Let $ABC$ is a triangle with $\angle BAC=\frac{\pi}{6}$ and the circumradius equal to 1. If $X$ is a point inside or in its boundary let $m(X)=\min(AX,BX,CX).$ Find all the angles of this triangle if $\max(m(X))=\frac{\sqrt{3}}{3}.$

We know that $R^3 = \{(x_1, x_2, x_3) | x_i \in R, i = 1, 2, 3\}$ is a vector space regarding the laws of composition $(x_1, x_2, x_3) + (y_1, y_2, y_3) = (x_1 + y_1, x_2 + y_2, x_3 + y_3)$, $\lambda (x_1, x_2, x_3) = (\lambda x_1, \lambda x_2, \lambda x_3)$, $\lambda \in R$. We consider the following subset of $R^3$ : $L =\{(x_1, x2, x_3) \in R^3 | x_1 + x_2 + x_3 = 0\}$. a) Prove that $L$ is a vector subspace of $R^3$ . b) In $R^3$ the following relation is defined $\overline{x} R \overline{y} \Leftrightarrow \overline{x} -\overline{y} \in L, \overline{x} , \overline{y} \in R^3$. Prove that it is an equivalence relation. c) Find two vectors of $R^3$ that belong to the same class as the vector $(-1, 3, 2)$.

At a turnament between $n$ persons, everyone playes exactly one time against everyone else, and at one game there is everytime a winner and a looser. Prove that one can arrange the participants in a chain$P_1 \to P_2 \to ... \to P_n$ such that the $i$-th person has won against the $(i+1)$-th person.

Given that $\sqrt{x+2y}-\sqrt{x-2y}=2,$ compute the minimum value of $x+y.$ [i]Proposed by Alex Li[/i]

Let $n>1$ be a positive integer. Each cell of an $n\times n$ table contains an integer. Suppose that the following conditions are satisfied: [list=1] [*] Each number in the table is congruent to $1$ modulo $n$. [*] The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^2$. [/list] Let $R_i$ be the product of the numbers in the $i^{\text{th}}$ row, and $C_j$ be the product of the number in the $j^{\text{th}}$ column. Prove that the sums $R_1+\hdots R_n$ and $C_1+\hdots C_n$ are congruent modulo $n^4$.

Suppose that the plane is tiled with an infinite checkerboard of unit squares. If another unit square is dropped on the plane at random with position and orientation independent of the checkerboard tiling, what is the probability that it does not cover any of the corners of the squares of the checkerboard? [hide=Solution] With probability $1$ the number of corners covered is $0$, $1$, or $2$ for example by the diameter of a square being $\sqrt{2}$ so it suffices to compute the probability that the square covers $2$ corners. This is due to the fact that density implies the mean number of captured corners is $1$. For the lattice with offset angle $\theta \in \left[0,\frac{\pi}{2}\right]$ consider placing a lattice uniformly randomly on to it and in particular say without loss of generality consider the square which covers the horizontal lattice midpoint $\left(\frac{1}{2},0 \right)$. The locus of such midpoint locations so that the square captures the $2$ points $(0,0),(1,0)$, is a rectangle. As capturing horizontally adjacent points does not occur when capturing vertically adjacent points one computes twice that probability as $\frac{4}{\pi} \int_0^{\frac{\pi}{2}} (1-\sin(\theta))(1-\cos(\theta)) d\theta=\boxed{\frac{2(\pi-3)}{\pi}}$ \\ [asy] draw((0,0)--(80,40)); draw((0,0)--(-40,80)); draw((80,40)--(40,120)); draw((-40,80)--(40,120)); draw((80,40)--(-20,40)); draw((-40,80)--(60,80)); draw((32*sqrt(5),16*sqrt(5))--(-8*sqrt(5),16*sqrt(5))); draw((40+8*sqrt(5),120-16*sqrt(5))--(40-32*sqrt(5),120-16*sqrt(5))); draw((12*sqrt(5),16*sqrt(5))--(12*sqrt(5)+2*(40-16*sqrt(5)),16*sqrt(5)+(40-16*sqrt(5)))); draw((12*sqrt(5),16*sqrt(5))--(12*sqrt(5)-(80-16*sqrt(5))/2,16*sqrt(5)+(80-16*sqrt(5)))); draw((40-12*sqrt(5),120-16*sqrt(5))--(40-12*sqrt(5)+(120-16*sqrt(5)-40)/2,120-16*sqrt(5)-(120-16*sqrt(5)-40))); draw((40-12*sqrt(5),120-16*sqrt(5))--(40-12*sqrt(5)-2*(120-16*sqrt(5)-80),120-16*sqrt(5)-(120-16*sqrt(5)-80))); [/asy] [/hide]

Let $n\ge 2$ be a positive integer. Find whether there exist $n$ pairwise nonintersecting nonempty subsets of $\{1, 2, 3, \ldots \}$ such that each positive integer can be expressed in a unique way as a sum of at most $n$ integers, all from different subsets.

