Circle $S$ with center $O$ and circle $S'$ intersect at points $A$ and $B$. Point $C$ is taken on the arc of a circle $S$ lying inside $S'$. Denote the intersection points of $AC$ and $BC$ with $S'$, other than $A$ and $B$, as $E$ and $D$, respectively. Prove that lines $DE$ and $OC$ are perpendicular.

Given a cube. On each edge of the cube, we write a number, either $1$ or $-1$. For each face of the cube, we multiply the four numbers on the edges of this face, and write the product on this face. Finally, we add all the eighteen numbers that we wrote down on the edges and face of the cube. What is the smallest possible sum that we can get?

Arranged in a circle are $2010$ digits, each of them equal to $1$, $2$, or $3$. For each positive integer $k$, it's known that in any block of $3k$ consecutive digits, each of the digits appears at most $k+10$ times. Prove that there is a block of several consecutive digits with the same number of $1$s, $2$s, and $3$s.

The segments connecting the interior point of a convex non-sided $n$-gon with its vertices divide the $n$-gon into $n$ congruent triangles. For what is the smallest $n$ that is possible?

Let $\{a_i\}$ be a strictly increasing sequence of positive integers. Define the sequence $\{s_k\}$ as $$s_k = \sum_{i=1}^{k}\frac{1}{[a_i,a_{i+1}]},$$ where $[a_i,a_{i+1}]$ is the least commun multiple of $a_i$ and $a_{i+1}$. Show that the sequence $\{s_k\}$ is convergent.

Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[ f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$ [i]Proposed by Pierre Bornsztein, France[/i]

Let $\Lambda= \{1, 2, \ldots, 2v-1,2v\}$ and $P=\{\alpha_1, \alpha_2, \ldots, \alpha_{2v-1}, \alpha_{2v}\}$ be a permutation of the elements of $\Lambda$. (a) Prove that $$\sum_{i=1}^v \alpha_{2i-1}\alpha_{2i} \leq \sum_{i=1}^v (2i-1)2i.$$ (b) Determine the largest positive integer $m$ such that we can partition the $m\times m$ square into $7$ rectangles for which every pair of them has no common interior points and their lengths and widths form the following sequence: $$1,2,3,4,5,6,7,8,9,10,11,12,13,14.$$

The first $32$ perfect squares, $1$, $4$, $9$, $16$, $25$, $\ldots$, $961$, $1024$ are combined together into one large number by appending their digits in succession, forming the number $N = 1491625\ldots9611024$. How many digits does $N$ have? $\textbf{(A) }84\qquad\textbf{(B) }85\qquad\textbf{(C) }86\qquad\textbf{(D) }87\qquad\textbf{(E) }88$

The equation $$(x-1)(x-2)\cdots(x-2016)=(x-1)(x-2)\cdots (x-2016)$$ is written on the board, with $2016$ linear factors on each side. What is the least possible value of $k$ for which it is possible to erase exactly $k$ of these $4032$ linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?

David and Evan are playing a game. Evan thinks of a positive integer $N$ between 1 and 59, inclusive, and David tries to guess it. Each time David makes a guess, Evan will tell him whether the guess is greater than, equal to, or less than $N$. David wants to devise a strategy that will guarantee that he knows $N$ in five guesses. In David's strategy, each guess will be determined only by Evan's responses to any previous guesses (the first guess will always be the same), and David will only guess a number which satisfies each of Evan's responses. How many such strategies are there? Note: David need not guess $N$ within his five guesses; he just needs to know what $N$ is after five guesses.

A $2.00 \text{L}$ balloon at $20.0^{\circ} \text{C}$ and $745 \text{mmHg}$ floats to an altitude where the temperature is $10.0^{\circ} \text{C}$ and the air pressure is $700 \text{mmHg}$. What is the new volume of the balloon? $ \textbf{(A)}\hspace{.05in}0.94 \text{L}\qquad\textbf{(B)}\hspace{.05in}1.06 \text{L}\qquad\textbf{(C)}\hspace{.05in}2.06 \text{L}\qquad\textbf{(D)}\hspace{.05in}2.20 \text{L}\qquad $

Find all pairs $(m, n)$ of positive integers $m$ and $n$ for which one has $$\sqrt{ m^2 - 4} < 2\sqrt{n} - m < \sqrt{ m^2 - 2}$$

Find all pairs of positive integers $(a,b)$ satisfying the following conditions: [list] [*] $a$ divides $b^4+1$, [*] $b$ divides $a^4+1$, [*] $\lfloor\sqrt{a}\rfloor=\lfloor \sqrt{b}\rfloor$. [/list] [i]Yang Liu[/i]

(a) Let $a$ and $b$ be positive integers such that $M(a, b) = a - \frac1b +b(b + \frac3a)$ is an integer. Prove that $M(a,b)$ is a square. (b) Find nonzero integers $a$ and $b$ such that $M(a,b)$ is a positive integer, but not a square.

