally-skilled players in a Pokemon bracket tournament. Because they are equally skilled, whenever any two of the players battle, they are equally likely to win. In the bracket tournament, the four players are randomly paired into two rounds, each round consisting of two players. The winners of the first two rounds then play each other in the final round. The winner of the final match ranks first; the loser of the final round ranks second; and the two losers of the previous rounds jointly rank third. What is the probability that Charmander plays Vulpix in a round, but ranks lower than Pikachu? $\textbf{(A) }\dfrac1{24}\qquad\textbf{(B) }\dfrac18\qquad\textbf{(C) }\dfrac13\qquad\textbf{(D) } \dfrac38 \qquad \textbf{(E) } \dfrac12$

Given the equation $\frac{ax^2-24x+b}{x^2-1} = x$. Find all the real numbers $a$ and $b$ for which you have two real solutions whose sum is equal to $12$.

Find the least positive integer $n$ greater than $1$ such that $n^3 -n^2$ is divisible by $7^2 \times 11$. [i]Proposed by Jacob Xu[/i]

Find the maximum and minimum values of $x^2 + 2y^2 + 3z^2$ for real $x, y, z$ satisfying $x^2 + y^2 + z^2 = 1$.

Let $ \{ a_1 , a_2 , \cdots, a_{10} \} = \{ 1, 2, \cdots , 10 \} $ . Find the maximum value of \[ \sum_{n=1}^{10}(na_n ^2 - n^2 a_n ) \]

Let $\Omega$ be the circumcircle of triangle $ABC$ such that the tangents to $\Omega$ in points $B$ and $C$ intersect in $P$. The squares $ABB_1B_2$ and $ACC_1C_2$ are constructed on the sides $AB$ and $AC$ in the exterior of triangle $ABC$, such that the lines $B_1B_2$ and $C_1C_2$ intersect in point $Q$. Prove that $P$, $A$, and $Q$ are collinear.

A triangle and a square are circumscribed around the unit circle. Prove that the intersection area is more than $3.4$. Is it possible to assert that it is more than $3.5$?

The distance between any two of five given points exceeds 2. Is it true that the distance between some two of these points exceeds 3 if these five points are in a) the plane; and b) three-dimensional space? [i]Alexey Tolpygo[/i]

Find the coordinates of the vertices of a square $ABCD$, knowing that $A$ is on the line $y -2x -6 = 0$, $C$ at $x = 0$ and $B$ is the point $(a, 0)$ , being $a = \log_{2/3}(16/81)$.

A fixed circle $\omega$ is inscribed into an angle with vertex $C$. An arbitrary circle passing through $C$, touches $\omega$ externally and meets the sides of the angle at points $A$ and $B$. Prove that the perimeters of all triangles $ABC$ are equal.

In a class of $14$ boys, each boy was asked how many classmates had the same first name. and how many colleagues had the same last name as them. The numbers $0, 1, 2, 3, 4, 5$ and $6$. Proves that there are two colleagues with the same first name and the same last name

The circle of center $I$ is inscribed in the convex quadrilateral $ABCD$. Let $M$ and $N$ be points on the segments $AI$ and $CI$, respectively, such that $\angle MBN = \frac 12 \angle ABC$. Prove that $\angle MDN = \frac 12 \angle ADC$.

A [i]substring [/i] of a number $n$ is a number formed by removing some digits from the beginning and end of $n$ (possibly a different number of digits is removed from each side). Find the sum of all prime numbers $p$ that have the property that any substring of $p$ is also prime.

A graph is called $7-chip$ if it obtained by removing at most three edges that have no vertex in common from a complete graph with seven vertices. Consider a complete graph $G$ with $v$ vertices which each edge of its is colored blue or red. Prove that there is either a blue path with $100$ edges or a red $7-chip$.

Find two angles which add to $180^{\circ}$ which difference is $1^{'}$

Let $a$ be such an integer, that $3a$ can be written in the form $x^2 + 2y^2$, with integers $x$ and $y$. Prove that the number $a$ can also be written in this form. [b]Additional problems:[/b] [b]a)[/b] Find a general (necessary and sufficent) criterion for an integer $n$ to be of that form. [b]b)[/b] In how many ways can the integer $n$ be represented in that way?

Let $f(x)=\cos(a_1+x)+{1\over2}\cos(a_2+x)+{1\over4}\cos(a_3+x)+\ldots+{1\over2^{n-1}}\cos(a_n+x)$, where $a_i$ are real constants and $x$ is a real variable. If $f(x_1)=f(x_2)=0$, prove that $x_1-x_2$ is a multiple of $\pi$.

Let $ABC$ be an equilateral triangle with side lenght is $1$ $cm$.Let $D \in [AB]$ is a point. Perpendiculars from $D$ to $[AC]$ and $[BC]$ intersects with $[AC]$ and $[BC]$ at points $E$ and $F$ respectively. Perpendiculars from $E$ and $F$ to $[AB]$ intersects with $[AB]$ at points $E_1$ and $F_1$. Prove that $$[E_1F_1]=\frac{3}{4}$$

Given a cuboid $Q$ with dimensions $a, b, c$, $a < b < c$. Find the length of the edge of a cube $K$ , which has parallel faces and a common center with the given cuboid so that the volume of the difference of the sets $Q \cup K$ and $Q \cap K$ is minimal.

Let $ m,n$ be positive integers, and let $ F$ be a family of $ m$-element subsets of $ \{1,2,...,n\}$ satisfying $ A\cap B \not \equal{} \emptyset$ for all $ A,B\in F$. Determine the maximum possible number of elements in $ F$.

Find the number of ways $66$ identical coins can be separated into three nonempty piles so that there are fewer coins in the first pile than in the second pile and fewer coins in the second pile than in the third pile.

The [i]cross[/i] of a convex $n$-gon is the quadratic mean of the lengths between the possible pairs of vertices. For example, the cross of a $3 \times 4$ rectangle is $\sqrt{ \dfrac{3^2 + 3^2 + 4^2 + 4^2 + 5^2 + 5^2}{6} } = \dfrac{5}{3} \sqrt{6}$. Suppose $S$ is a dodecagon ($12$-gon) inscribed in a unit circle. Find the greatest possible cross of $S$.

In a city, there are some buildings. We say the building $A$ is dominant to the building $B$ if the line that connects upside of $A$ to upside of $B$ makes an angle more than $45^{\circ}$ with earth. We want to make a building in a given location. Suppose none of the buildings are dominant to each other. Prove that we can make the building with a height such that again, none of the buildings are dominant to each other. (Suppose the city as a horizontal plain and each building as a perpendicular line to the plain.)