Point $ O $ is the center of the circumscribed circle of an acute triangle $ Abc $. A certain circle passes through the points $ B $ and $ C $ and intersects sides $ AB $ and $ AC $ of a triangle. On its arc lying inside the triangle, points $ D $ and $ E $ are chosen so that the segments $ BD $ and $ CE $ pass through the point $ O $. Perpendicular $ DD_1 $ to $ AB $ side and perpendicular $ EE_1 $ to $ AC $ side intersect at $ M $. Prove that the points $ A $, $ M $ and $ O $ lie on the same straight line.

In rectangular coordinate system, define that if and only if both $x$-axis and $y$-axis of a point are integers, we call it integral point. Please color all intengral points in white, red and black, satisfying: (1) Points in every color appear on infinitely many lines that are parallel to $x$-axis. (2) For any white point $A$, red point $B$, black point $C$, we can find another red point $D$, such that $ABCD$ is a parallelogram.

Let $p$ be a prime such that $3|p+1$. Show that $p|a-b$ if and only if $p|a^3-b^3$

[b]p1.[/b] Find all triples of positive integers such that the sum of their reciprocals is equal to one. [b]p2.[/b] Prove that $a(a + 1)(a + 2)(a + 3)$ is divisible by $24$. [b]p3.[/b] There are $20$ very small red chips and some blue ones. Find out whether it is possible to put them on a large circle such that (a) for each chip positioned on the circle the antipodal position is occupied by a chip of different color; (b) there are no two neighboring blue chips. [b]p4.[/b] A $12$ liter container is filled with gasoline. How to split it in two equal parts using two empty $5$ and $8$ liter containers? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the largest prime number $p$ such that when $2012!$ is written in base $p$, it has at least $p$ trailing zeroes. [i]Author: Alex Zhu[/i]

The number of scalene triangles having all sides of integral lengths, and perimeter less than $ 13$ is: $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 18$

A function $f(n)$ is defined on the set of positive integers is said to be multiplicative if $f(mn)=f(m)f(n)$ whenever $m$ and $n$ have no common factors greater than $1$. Are the following functions multiplicative? Justify your answer. (a) $g(n)=5^k$ where $k$ is the number of distinct primes which divide $n$. (b) $h(n)=\begin{cases} 0 & \text{if} \ n \ \text{is divisible by} \ k^2 \ \text{for some integer} \ k>1 \\ 1 & \text{otherwise} \end{cases}$

Let $ABC$ be a triangle with circumcircle $w_1$ and incenter $I$. Suppose $w_2$ is a circle tangent to $AB,AC$ at $X,Y$, and internally tangent to $w$ at $D$. Let the parallel to the exterior angle bisector of $A$ through $D$ meet $w_2$ at $P$. Show that $AP, DI$ intersect on $w_2$.

Let $n$ and $k$ be two integers with $n>k\geqslant 1$. There are $2n+1$ students standing in a circle. Each student $S$ has $2k$ [i]neighbors[/i] - namely, the $k$ students closest to $S$ on the left, and the $k$ students closest to $S$ on the right. Suppose that $n+1$ of the students are girls, and the other $n$ are boys. Prove that there is a girl with at least $k$ girls among her neighbors. [i]Proposed by Gurgen Asatryan, Armenia[/i]

The function $f:\mathbb{N}\rightarrow \mathbb{N}$ is [b]peruvian[/b] if it satifies the following two properties: $\triangleright f$ is strictly increasing. $\triangleright$ The numbers $a_1,a_2,a_3,\dots$ where $a_1=f(1)$ and $a_{n+1}=f(a_n)$ for every $n\geq 1$, are in arithmetic progression. Determine all peruvian functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that $f(1)=3$.

Let $\alpha, \beta, \gamma, \delta$ be the interior angles of a convex quadrilateral, If $$ \cos\alpha + \cos\beta + \cos\gamma, + \cos\delta = 0 , $$ then this quadrilateral is cyclic or a trapezium. Prove it.

Let $p\ge5$ be a prime number. Prove that there exists $a\in\{1,2,\ldots,p-2\}$ satisfying $p^2\nmid a^{p-1}-1$ and $p^2\nmid(a+1)^{p-1}-1$.