Let $S=\{1,2,3,\cdots,100\}$. Find the maximum value of integer $k$, such that there exist $k$ different nonempty subsets of $S$ satisfying the condition: for any two of the $k$ subsets, if their intersection is nonemply, then the minimal element of their intersection is not equal to the maximal element of either of the two subsets.

Find the smallest real number $M$ such that $$\sum_{k = 1}^{99}\frac{a_{k+1}}{a_k+a_{k+1}+a_{k+2}} < M$$ for all positive real numbers $a_1, a_2, \dots, a_{99}$. ($a_{100} = a_1, a_{101} = a_2$)

Find all functions $f:\mathbb Z\to \mathbb Z$ such that for all surjective functions $g:\mathbb Z\to \mathbb Z$, $f+g$ is also surjective. (A function $g$ is surjective over $\mathbb Z$ if for all integers $y$, there exists an integer $x$ such that $g(x)=y$.) [i]Proposed by Sean Li[/i]

Which of the followings is false for the sequence $9,99,999,\dots$? $\textbf{(A)}$ The primes which do not divide any term of the sequence are finite. $\textbf{(B)}$ Infinitely many primes divide infinitely many terms of the sequence. $\textbf{(C)}$ For every positive integer $n$, there is a term which is divisible by at least $n$ distinct prime numbers. $\textbf{(D)}$ There is an inteter $n$ such that every prime number greater than $n$ divides infinitely many terms of the sequence. $\textbf{(E)}$ None of above

Given a set $S$ that consists of $n \geq 3$ positive integers. It is known that if for some (not necessarily distinct) numbers $a,b,c,d$ from $S$ the equality $a-b=2(c-d)$ holds, then $a=b$ and $c=d$. Let $M$ be the biggest element in $S$. a) Prove that $M > \frac{n^2}{3}$. b) For $n=1024$ find the biggest possible value of $M$. [i]M. Zorka, Y. Sheshukou[/i]

Given an acute-angled triangle $ABC$ in which $|AB| <|AC|$. Point $D$ is the midpoint of the shorter arc $BC$ of its circumcircle. The point $I$ is the center of its incircle, and the point $J$ is symmetric point of $I$ wrt line $BC$. The line $DJ$ intersects the circumcircle of the triangle $ABC$ at the point $E$ belonging to the arc $AB$. Prove that $|AI |= |IE|$.

Let be the set $ \Lambda = \left\{ \lambda\in\text{Hol} \left[ \mathbb{C}\longrightarrow\mathbb{C} \right] |z\in\mathbb{C}\implies |\lambda (z)|\le e^{|\text{Im}(z)|} \right\} . $ Show that: $ \text{(1)}\sin\in\Lambda $ $ \text{(2)}\sum_{p\in\mathbb{Z}}\frac{1}{(1+2p)^2} =\frac{\pi^2}{4} $ $ \text{(3)} f\in\Lambda\implies \left| f'(0) \right|\le 1 $

Find the largest number among the following numbers: $ \textbf{(A) }\tan47^{\circ}+\cos47^{\circ}\qquad\textbf{(B) }\cot 47^{\circ}+\sqrt{2}\sin 47^{\circ}\qquad\textbf{(C) }\sqrt{2}\cos47^{\circ}+\sin47^{\circ}\qquad\textbf{(D) }\tan47^{\circ}+\cot47^{\circ}\qquad\textbf{(E) }\cos47^{\circ}+\sqrt{2}\sin47^{\circ} $

let $m,n\in \mathbb{N}$ and $p(x),q(x),h(x)$ are polynomials with real Coefficients such that $p(x)$ is Descending. and for all $x\in \mathbb{R}$ $p(q(nx+m)+h(x))=n(q(p(x))+h(x))+m$ . prove that dont exist function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x\in \mathbb{R}$ $f(q(p(x))+h(x))=f(x)^{2}+1$

In triangle $ABC$, the median from vertex $A$ is perpendicular to the median from vertex $B$. If the lengths of sides $AC$ and $BC$ are $6$ and $7$ respectively, then the length of side $AB$ is $\textbf{(A) }\sqrt{17}\qquad\textbf{(B) }4\qquad\textbf{(C) }4\dfrac{1}{2}\qquad\textbf{(D) }2\sqrt{5}\qquad \textbf{(E) }4\dfrac{1}{4}$

Ivan has a $n \times n$ board. He colors some of the squares black such that every black square has exactly two neighbouring square that are also black. Let $d_n$ be the maximum number of black squares possible, prove that there exist some real constants $a$, $b$, $c\ge 0$ such that; $$an^2-bn\le d_n\le an^2+cn.$$ [i]Proposed by Ivan Chan Kai Chin[/i]