Given $X \subset \mathbb{N}$, define $d(X)$ as the largest $c \in [0, 1]$ such that for any $a < c$ and $n_0\in \mathbb{N}$, there exists $m, r \in \mathbb{N}$ with $r \geq n_0$ and $\frac{\mid X \cap [m, m+r)\mid}{r} \geq a$. Let $E, F \subset \mathbb{N}$ such that $d(E)d(F) > 1/4$. Prove that for any prime $p$ and $k\in\mathbb{N}$, there exists $m \in E, n \in F$ such that $m\equiv n \pmod{p^k}$

Find a sequence of $ 2012 $ distinct integers bigger than $ 0 $ such that their sum is a perfect square and their product is a perfect cube.

Define a sequence ${{a_n}}_{n\ge1}$ such that $a_1=1$ , $a_2=2$ and $a_{n+1}$ is the smallest positive integer $m$ such that $m$ hasn't yet occurred in the sequence and also $gcd(m,a_n)\neq{1}$. Show that all positive integers occur in the sequence.

Seyed has 998 white coins, a red coin, and an unusual coin with one red side and one white side. He can not see the color of the coins instead he has a scanner which checks if all of the coin sides touching the scanner glass are white. Is there any algorithm to find the red coin by using the scanner at most 17 times? [i]Proposed by Seyed Reza Hosseini[/i]

Let $x =\sqrt{a}+\sqrt{b}$, where $a$ and $b$ are natural numbers, $x$ is not an integer, and $x < 1976$. Prove that the fractional part of $x$ exceeds $10^{-19.76}$.

A $\emph{planar}$ graph is a connected graph that can be drawn on a sphere without edge crossings. Such a drawing will divide the sphere into a number of faces. Let $G$ be a planar graph with $11$ vertices of degree $2$, $5$ vertices of degree $3$, and $1$ vertex of degree $7$. Find the number of faces into which $G$ divides the sphere.

Let $0 < \theta < 2\pi$ be a real number for which $\cos (\theta) + \cos (2\theta) + \cos (3\theta) + ...+ \cos (2020\theta) = 0$ and $\theta =\frac{\pi}{n}$ for some positive integer $n$. Compute the sum of the possible values of $n \le 2020$.

Let $ n$ be a positive integer, and let $ x$ and $ y$ be a positive real number such that $ x^n \plus{} y^n \equal{} 1.$ Prove that \[ \left(\sum^n_{k \equal{} 1} \frac {1 \plus{} x^{2k}}{1 \plus{} x^{4k}} \right) \cdot \left( \sum^n_{k \equal{} 1} \frac {1 \plus{} y^{2k}}{1 \plus{} y^{4k}} \right) < \frac {1}{(1 \minus{} x) \cdot (1 \minus{} y)}. \] [i]Author: Juhan Aru, Estonia[/i]

Points $Y,Z$ lie on the smaller arc $BC$ of the circumcircle of an acute triangle $\bigtriangleup ABC$ ($Y$ lies on the smaller arc $BZ$). Let $X$ be a point such that the triangles $\bigtriangleup ABC,\bigtriangleup XYZ$ are similar (in this exact order) with $A,X$ lying on the same side of $YZ$. Lines $XY,XZ$ intersect sides $AB,AC$ at points $E,F$ respectively. Let $K$ be the intersection of lines $BY,CZ$. Prove that one of the intersections of the circumcircles of triangles $\bigtriangleup AEF,\bigtriangleup KBC$ lie on the line $KX$. [i]Proposed by Amirparsa Hosseini Nayeri - Iran[/i]

An island has $10$ cities, where some of the possible pairs of cities are connected by roads. A [i]tour route[/i] is a route starting from a city, passing exactly eight out of the other nine cities exactly once each, and returning to the starting city. (In other words, it is a loop that passes only nine cities instead of all ten cities.) For each city, there exists a tour route that doesn't pass the given city. Find the minimum number of roads on the island.

Construct a set $A\subset[0,1]\times[0,1]$ such that $A$ is dense in $[0,1]\times[0,1]$ and every vertical and every horizontal line intersects $A$ in at most one point.

There is a point $D$ on side $AC$ of acute triangle $\triangle ABC$. Let $AM$ be the median drawn from $A$ (so $M$ is on $BC$) and $CH$ be the altitude drawn from $C$ (so $H$ is on $AB$). Let $I$ be the intersection of $AM$ and $CH$, and let $K$ be the intersection of $AM$ and line segment $BD$. We know that $AK=8$, $BK=8$, and $MK=6$. Find the length of $AI$.

Determine, for any positive real number $a$, the number of solutions $(x,y)$ to the system of equations $$\begin{cases} |x|+|y|= 1 \\ x^2 + y^2 = a \end{cases}$$ where $x$ and $y$ are real numbers.

A particular coin can land on heads (H), on tails (T), or in the middle (M), each with probability $\frac{1}{3}$. Find the expected number of flips necessary to observe the contiguous sequence HMMTHMMT...HMMT, where the sequence HMMT is repeated 2016 times.

