In the Cartesian plane is given a set of points with integer coordinate \[ T=\{ (x;y)\mid x,y\in\mathbb{Z} ; \ |x|,|y|\leq 20 ; \ (x;y)\ne (0;0)\} \] We colour some points of $ T $ such that for each point $ (x;y)\in T $ then either $ (x;y) $ or $ (-x;-y) $ is coloured. Denote $ N $ to be the number of couples $ {(x_1;y_1),(x_2;y_2)} $ such that both $ (x_1;y_1) $ and $ (x_2;y_2) $ are coloured and $ x_1\equiv 2x_2 \pmod {41}, y_1\equiv 2y_2 \pmod {41} $. Find the all possible values of $ N $.

A polynomial $P$ of degree $2015$ satisfies the equation $P(n)=\frac{1}{n^2}$ for $n=1, 2, \dots, 2016$. Find $\lfloor 2017P(2017)\rfloor$.

Let $ABCD$ be a tetrahedron. Denote by $S_1$ the inscribed sphere inside it, which is tangent to all four faces. Denote by $S_2$ the outer escribed sphere outside $ABC$, tangent to face $ABC$ and to the planes containing faces $ABD,ACD,BCD$. Let $K$ be the tangency point of $S_1$ to the face $ABC$, and let $L$ be the tangency point of $S_2$ to the face $ABC$. Let $T$ be the foot of the perpendicular from $D$ to the face $ABC$. Prove that $L,T,K$ lie on one line.

If $$\Omega_n=\sum \limits_{k=1}^n \left(\int \limits_{-\frac{1}{k}}^{\frac{1}{k}}(2x^{10} + 3x^8 + 1)\cos^{-1}(kx)dx\right)$$Then find $$\Omega=\lim \limits_{n\to \infty}\left(\Omega_n-\pi H_n\right)$$

When Eva counts, she skips all numbers containing a digit divisible by 3. For example, the first ten numbers she counts are 1, 2, 4, 5, 7, 8, 11, 12, 14, 15. What is the $100^{\text{th}}$ number she counts? [i]Proposed by Eugene Chen[/i]

A polynomial $P(x, y, z)$ in three variables with real coefficients satisfies the identities $$P(x, y, z)=P(x, y, xy-z)=P(x, zx-y, z)=P(yz-x, y, z).$$ Prove that there exists a polynomial $F(t)$ in one variable such that $$P(x,y,z)=F(x^2+y^2+z^2-xyz).$$

In an acute triangle $ABC$ the point $O{}$ is the circumcenter, $H_1$ is the foot of the perpendicular from $A{}$ onto $BC$, and $M_H$ and $N_H$ are the projections of $H_1$ on $AC$ and $AB{}$, respectively. Prove that the polyline $M_HON_H$ divides the triangle $ABC$ in two figures of equal area. [i]Proposed by I. A. Kushner[/i]

A sphere with center $O$ has radius $6$. A triangle with sides of length $15, 15,$ and $24$ is situated in space so that each of its sides is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle? $ \textbf{(A) }2\sqrt{3}\qquad \textbf{(B) }4\qquad \textbf{(C) }3\sqrt{2}\qquad \textbf{(D) }2\sqrt{5}\qquad \textbf{(E) }5\qquad $

Let \(ABC\) be a right-angled triangle with \(\angle B=90^{\circ}\). Let \(I\) be the incentre if \(ABC\). Extend \(AI\) and \(CI\); let them intersect \(BC\) in \(D\) and \(AB\) in \(E\) respectively. Draw a line perpendicular to \(AI\) at \(I\) to meet \(AC\) in \(J\), draw a line perpendicular to \(CI\) at \(I\) to meet \(AC\) at \(K\). Suppose \(DJ=EK\). Prove that \(BA=BC\).

Let A be a subset of the set $\{1,2, ...,n\}$ with at least $100\sqrt n$ elements. Prove that there is a four-element arithmetic sequence in which each element is the sum of two different elements of the set A.

In triangle $ABC$, $AB = 28$, $AC = 36$, and $BC = 32$. Let $D$ be the point on segment $BC$ satisfying $\angle BAD = \angle DAC$, and let $E$ be the unique point such that $DE \parallel AB$ and line $AE$ is tangent to the circumcircle of $ABC$. Find the length of segment $AE$. [i]Ray Li[/i]

Let G, O, D, I, and T be digits that satisfy the following equation: \begin{tabular}{ccccc} &G&O&G&O\\ +&D&I&D&I\\ \hline G&O&D&O&T \end{tabular} (Note that G and D cannot be $0$, and that the five variables are not necessarily different.) Compute the value of GODOT.

