Let $a,b,c$ be the lengths of the sides of a triangle, and $\alpha, \beta, \gamma$ respectively, the angles opposite these sides. Prove that if \[ a+b=\tan{\frac{\gamma}{2}}(a\tan{\alpha}+b\tan{\beta}) \] the triangle is isosceles.

Minneapolis-St. Paul International Airport is $ 8$ miles southwest of downtown St. Paul and $ 10$ miles southeast of downtown Minneapolis. Which of the following is closest to the number of miles between downtown St. Paul and downtown Minneapolis? $ \textbf{(A)}\ 13\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 15\qquad \textbf{(D)}\ 16\qquad \textbf{(E)}\ 17$

For any integer $n\geq 2$, let $N(n)$ be the maxima number of triples $(a_i, b_i, c_i)$, $i=1, \ldots, N(n)$, consisting of nonnegative integers $a_i$, $b_i$ and $c_i$ such that the following two conditions are satisfied: [list][*] $a_i+b_i+c_i=n$ for all $i=1, \ldots, N(n)$, [*] If $i\neq j$ then $a_i\neq a_j$, $b_i\neq b_j$ and $c_i\neq c_j$[/list] Determine $N(n)$ for all $n\geq 2$. [i]Proposed by Dan Schwarz, Romania[/i]

Let $ABC$ be a triangle inscribed in the parabola $y=x^2$ such that the line $AB \parallel$ the axis $Ox$. Also point $C$ is closer to the axis $Ox$ than the line $AB$. Given that the length of the segment $AB$ is 1 shorter than the length of the altitude $CH$ (of the triangle $ABC$). Determine the angle $\angle{ACB}$ .

Let $P=(x^4-40x^2+144)(x^3-16x)$. $a)$ Factor $P$ as a product of irreducible polynomials. $b)$ We write down the values of $P(10)$ and $P(91)$. What is the greatest common divisor of the two numbers?

In a tennis tournament, $n$ players want to make $2$ vs $2$ matches such that each player has each of the other players as opponents exactly once. Find all possible values of $n$.

Find all prime numbers $p$ and $q$ such that $pq-p-q+3$ is a perfect square.

For positive integers $n, k, l$, we define the number of $l$-tuples of positive integers $(a_1,a_2,\cdots a_l)$ satisfying the following as $Q(n,k,l)$. (i): $n=a_1+a_2+\cdots +a_l$ (ii): $a_1>a_2>\cdots > a_l > 0$. (iii): $a_l$ is an odd number. (iv): There are $k$ odd numbers out of $a_i$. For example, from $9=8+1=6+3=6+2+1$, we have $Q(9,1,1)=1$, $Q(9,1,2)=2$, $Q(9,1,3)=1$. Prove that if $n>k^2$, $\sum_{l=1}^n Q(n,k,l)$ is $0$ or an even number.

Show that $\cos 1^{\circ}$ is irrational.

Prove that for arbitary integer $ n > 16$, there exists the set $ S$ that contains $ n$ positive integers and has the following property:if the subset $ A$ of $ S$ satisfies for arbitary $ a,a'\in A, a\neq a', a \plus{} a'\notin S$ holds, then $ |A|\leq4\sqrt n.$

Given $\triangle{ABC}$ with $\angle{B}=60^{\circ}$ and $\angle{C}=30^{\circ}$, let $P,Q,R$ be points on the sides $BA,AC,CB$ respectively such that $BPQR$ is an isosceles trapezium with $PQ \parallel BR$ and $BP=QR$.\\ Find the maximum possible value of $\frac{2[ABC]}{[BPQR]}$ where $[S]$ denotes the area of any polygon $S$.

Three sequences ${a_n},{b_n},{c_n}$ satisfy the following conditions. [list] [*]$a_1=2,\,b_1=4,\,c_1=5$ [*]$\forall n,\; a_{n+1}=b_n+\frac{1}{c_n}, \, b_{n+1}=c_n+\frac{1}{a_n}, \, c_{n+1}=a_n+\frac{1}{b_n}$ [/list] Prove that for all positive integers $n$, $ $ $ $ $max(a_n,b_n,c_n)>\sqrt{2n+13}$.

