Evaluate \[\int_0^{\frac{\pi}{4}} \frac{x\cos 5x}{\cos x}dx\]

Three flies are crawling along the perimeter of the triangle $ABC$ in such a way, that the centre of their masses is a constant point. One of the flies has already passed along all the perimeter. Prove that the centre of the flies' masses coincides with the centre of masses of the triangle $ABC$ . (The centre of masses for the triangle is the point of medians intersection.

$ 1 \minus{} 2 \plus{} 3 \minus{} 4 \plus{} \cdots \minus{} 98 \plus{} 99 \equal{}$ $ \textbf{(A)}\minus{}\! 50 \qquad \textbf{(B)}\minus{}\! 49 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 49 \qquad \textbf{(E)}\ 50$

Let $ABC$ be a triangle and let $M$ and $N$ denote the midpoints of $\overline{AB}$ and $\overline{AC}$, respectively. Let $X$ be a point such that $\overline{AX}$ is tangent to the circumcircle of triangle $ABC$. Denote by $\omega_B$ the circle through $M$ and $B$ tangent to $\overline{MX}$, and by $\omega_C$ the circle through $N$ and $C$ tangent to $\overline{NX}$. Show that $\omega_B$ and $\omega_C$ intersect on line $BC$. [i]Merlijn Staps[/i]

Let $n$ be a positive integer, set $S_n = \{ (a_1,a_2,\cdots,a_{2^n}) \mid a_i=0 \ \text{or} \ 1, 1 \leq i \leq 2^n\}$. For any two elements $a=(a_1,a_2,\cdots,a_{2^n})$ and $b=(b_1,b_2,\cdots,b_{2^n})$ of $S_n$, define \[ d(a,b)= \sum_{i=1}^{2^n} |a_i - b_i| \] We call $A \subseteq S_n$ a $\textsl{Good Subset}$ if $d(a,b) \geq 2^{n-1}$ holds for any two distinct elements $a$ and $b$ of $A$. How many elements can the $\textsl{Good Subset}$ of $S_n$ at most have?

A point $A_1$ is marked inside an acute non-isosceles triangle $ABC$ such that $\angle A_1AB = \angle A_1BC$ and $\angle A_1AC=\angle A_1CB$. Points $B_1$ and $C_1$ are defined same way. Let $G$ be the gravity center if the triangle $ABC$. Prove that the points $A_1,B_1,C_1,G$ are concyclic.

Find all natural numbers \(n\) for which there exists an even natural number \(a\) such that the number \[ (a - 1)(a^2 - 1)\cdots(a^n - 1) \] is a perfect square.

Let be a circle centeted at $ O, $ and $ A,B,C, $ points situated on this circle. Show that if $$ \left|\overrightarrow{OA} +\overrightarrow{OB}\right| = \left|\overrightarrow{OB} +\overrightarrow{OC}\right| = \left|\overrightarrow{OC} +\overrightarrow{OA}\right| , $$ then $ A=B=C, $ or $ ABC $ is an equilateral triangle.

Five edges of a triangular pyramid are equal to $1$. Find the sixth edge if it is known that the radius of the ball circumscribed about this pyramid is equal to $1$.

$ \frac{1}{10}+\frac{2}{20}+\frac{3}{30}= $ $ \text{(A)}\ .1\qquad\text{(B)}\ .123\qquad\text{(C)}\ .2\qquad\text{(D)}\ .3\qquad\text{(E)}\ .6 $

Does there exist a a sequence $a_{0},a_{1},a_{2},\dots$ in $\mathbb N$, such that for each $i\neq j, (a_{i},a_{j})=1$, and for each $n$, the polynomial $\sum_{i=0}^{n}a_{i}x^{i}$ is irreducible in $\mathbb Z[x]$? [i]By Omid Hatami[/i]

There are $60$ towns in $Graphland$ every two countries of which are connected by only a directed way. Prove that we can color four towns to red and four towns to green such that every way between green and red towns are directed from red to green

Let $p$ and $q$ be prime numbers such that $p^2+pq+q^2$ is perfect square. Prove that $p^2-pq+q^2$ is prime

