Prove that for every $x$ such that $\sin x \ne 0$, there is such natural $n$, which $$ | \sin nx| \ge \frac{\sqrt3}{2}.$$

Compute the sum of all real numbers $x$ such that \[2x^6-3x^5+3x^4+x^3-3x^2+3x-1=0.\]

Point $D$ is an arbitary point on side $BC$ of triangle $ABC$. $I$,$I_1$ and$I_2$ are the incenters of triangles $ABC$,$ABD$ and $ACD$ respectively. $M\not=A$ and $N\not=A$ are the intersections of circumcircle of triangle $ABC$ and circumcircles of triangles $IAI_1$ and $IAI_2$ respectively. Prove that regardless of point $D$, line $MN$ goes through a fixed point.

There are two piles with $72$ and $30$ candies. Two students alternate taking candies from one of the piles. Each time the number of candies taken from a pile must be a multiple of the number of candies in the other pile. Which student can always assure taking the last candy from one of the piles?

What is the remainder when $2^{2013}$ is divided by $5$? $\text{(A) }0\qquad\text{(B) }1\qquad\text{(C) }2\qquad\text{(D) }3\qquad\text{(E) }4$

A circle of radius $5$ is inscribed in a rectangle as shown. The ratio of the the length of the rectangle to its width is $2\ :\ 1$. What is the area of the rectangle? [asy] draw((0,0)--(0,10)--(20,10)--(20,0)--cycle); draw(circle((10,5),5)); [/asy] $ \textbf{(A)}\ 50\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 125\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 200 $

We have $ n \geq 2$ lamps $ L_{1}, . . . ,L_{n}$ in a row, each of them being either on or off. Every second we simultaneously modify the state of each lamp as follows: if the lamp $ L_{i}$ and its neighbours (only one neighbour for $ i \equal{} 1$ or $ i \equal{} n$, two neighbours for other $ i$) are in the same state, then $ L_{i}$ is switched off; – otherwise, $ L_{i}$ is switched on. Initially all the lamps are off except the leftmost one which is on. $ (a)$ Prove that there are infinitely many integers $ n$ for which all the lamps will eventually be off. $ (b)$ Prove that there are infinitely many integers $ n$ for which the lamps will never be all off.

Find all the real number sequences $\{b_n\}_{n \geq 1}$ and $\{c_n\}_{n \geq 1}$ that satisfy the following conditions: (i) For any positive integer $n$, $b_n \leq c_n$; (ii) For any positive integer $n$, $b_{n+1}$ and $c_{n+1}$ is the two roots of the equation $x^2+b_nx+c_n=0$.

What is the value of $\frac{(1+\frac{1}{3})(1+\frac{1}{5})(1+\frac{1}{7})}{\sqrt{(1-\frac{1}{3^2})(1-\frac{1}{5^2})(1-\frac{1}{7^2})}}?$ $\textbf{(A) }\sqrt{3} \qquad \textbf{(B) }2 \qquad \textbf{(C) }\sqrt{15} \qquad \textbf{(D) }4 \qquad \textbf{(E) }\sqrt{105}$

A $6n$-digit number is divisible by $7$. Prove that if its last digit is moved to the beginning of the number then the new number is also divisible by $7$.

The unknown real numbers $x_1,x_2,\dots,x_n$ satisfy $x_1 < x_2 < \cdots < x_n,$ where $n \geq 3$. The numbers $s$, $t$ and $d_1,d_2,\dots,d_{n - 2}$ are given, such that \[ \begin{aligned} s & = \sum\limits_{i = 1}^nx_i, \\ t & = \sum\limits_{i = 1}^nx_i^2,\\ d_i & = x_{i + 2} - x_i,\ \ i = 1,2,\dots,n - 2. \end{aligned} \] For which $n$ is this information always sufficient to determine $x_1,x_2,\dots,x_n$ uniquely?

Determine the number of triples $0 \le k,m,n \le 100$ of integers such that \[ 2^mn - 2^nm = 2^k. \]

The diagonals of a cyclic quadrilateral $ABCD$ intersect at $P$, and there exist a circle $\Gamma$ tangent to the extensions of $AB,BC,AD,DC$ at $X,Y,Z,T$ respectively. Circle $\Omega$ passes through points $A,B$, and is externally tangent to circle $\Gamma$ at $S$. Prove that $SP\perp ST$.