Let $ b$ be an integer greater than $ 5$. For each positive integer $ n$, consider the number \[ x_n = \underbrace{11\cdots1}_{n \minus{} 1}\underbrace{22\cdots2}_{n}5, \] written in base $ b$. Prove that the following condition holds if and only if $ b \equal{} 10$: [i]there exists a positive integer $ M$ such that for any integer $ n$ greater than $ M$, the number $ x_n$ is a perfect square.[/i] [i]Proposed by Laurentiu Panaitopol, Romania[/i]

There are $n ( \ge 3) $ students in a class. Some students are friends each other, and friendship is always mutual. There are $ s ( \ge 1 ) $ couples of two students who are friends, and $ t ( \ge 1 ) $ triples of three students who are each friends. For two students $ x, y $ define $ d(x,y)$ be the number of students who are both friends with $ x $ and $ y $. Prove that there exist three students $ u, v, w $ who are each friends and satisfying \[ d(u,v) + d(v,w) + d(w,u) \ge \frac{9t}{s} . \]

William was bored at the math lesson, so he drew a circle and $n\ge3$ empty cells around the circumference. In every cell he wrote a positive number. Later on he erased the numbers and in every cell wrote the geometric mean of the numbers previously written in the two neighboring cells. Show that there exists a cell whose number was not replaced by a larger number.

Determine all pairs $(a,b)$ of positive integers for which there exist positive integers $g$ and $N$ such that $$\gcd (a^n+b,b^n+a)=g$$ holds for all integers $n\geqslant N.$ (Note that $\gcd(x, y)$ denotes the greatest common divisor of integers $x$ and $y.$) [i]Proposed by Valentio Iverson, Indonesia[/i]

Assume that it is possible to color more than half of the surfaces of a given polyhedron so that no two colored surfaces have a common edge. (a) Describe one polyhedron with the above property. (b) Prove that one cannot inscribe a sphere touching all the surfaces of a polyhedron with the above property.

Triangles $ ABC$ and $ ADC$ are isosceles with $ AB \equal{} BC$ and $ AD \equal{} DC$. Point $ D$ is inside $ \triangle ABC$, $ \angle ABC \equal{} 40^\circ$, and $ \angle ADC \equal{} 140^\circ$. What is the degree measure of $ \angle BAD$? $ \textbf{(A)}\ 20 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 50 \qquad \textbf{(E)}\ 60$

Let $ABCD$ be a convex quadrilateral with angles $\sphericalangle A, \sphericalangle C\geq90^{\circ}$. On sides $AB,BC,CD$ and $DA$, consider the points $K,L,M$ and $N$ respectively. Prove that the perimeter of $KLMN$ is greater than or equal to $2\cdot AC$.

Call a set of some cells in infinite chess field as board. Set of rooks on the board call as awesome if no one rook can beat another, but every empty cell is under rook attack. There are awesome set with $2008$ rooks and with $2010$ rooks. Prove, that there are awesome set with $2009$ rooks.

Find the lengths of the principal axes of the ellipsoid \[\sum_{i\le j}x_i x_j=1\]

The sequence $ (x_n)$, $ n\in\mathbb{N}^*$ is defined by $ |x_1|<1$, and for all $ n \ge 1$, \[ x_{n\plus{}1} \equal{}\frac{\minus{}x_n \plus{}\sqrt{3\minus{}3x_n^2}}{2}\] (a) Find the necessary and sufficient condition for $ x_1$ so that each $ x_n > 0$. (b) Is this sequence periodic? And why?

[asy]draw((0,0)--(1,0)--(1,1)--(2,1)--(2,2)--(3,2)--(3,3)--(2,3)--(2,4)--(1,4)--(1,5)--(0,5)--(0,4)--(-1,4)--(-1,1)--(0,1)--cycle); draw((0,1)--(1,1)); draw((-1,2)--(2,2)); draw((-1,3)--(2,3)); draw((0,4)--(1,4)); draw((0,1)--(0,4)); draw((1,1)--(1,4)); draw((2,2)--(2,3)); draw((0,1)--(1,1)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(0,4)--cycle); draw((0,1)--(1,1)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(0,4)--cycle); draw((0,1)--(1,1)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(0,4)--cycle); draw((0,1)--(1,1)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(0,4)--cycle); draw((0,1)--(1,1)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(0,4)--cycle); draw((0,1)--(1,1)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(0,4)--cycle); label("H",(0.5,0.2),N); label("G",(1.5,1.2),N); label("F",(-0.5,1.2),N); label("E",(2.5,2.2),N); label("D",(-0.5,2.2),N); label("C",(1.5,3.2),N); label("B",(-0.5,3.2),N); label("A",(0.5,4.2),N);[/asy] Suppose one of the eight lettered identical squares is included with the four squares in the T-shaped figure outlined. How many of the resulting figures can be folded into a topless cubical box? \[ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6 \]

The polynomial $A(x) = x^2 + ax + b$ with integer coefficients has the following property: for each prime $p$ there is an integer $k$ such that $A(k)$ and $A(k + 1)$ are both divisible by $p$. Proof that there is an integer $m$ such that $A(m) = A(m + 1) = 0$.

Mari and Juri ordered a round pizza. Juri cut the pizza into four pieces by two straight cuts, none of which passed through the centre point of the pizza. Mari can choose two pieces not aside of these four, and Juri gets the rest two pieces. Prove that if Mari chooses the piece that covers the centre point of the pizza, she will get more pizza than Juri.