A point in the plane with a cartesian coordinate system is called a [i]mixed point[/i] if one of its coordinates is rational and the other one is irrational. Find all polynomials with real coefficients such that their graphs do not contain any mixed point.

assume that A is a finite subset of prime numbers, and a is an positive integer. prove that there are only finitely many positive integers m s.t: prime divisors of a^m-1 are contained in A.

The numbers $1,2,...,64 $ are written on the unit squares of a table $8 \times 8$. The two smallest numbers of every row are marked black and the two smaller numbers of every comlumn are marked white. Prove or disprove that there are at least k numbers on the table that are marked both black and white when: a) $k=3$ b) $k=4$ c) $k=5$ .

Let $z_1$ and $z_2$ be complex numbers. Prove that \[|z_1+z_2|+|z_1-z_2|\leqslant |z_1|+|z_2|+\max\{|z_1|,|z_2|\}.\][i]Vlad Cerbu and Sorin Rădulescu[/i]

Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

Determine all real solutions to the following system of equations: $$ \begin{cases} y = 4x^3 + 12x^2 + 12x + 3\\ x = 4y^3 + 12y^2 + 12y + 3. \end{cases} $$

For any positive integer $n$, define $f(n)$ to be the smallest positive integer that does not divide $n$. For example, $f(1)=2$, $f(6)=4$. Prove that for any positive integer $n$, either $f(f(n))$ or $f(f(f(n)))$ must be equal to $2$.

Let $S$ be the sum of all $x$ such that $1\leq x\leq 99$ and \[\{x^2\}=\{x\}^2.\] Compute $\lfloor S\rfloor$.

What is the sum of all two-digit positive integer $n<50$ for which the sum of the squares of first $n$ positive integers is not a divisor of $(2n)!$ ?

Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. A line $\ell$ intersects the lines $AI$, $BI$, and $CI$ at points $D$, $E$, and $F$, respectively, distinct from the points $A$, $B$, $C$, and $I$. The perpendicular bisectors $x$, $y$, and $z$ of the segments $AD$, $BE$, and $CF$, respectively determine a triangle $\Theta$. Show that the circumcircle of the triangle $\Theta$ is tangent to $\Omega$.

Let $k$ and $n$ be positive integers. Determine the smallest integer $N \ge k$ such that the following holds: If a set of $N$ integers contains a complete residue modulo $k$, then it has a non-empty subset whose sum of elements is divisible by $n$.

Find the number of $6$-tuples $(a_1,a_2, a_3,a_4, a_5,a_6)$ of distinct positive integers satisfying the following two conditions: (a) $a_1 + a_2 + a_3 + a_4 + a_5 + a_6 = 30$ (b) We can write $a_1,a_2, a_3,a_4, a_5,a_6$ on sides of a hexagon such that after a finite number of time choosing a vertex of the hexagon and adding $1$ to the two numbers written on two sides adjacent to the vertex, we obtain a hexagon with equal numbers on its sides. Lê Anh Vinh

In trapezoid $ ABCD$ we have $ \overline{AB}$ parallel to $ \overline{DC}$, $ E$ as the midpoint of $ \overline{BC}$, and $ F$ as the midpoint of $ \overline{DA}$. The area of $ ABEF$ is twice the area of $ FECD$. What is $ AB/DC$? $ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 8$

For a given rational number $r$, find all integers $z$ such that \[2^z + 2 = r^2\mbox{.}\] [i](Swiss Mathematical Olympiad 2011, Final round, problem 7)[/i]

If $a,b,c,d$ are Distinct Real no. such that $a = \sqrt{4+\sqrt{5+a}}$ $b = \sqrt{4-\sqrt{5+b}}$ $c = \sqrt{4+\sqrt{5-c}}$ $d = \sqrt{4-\sqrt{5-d}}$ Then $abcd = $

For every integer $ n \ge 3$ and any given angle $ \alpha$ with $ 0 < \alpha < \pi$, let $ P_n(x) \equal{} x^n \sin\alpha \minus{} x \sin n\alpha \plus{} \sin(n \minus{} 1)\alpha$. (a) Prove that there is a unique polynomial of the form $ f(x) \equal{} x^2 \plus{} ax \plus{} b$ which divides $ P_n(x)$ for every $ n \ge 3$. (b) Prove that there is no polynomial $ g(x) \equal{} x \plus{} c$ which divides $ P_n(x)$ for every $ n \ge 3$.

Let $P$ be a point inside of equilateral $\triangle ABC$ such that $m(\widehat{APB})=150^\circ$, $|AP|=2\sqrt 3$, and $|BP|=2$. Find $|PC|$.

Let $ABCD$ be a trapezoid with $AB \parallel CD$ and $BC \perp CD$. There exists a point $P$ on $BC$ such that $\triangle{PAD}$ is equilateral. If $PB = 20$ and $PC = 23$, the area of $ABCD$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ where $b$ is square-free and $a$ and $c$ are relatively prime. Find $a+b+c$. [i]Proposed by Andy Xu[/i]