Let $ABC$ be the triangle and $I$ the center of the circle inscribed in this triangle. The point $M$, located on the tangent taken to the point $B$ to the circumscribed circle of the triangle $ABC$, satisfies the relation $AB = MB$. Point $N$, located on the tangent taken to point $C$ to the same circle, satisfies the relation $AC = NC$. Points $M, A$ and $N$ lie on the same side of the line $BC$. Prove that $$\angle BAC + \angle MIN = 180^o.$$

At the bottom-left corner of a $2014\times 2014$ chessboard, there are some green worms and at the top-left corner of the same chessboard, there are some brown worms. Green worms can move only to right and up, and brown worms can move only to right and down. After a while, the worms make some moves and all of the unit squares of the chessboard become occupied at least once throughout this process. Find the minimum total number of the worms.

From which two statements about the trapezoid follows the third: 1) the trapezoid is tangential, 2) the trapezoid is right, 3) its area is equal to the product of the bases?

In circle $\Omega$, let $\overline{AB}=65$ be the diameter and let points $C$ and $D$ lie on the same side of arc $\overarc{AB}$ such that $CD=16$, with $C$ closer to $B$ and $D$ closer to $A$. Moreover, let $AD, BC, AC,$ and $BD$ all have integer lengths. Two other circles, circles $\omega_1$ and $\omega_2$, have $\overline{AC}$ and $\overline{BD}$ as their diameters, respectively. Let circle $\omega_1$ intersect $AB$ at a point $E \neq A$ and let circle $\omega_2$ intersect $AB$ at a point $F \neq B$. Then $EF=\frac{m}{n}$, for relatively prime integers $m$ and $n$. Find $m+n$. [asy] size(7cm); pair A=(0,0), B=(65,0), C=(117/5,156/5), D=(125/13,300/13), E=(23.4,0), F=(9.615,0); draw(A--B--C--D--cycle); draw(A--C); draw(B--D); dot("$A$", A, SW); dot("$B$", B, SE); dot("$C$", C, NE); dot("$D$", D, NW); dot("$E$", E, S); dot("$F$", F, S); draw(circle((A + C)/2, abs(A - C)/2)); draw(circle((B + D)/2, abs(B - D)/2)); draw(circle((A + B)/2, abs(A - B)/2)); label("$\mathcal P$", (A + B)/2 + abs(A - B)/2 * dir(-45), dir(-45)); label("$\mathcal Q$", (A + C)/2 + abs(A - C)/2 * dir(-210), dir(-210)); label("$\mathcal R$", (B + D)/2 + abs(B - D)/2 * dir(70), dir(70)); [/asy] [i]Proposed by [b]AOPS12142015[/b][/i]

A positive integer with 3 digits $\overline{ABC}$ is $Lusophon$ if $\overline{ABC}+\overline{CBA}$ is a perfect square. Find all $Lusophon$ numbers.

With all angles measured in degrees, the product $\prod_{k=1}^{45} \csc^2(2k-1)^\circ=m^n$, where $m$ and $n$ are integers greater than 1. Find $m+n$.

Find all the functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $$|f(x)-f(y)| = 2 |x - y| $$

Max divided a natural number $p$ by a natural number $q \leq 100$. In the decimal representation of the quotient he calculated, the sequence of digits $1982$ occurs somewhere after the decimal point. Show that Max made a computational mistake.

In which set are both elements metalloids? ${ \textbf{(A)}\ \text{Cr and Mo}\qquad\textbf{(B)}\ \text{Ge and As}\qquad\textbf{(C)}\ \text{Sn and Pb}\qquad\textbf{(D)}}\ \text{Se and Br}\qquad $

Let $r$ and $s$ be two rational numbers. Find all functions $f: \mathbb Q \to \mathbb Q$ such that for all $x,y\in\mathbb Q$ we have \[ f(x+f(y)) = f(x+r)+y+s. \]

We consider the following system with $q=2p$: \[\begin{matrix} a_{11}x_{1}+\ldots+a_{1q}x_{q}=0,\\ a_{21}x_{1}+\ldots+a_{2q}x_{q}=0,\\ \ldots ,\\ a_{p1}x_{1}+\ldots+a_{pq}x_{q}=0,\\ \end{matrix}\] in which every coefficient is an element from the set $\{-1,0,1\}$$.$ Prove that there exists a solution $x_{1}, \ldots,x_{q}$ for the system with the properties: [b]a.)[/b] all $x_{j}, j=1,\ldots,q$ are integers$;$ [b]b.)[/b] there exists at least one j for which $x_{j} \neq 0;$ [b]c.)[/b] $|x_{j}| \leq q$ for any $j=1, \ldots ,q.$

Given function $f(x) = \dfrac{1}{\sqrt[3]{1-x^3}}$, find $\underbrace{f(... f(f(19))...)}_{95}$. .

Points $A' ,B' ,C'$ on the respective sides $BC,CA,AB$ of triangle $ABC$ satisfy $\frac{AC' }{AB} = \frac{BA' }{BC} = \frac{CB' }{CA} = k$. The lines $AA' ,BB' ,CC' $ form a triangle $A_1B_1C_1$ (possibly degenerate). Given $k$ and the area $S$ of $\triangle ABC$, compute the area of $\triangle A_1B_1C_1$.