Let $ABC$ be a triangle with $AB > AC$. Point $P \in (AB)$ is such that $\angle ACP = \angle ABC$. Let $D$ be the reflection of $P$ into the line $AC$ and let $E$ be the point in which the circumcircle of $BCD$ meets again the line $AC$. Prove that $AE = AC$.

Let $ S$ be the set of natural numbers $ n$ satisfying the following conditions: $ (i)$ $ n$ has $ 1000$ digits, $ (ii)$ all the digits of $ n$ are odd, and $ (iii)$ any two adjacent digits of $ n$ differ by $ 2$. Determine the number of elements of $ S$.

Evaluate \[\lim_{n\to\infty} \int_0^{\pi} x^2 |\sin nx| dx\]

Let $S$ be a finite set of points on a plane, where no three points are collinear, and the convex hull of $S$, $\Omega$, is a $2016-$gon $A_1A_2\ldots A_{2016}$. Every point on $S$ is labelled one of the four numbers $\pm 1,\pm 2$, such that for $i=1,2,\ldots , 1008,$ the numbers labelled on points $A_i$ and $A_{i+1008}$ are the negative of each other. Draw triangles whose vertices are in $S$, such that any two triangles do not have any common interior points, and the union of these triangles is $\Omega$. Prove that there must exist a triangle, where the numbers labelled on some two of its vertices are the negative of each other.

Find all positive integers $n$ such that the number $n^5+79$ has all the same digits when it is written in decimal represantation.

Let $m$ boxes be given, with some balls in each box. Let $n < m$ be a given integer. The following operation is performed: choose $n$ of the boxes and put $1$ ball in each of them. Prove: [i](a) [/i]If $m$ and $n$ are relatively prime, then it is possible, by performing the operation a finite number of times, to arrive at the situation that all the boxes contain an equal number of balls. [i](b)[/i] If $m$ and $n$ are not relatively prime, there exist initial distributions of balls in the boxes such that an equal distribution is not possible to achieve.

A circle of radius $ 1$ is internally tangent to two circles of radius $ 2$ at points $ A$ and $ B$, where $ AB$ is a diameter of the smaller circle. What is the area of the region, shaded in the gure, that is outside the smaller circle and inside each of the two larger circles? [asy]size(200);defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair B = (0,1); pair A = (0,-1); label("$B$",B,NW);label("$A$",A,2S); draw(Circle(A,2));draw(Circle(B,2)); fill((-sqrt(3),0)..B..(sqrt(3),0)--cycle,gray); fill((-sqrt(3),0)..A..(sqrt(3),0)--cycle,gray); draw((-sqrt(3),0)..B..(sqrt(3),0)); draw((-sqrt(3),0)..A..(sqrt(3),0)); path circ = Circle(origin,1); fill(circ,white); draw(circ); dot(A);dot(B); pair A1 = B + dir(45)*2; pair A2 = dir(45); pair A3 = dir(-135)*2 + A; draw(B--A1,EndArrow(HookHead,2)); draw(origin--A2,EndArrow(HookHead,2)); draw(A--A3,EndArrow(HookHead,2)); label("$2$",midpoint(B--A1),NW); label("$1$",midpoint(origin--A2),NW); label("$2$",midpoint(A--A3),NW);[/asy]$ \textbf{(A)}\ \frac {5}{3}\pi \minus{} 3\sqrt {2}\qquad \textbf{(B)}\ \frac {5}{3}\pi \minus{} 2\sqrt {3}\qquad \textbf{(C)}\ \frac {8}{3}\pi \minus{} 3\sqrt {3}\qquad\textbf{(D)}\ \frac {8}{3}\pi \minus{} 3\sqrt {2}$ $ \textbf{(E)}\ \frac {8}{3}\pi \minus{} 2\sqrt {3}$

Let $a_0$ be a positive integer and $a_{n + 1} =\sqrt{a_n^2 + 1}$, for all $n \ge 0$. 1) Prove that for all $a_0$ the sequence contains infinitely many integers and infinitely many irrational numbers. 2) Is there an $a_0$ for which $a_{2010}$ is an integer?

In an $m\times n$ rectangular chessboard,there is a stone in the lower leftmost square. Two persons A,B move the stone alternately. In each step one can move the stone upward or rightward any number of squares. The one who moves it into the upper rightmost square wins. Find all $(m,n)$ such that the first person has a winning strategy.

Let $S_1$ and $S_2$ be two circumferences, with centers $O_1$ and $O_2$ respectively, and secants on $M$ and $N$. The line $t$ is the common tangent to $S_1$ and $S_2$ closer to $M$. The points $A$ and $B$ are the intersection points of $t$ with $S_1$ and $S_2$, $C$ is the point such that $BC$ is a diameter of $S_2$, and $D$ the intersection point of the line $O_1O_2$ with the perpendicular line to $AM$ through $B$. Show that $M$, $D$ and $C$ are collinear.

Find all surjective functions $ f: \mathbb{N} \to \mathbb{N}$ such that for every $ m,n \in \mathbb{N}$ and every prime $ p,$ the number $ f(m + n)$ is divisible by $ p$ if and only if $ f(m) + f(n)$ is divisible by $ p$. [i]Author: Mohsen Jamaali and Nima Ahmadi Pour Anari, Iran[/i]