Chandler the Octopus is at a tentacle party! At this party, there is $1$ creature with $2$ tentacles, $2$ creatures with $3$ tentacles, $3$ creatures with $4$ tentacles, all the way up to $14$ creatures with $15$ tentacles. Each tentacle is distinguishable from all other tentacles. For some $2\le m < n \le 15$, a creature with m tentacles “meets” a creature with n tentacles; “meeting” another creature consists of shaking exactly 1 tentacle with each other. Find the number of ways there are to pick distinct $m < n$ between $2$ and $15$, inclusive, and then to pick a creature with $m$ tentacles to “meet” a selected creature with $n$ tentacles. [i]Proposed by Armaan Tipirneni, Richard Chen, and Denise the Octopus[/i]

In the figure below, we know that $AB = CD$ and $BC = 2AD$. Prove that $\angle BAD = 30^o$. [img]https://3.bp.blogspot.com/-IXi_8jSwzlU/W1R5IydV5uI/AAAAAAAAIzo/2sREnDEnLH8R9zmAZLCkVCGeMaeITX9YwCK4BGAYYCw/s400/IGO%2B2015.el3.png[/img]

For how many primes $ p$, there exits unique integers $ r$ and $ s$ such that for every integer $ x$ $ x^{3} \minus{} x \plus{} 2\equiv \left(x \minus{} r\right)^{2} \left(x \minus{} s\right)\pmod p$? $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None}$

What is the area of the figure in the complex plane enclosed by the origin and the set of all points $\tfrac{1}{z}$ such that $(1-2i)z+(-2i-1)\overline{z}=6i$?

A rectangle is divided into $9$ small rectangles if by parallel lines to its sides, as shown in the figure. [img]https://cdn.artofproblemsolving.com/attachments/e/d/1fd545862a3c7950249ec54a631c74e59fb9ed.png[/img] The four numbers written indicate the areas of the four corresponding rectangles. Prove that the total area of the rectangle is greater than or equal to $90$.

A pair of positive integers $(a,b)$ is called an [b]average couple[/b] if there exist positive integers $k$ and $c_1, \dots, c_k$ for which \[\frac{c_1+c_2+\cdots+c_k}{k}=a\qquad \text{and} \qquad \frac{s(c_1)+s(c_2)+\cdots+s(c_k)}{k}=b\] where $s(n)$ denotes the sum of digits of $n$ in decimal representation. Find the number of average couples $(a,b)$ for which $a,b<10^{10}$.

Two players $A$ and $B$ play the following game. Before the game starts, $A$ chooses 1000 not necessarily different odd primes, and then $B$ chooses half of them and writes them on a blackboard. In each turn a player chooses a positive integer $n$, erases some primes $p_1$, $p_2$, $\dots$, $p_n$ from the blackboard and writes all the prime factors of $p_1 p_2 \dotsm p_n - 2$ instead (if a prime occurs several times in the prime factorization of $p_1 p_2 \dotsm p_n - 2$, it is written as many times as it occurs). Player $A$ starts, and the player whose move leaves the blackboard empty loses the game. Prove that one of the two players has a winning strategy and determine who. Remark: Since 1 has no prime factors, erasing a single 3 is a legal move.

A design $X$ is an array of the digits $1,2,..., 9$ in the shape of an $X$, for example, [img]https://cdn.artofproblemsolving.com/attachments/8/e/d371a2cd442cb7a8784e1cc7635344df722e20.png[/img] We will say that a design $X$ is [i]balanced [/i] if the sum of the numbers of each of the diagonals match. Determine the number of designs $X$ that are balanced.

Let $n$ be a positive integer. On the blackboard all quadratic polynomials with positive integer coefficients, that do not exceed $n$, without real roots are written Find all $n$ for which the number of written polynomials is even [i]A. Voidelevich[/i]

An arbitrary point $M$ is chosen inside the triangle $ABC$. Prove that $MA + MB + MC \le max (AB + BC, BC + AC, AC + AB)$. (N. Sedrakyan)

In the accompanying figure, semicircles with centers$ A$ and $B$ have radii $4$ and $2$, respectively. Furthermore, they are internally tangent to the circle of diameter $PQ$. Also the semicircles with centers $ A$ and $ B$ are externally tangent to each other. The circle with center $C$ is internally tangent to the semicircle with diameter $PQ$ and externally tangent to the others two semicircles. Determine the value of the radius of the circle with center $C$. [img]https://cdn.artofproblemsolving.com/attachments/c/b/281b335f6a2d6230a5b79060e6d85d6ca6f06c.png[/img]

Compute $\sum_{k=0}^{2n} (-1)^k a_k^2$ where $a_k$ are the coefficients in the expansion \[(1- \sqrt 2 x +x^2)^n =\sum_{k=0}^{2n} a_k x^k.\]