Let $n$ be positive integers and $t$ be a positive real number. Evaluate $\int_0^{\frac{2n}{t}\pi} |x\sin\ tx|\ dx.$

Let $k,n>1$ be integers such that the number $p=2k-1$ is prime. Prove that, if the number $\binom{n}{2}-\binom{k}{2}$ is divisible by $p$, then it is divisible by $p^2$.

For each $ n \in \mathbb{N}$ we have two numbers $ p_n, q_n$ with the following property: For exactly $ n$ distinct integer numbers $ x$ the number \[ x^2 \plus{} p_n \cdot x \plus{} q_n\] is the square of a natural number. (Note the definition of natural numbers includes the zero here.)

Let $n$ be a positive integer. Prove that \[\frac{1}{3}+\cdots+\frac{1}{2n+1}\] is not an integer.

Solve the following system of equations: $$x+\frac{1}{x^3}=2y,\quad y+\frac{1}{y^3}=2z,\quad z+\frac{1}{z^3}=2w,\quad w+\frac{1}{w^3}=2x.$$

There's a convex $3n$-polygon on the plane with a robot on each of it's vertices. Each robot fires a laser beam toward another robot. On each of your move,you select a robot to rotate counter clockwise until it's laser point a new robot. Three robots $A$, $B$ and $C$ form a triangle if $A$'s laser points at $B$, $B$'s laser points at $C$, and $C$'s laser points at $A$. Find the minimum number of moves that can guarantee $n$ triangles on the plane.

Positive real numbers $a,b,c$ satisfy $a+b+c=1$. Prove that $$(a+1)\sqrt{2a(1-a)} + (b+1)\sqrt{2b(1-b)} + (c+1)\sqrt{2c(1-c)} \geq 8(ab+bc+ca).$$ Also, find the values of $a,b,c$ for which the equality happens.

Let $ABC$ be a non-rectangle triangle with $M$ the middle of $BC$. Let $D$ be a point on the line $AB$ such that $CA=CD$ and let $E$ be a point on the line $BC$ such that $EB=ED$. The parallel to $ED$ passing through $A$ intersects the line $MD$ at the point $I$ and the line $AM$ intersects the line $ED$ at the point $J$. Show that the points $C, I$ and $J$ are aligned.

Let $p(x)$ be a polynomial of degree $n$ with real coefficients such that $p(x) \ge 0$ for all $x$. Prove that $p(x)+ p'(x)+ p''(x)+...+ p^{(n)}(x) \ge 0$.

Two positive integers $p,q \in \mathbf{Z}^{+}$ are given. There is a blackboard with $n$ positive integers written on it. A operation is to choose two same number $a,a$ written on the blackboard, and replace them with $a+p,a+q$. Determine the smallest $n$ so that such operation can go on infinitely.

There are eight rooms on the first floor of a hotel, with four rooms on each side of the corridor, symmetrically situated (that is each room is exactly opposite to one other room). Four guests have to be accommodated in four of the eight rooms (that is, one in each) such that no two guests are in adjacent rooms or in opposite rooms. In how many ways can the guests be accommodated?

Let $a>1$ be an odd positive integer. Find the least positive integer $n$ such that $2^{2000}$ is a divisor of $a^n-1$. [i]Mircea Becheanu [/i]

Given 5 nonzero vectors in three-dimensional Euclidean space, prove that the sum of their pairwise angles is at most $6\pi$.

Let $ABC$ be a triangle with $\angle BAC \neq 90^{\circ}.$ Let $O$ be the circumcenter of the triangle $ABC$ and $\Gamma$ be the circumcircle of the triangle $BOC.$ Suppose that $\Gamma$ intersects the line segment $AB$ at $P$ different from $B$, and the line segment $AC$ at $Q$ different from $C.$ Let $ON$ be the diameter of the circle $\Gamma.$ Prove that the quadrilateral $APNQ$ is a parallelogram.

Let $f(x)=\frac{\sin x}{x}$, for $x>0$, and let $n$ be a positive integer. Prove that $|f^{(n)}(x)|<\frac{1}{n+1}$, where $f^{(n)}$ denotes the $n^{\mathrm{th}}$ derivative of $f$. (Proposed by Alexander Bolbot, State University, Novosibirsk)

Prove that the function $f (x)$ satisfying the relation $|f (x) - f (y) | \le | x - y|^a$ for any real numbers $x, y$ and some number $a> 1$ is constant.

Let $S$ be the set of all positive integer divisors of $100,000.$ How many numbers are the product of two distinct elements of $S?$ $\textbf{(A) }98\qquad\textbf{(B) }100\qquad\textbf{(C) }117\qquad\textbf{(D) }119\qquad\textbf{(E) }121$

Compute $$\int_0^4\frac{x^4-4x+4}{1+2017^{x-2}}dx.$$

Given $1990$ piles of stones, containing $1, 2, 3, ... , 1990$ stones. A move is to take an equal number of stones from one or more piles. How many moves are needed to take all the stones?