A rectangular garden 50 feet long and 10 feet wide is enclosed by a fence. To make the garden larger, while using the same fence, its shape is changed to a square. By how many square feet does this enlarge the garden? $ \text{(A)}\ 100\qquad\text{(B)}\ 200\qquad\text{(C)}\ 300\qquad\text{(D)}\ 400\qquad\text{(E)}\ 500 $

Call the ordered pair of distinct circles $(\omega, \gamma)$ scribable if there exists a triangle with circumcircle $\omega$ and incircle $\gamma$. Prove that among $n$ distinct circles there are at most $(n/2)^2$ scribable pairs. [i]Proposed by Daniel Liu

Let $n$ be a positive integer and let $a_1, \ldots, a_{n-1} $ be arbitrary real numbers. Define the sequences $u_0, \ldots, u_n $ and $v_0, \ldots, v_n $ inductively by $u_0 = u_1 = v_0 = v_1 = 1$, and $u_{k+1} = u_k + a_k u_{k-1}$, $v_{k+1} = v_k + a_{n-k} v_{k-1}$ for $k=1, \ldots, n-1.$ Prove that $u_n = v_n.$

Let $p,q \in \mathbb N^{\ast}$, $p,q \geq 2$. We say that a set $X$ has the property $\left( \mathcal S \right)$ if no matter how we choose $p$ subsets $B_i \subset X$, $i = \overline{1,n}$, not necessarily distinct, each with $q$ elements, there is a subset $Y \subset X$ with $p$ elements s.t. the intersection of $Y$ with each of the $B_i$'s has an element at most, $i=\overline{1,p}$. Prove that: (a) if $p=4,q=3$ then any set composed of $9$ elements doesn't have $\left( \mathcal S \right)$; (b) any set $X$ composed of $pq-q$ elements doesn't have the property $\left( \mathcal S \right)$; (c) any set $X$ composed of $pq-q+1$ elements has the property $\left( \mathcal S \right)$. [i]Dan Schwarz[/i]

Find all $a,b \in \mathbb{N}$ such that $$2049^ba^{2048}-2048^ab^{2049}=1.$$ [i]Proposed by fattypiggy123 and 61plus[/i]

The parallelogram $ABCD$ has $AB=a,AD=1,$ $\angle BAD=A$, and the triangle $ABD$ has all angles acute. Prove that circles radius $1$ and center $A,B,C,D$ cover the parallelogram if and only \[a\le\cos A+\sqrt3\sin A.\]

This problem is given by my teacher. :wink: [size=120]Seven skiers numbered 1,2,3,4,5,6,7 set out in turn at the starting point,each one slides the same distance at a constant speed. During this period,everyone just had two "beyond" experience.(going beyond one skier or be went beyond by another skier is called a "beyond" experience). When the race ended,we would decide the rank according to the order that skiers reached the ending. Prove that:there are two different rank at most.[/size]

Let $ABC$ be a triangle with $AB=20$ and $AC=18$. $E$ is on segment $AC$ and $F$ is on segment $AB$ such that $AE=AF=8$. Let $BE$ and $CF$ intersect at $G$. Given that $AEGF$ is cyclic, then $BC=m\sqrt{n}$ for positive integers $m$ and $n$ such that $n$ is not divisible by the square of any prime. Compute $100m+n$. [i] Proposed by James Lin[/i]

3. We have a $n \times n$ board. We color the unit square $(i,j)$ black if $i=j$, red if $i<j$ and green if $i>j$. Let $a_{i,j}$ be the color of the unit square $(i,j)$. In each move we switch two rows and write down the $n$-tuple $(a_{1,1},a_{2,2},\cdots,a_{n,n})$. How many $n$-tuples can we obtain by repeating this process? (note that the order of the numbers are important)

Let $\lambda$ the positive root of the equation $t^2-1998t-1=0$. It is defined the sequence $x_0,x_1,x_2,\ldots,x_n,\ldots$ by $x_0=1,\ x_{n+1}=\lfloor\lambda{x_n}\rfloor\mbox{ for }n=1,2\ldots$ Find the remainder of the division of $x_{1998}$ by $1998$. Note: $\lfloor{x}\rfloor$ is the greatest integer less than or equal to $x$.

Set $M$ contains $n \ge 2$ positive integers. It's known that for any two different $a, b \in M$, $a^2+1$ is divisible by $b$. What is the largest possible value of $n$? [i]Proposed by Oleksiy Masalitin[/i]

For a positive reals $ x_{1},...,x_{n} $ prove inequlity: $ \frac{1}{x_{1}+1}+...+\frac{1}{x_{n}+1}\le \frac{n}{1+\frac{n}{\frac{1}{x_{1}}+...+\frac{1}{x_{n}}}}$