Let $\Gamma_1$, $\Gamma_2$, $\Gamma_3$, $\Gamma_4$ be distinct circles such that $\Gamma_1$, $\Gamma_3$ are externally tangent at $P$, and $\Gamma_2$, $\Gamma_4$ are externally tangent at the same point $P$. Suppose that $\Gamma_1$ and $\Gamma_2$; $\Gamma_2$ and $\Gamma_3$; $\Gamma_3$ and $\Gamma_4$; $\Gamma_4$ and $\Gamma_1$ meet at $A$, $B$, $C$, $D$, respectively, and that all these points are different from $P$. Prove that \[ \frac{AB\cdot BC}{AD\cdot DC}=\frac{PB^2}{PD^2}. \]

Let $a,b,c$ be three different positive integers. Show that the sequence \[a+b+c,ab+bc+ca,3abc\] could be neither an arithmetic nor geometric progression. [i]Fajar Yuliawan, Bandung[/i]

Aileen plays badminton where she and her opponent stand on opposite sides of a net and attempt to bat a birdie back and forth over the net. A player wins a point if their opponent fails to bat the birdie over the net. When Aileen is the server (the first player to try to hit the birdie over the net), she wins a point with probability $\frac{9}{10}$ . Each time Aileen successfully bats the birdie over the net, her opponent, independent of all previous hits, returns the birdie with probability $\frac{3}{4}$ . Each time Aileen bats the birdie, independent of all previous hits, she returns the birdie with probability $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

In the plane there are given the lines $\ell_1$, $\ell_2$, the circle $\mathcal{C}$ with its center on the line $\ell_1$ and a second circle $\mathcal{C}_1$ which is tangent to $\ell_1$, $\ell_2$ and $\mathcal{C}$. Find the locus of the tangent point between $\mathcal{C}$ and $\mathcal{C}_1$ while the center of $\mathcal{C}$ is variable on $\ell_1$. [i]Mircea Becheanu[/i]

Call $A \in \mathbb{N}^0$ be $number of year$ if all digits of $A$ equals $0$, $1$ or $2$ (in decimal representation). Prove that exist infinity $N \in \mathbb{N}$, such that $N$ can't presented as $A^2+B$ where $A \in \mathbb{N}^0 ; B$- $number of year$.

In a right-angled triangle let $r$ be the inradius and $s_a,s_b$ be the lengths of the medians of the legs $a,b$. Prove the inequality \[ \frac{r^2}{s_a^2+s_b^2} \leq \frac{3-2 \sqrt2}{5}. \]

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all $x, y\in\mathbb{R}$, we have \[f(x+y)+f(x)f(y)=f(xy)+(y+1)f(x)+(x+1)f(y).\]

A quadratic polynomial $p(x)$ with real coefficients and leading coefficient $1$ is called disrespectful if the equation $p(p(x)) = 0$ is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial $\tilde{p}(x)$ for which the sum of the roots is maximized. What is $\tilde{p}(1)?$ $\textbf{(A) }\dfrac5{16} \qquad \textbf{(B) }\dfrac12 \qquad \textbf{(C) }\dfrac58 \qquad \textbf{(D) }1 \qquad \textbf{(E) }\dfrac98$

Let $E$ be a point inside the rhombus $ABCD$ such that $|AE|=|EB|, m(\widehat{EAB}) = 11^{\circ}$, and $m(\widehat{EBC}) = 71^{\circ}$. Find $m(\widehat{DCE})$. $\textbf{(A)}\ 72^{\circ} \qquad\textbf{(B)}\ 71^{\circ} \qquad\textbf{(C)}\ 70^{\circ} \qquad\textbf{(D)}\ 69^{\circ} \qquad\textbf{(E)}\ 68^{\circ}$

In a country, the time for presidential elections has approached. There are exactly 20 million voters in the country, of which only one percent supports the current president, Miraflores. Naturally, he wants to be elected again, but on the other hand, he wants the elections to seem democratic. Miraflores established the following voting process: all the voters are divided into several equal groups, then each of these groups is again divided into a number of equal groups, and so on. In the smallest groups, a representative is chosen. Then, the chosen electors choose representatives in the second-smallest groups, to vote in an even larger group, and so on. Finally, the representatives of the largest groups choose the president. Miraflores divides voters into groups as he wants and instructs his supporters how to vote. Will he be able to organize the elections in such a way that he will be elected president? (If the votes are equal, the opposition wins.) [i]From the 32nd Moscow Mathematical Olympiad[/i]

Let $x,y$ and $z$ be real numbers satisfying the system \begin{align*} \log_2(xyz-3+\log_5 x) &= 5 \\ \log_3(xyz-3+\log_5 y) &= 4 \\ \log_4(xyz-3+\log_5 z) &= 4. \end{align*} Find the value of $|\log_5 x|+|\log_5 y|+|\log_5 z|$.

Does there exist a piecewise linear continuous function $f:\mathbb{R}\to \mathbb{R}$ such that for any two-way infinite sequence $a_n\in[0,1]$, $n\in\mathbb{Z}$, there exists an $x\in\mathbb{R}$ with \[ \limsup_{K\to \infty} \frac{\#\{k\le K\,:\, k\in\mathbb{N},f^k(x)\in[n,n+1)\}}{K}=a_n \] for all $n\in\mathbb{Z}$, where $f^k=f\circ f\circ \dots\circ f$ stands for the $k$-fold iterate of $f$?

Given the positive numbers $a$ and $b$ and the natural number $n$, find the greatest among the $n + 1$ monomials in the binomial expansion of $\left(a+b\right)^n$.