In a tournament there are six teams that play each other twice. A team earns 3 points for a win, 1 point for a draw, and 0 points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams? $\textbf{(A) }22\qquad\textbf{(B) }23\qquad\textbf{(C) }24\qquad\textbf{(D) }26\qquad\textbf{(E) }30$

Determine whether it is possible to partition $\mathbb{Z}$ into triples $(a,b,c)$ such that, for every triple, $|a^3b + b^3c + c^3a|$ is perfect square.

Let $ ABC$ be an acute triangle with orthocenter $ H$ and let $ X$ be an arbitrary point in its plane. The circle with diameter $ HX$ intersects the lines $ AH$ and $ AX$ at $ A_{1}$ and $ A_{2}$, respectively. Similarly, define $ B_{1}$, $ B_{2}$, $ C_{1}$, $ C_{2}$. Prove that the lines $ A_{1}A_{2}$, $ B_{1}B_{2}$, $ C_{1}C_{2}$ are concurrent. [hide][i]Remark[/i]. The triangle obviously doesn't need to be acute.[/hide]

[u]Round 5[/u] [b]p13.[/b] A $2016 \times 2016$ chess board is cut into $k \ge 1$ rectangle(s) with positive integer sidelengths. Let $p$ be the sum of the perimeters of all $k$ rectangles. Additionally, let $m$ and $M$ be the minimum and maximum possible value of $\frac{p}{k}$, respectively. Determine the ordered pair $(m,M)$. [b]p14.[/b] For nonnegative integers $n$, let $f (n)$ be the product of the digits of $n$. Compute $\sum^{1000}_{i=1}f (i )$. [b]p15.[/b] How many ordered pairs of positive integers $(m,n)$ have the property that $mn$ divides $2016$? [u]Round 6[/u] [b]p16.[/b] Let $a,b,c$ be distinct integers such that $a +b +c = 0$. Find the minimum possible positive value of $|a^3 +b^3 +c^3|$. [b]p17.[/b] Find the greatest positive integer $k$ such that $11^k -2^k$ is a perfect square. [b]p18.[/b] Find all ordered triples $(a,b,c)$ with $a \le b \le c$ of nonnegative integers such that $2a +2b +2c = ab +bc +ca$. [u]Round 7[/u] [b]p19.[/b] Let $f :N \to N$ be a function such that $f ( f (n))+ f (n +1) = n +2$ for all positive integers $n$. Find $f (20)+ f (16)$. [b]p20.[/b] Let $\vartriangle ABC$ be a triangle with area $10$ and $BC = 10$. Find the minimum possible value of $AB \cdot AC$. [b]p21.[/b] Let $\vartriangle ABC$ be a triangle with sidelengths $AB = 19$, $BC = 24$, $C A = 23$. Let $D$ be a point on minor arc $BC$ of the circumcircle of $\vartriangle ABC$ such that $DB =DC$. A circle with center $D$ that passes through $B$ and $C$ interests $AC$ again at a point $E \ne C$. Find the length of $AE$. [u]Round 8[/u] [b]p22.[/b] Let $m =\frac12 \sqrt{2+\sqrt{2+... \sqrt2}}$, where there are $2014$ square roots. Let $f_1(x) =2x^2 -1$ and let $f_n(x) = f_1( f_{n-1}(x))$. Find $f_{2015}(m)$. [b]p23.[/b] How many ordered triples of integers $(a,b,c)$ are there such that $0 < c \le b \le a \le 2016$, and $a +b-c = 2016$? [b]p24.[/b] In cyclic quadrilateral $ABCD$, $\angle B AD = 120^o$,$\angle ABC = 150^o$,$CD = 8$ and the area of $ABCD$ is $6\sqrt3$. Find the perimeter of $ABCD$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3158461p28714996]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3162282p28763571]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides $ 7$ times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to his distance from the stadium? $ \textbf{(A)}\ \frac {2}{3}\qquad \textbf{(B)}\ \frac {3}{4}\qquad \textbf{(C)}\ \frac {4}{5}\qquad \textbf{(D)}\ \frac {5}{6}\qquad \textbf{(E)}\ \frac {6}{7}$

The catheti $AC$ and $BC$ in a right-angled triangle $ABC$ have lengths $b$ and $a$, respectively. A circle centered at $C$ is tangent to hypotenuse $AB$ at point $D$. The tangents to the circle through points $A$ and $B$ intersect the circle at points $E$ and $F$, respectively (where $E$ and $F$ are both different from $D$). Express the length of the segment $EF$ in terms of $a$ and $b$.