What is the smallest perfect square larger than $1$ with a perfect square number of positive integer factors? [i]Ray Li[/i]

Let $AB$ be a line segment with length 2, and $S$ be the set of points $P$ on the plane such that there exists point $X$ on segment $AB$ with $AX = 2PX$. Find the area of $S$.

Find the smallest positive integer $n$ such that if we color in red $n$ arbitrary vertices of the cube , there will be a vertex of the cube which has the three vertices adjacent to it colored in red.

Let $n > 1$ be an integer. In a [i]configuration[/i] of an $n \times n$ board, each of the $n^2$ cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate $90^{\circ}$ counterclockwise. We call a cell [i]good[/i] if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of $n$, the maximum number of good cells over all possible starting configurations. [i]Proposed by Melek Güngör, Turkey[/i]

Let $k$ be a circle and $l$–line that is tangent to $k$ in point $P$. On $l$ from the two sides of $P$ are chosen arbitrary points $A$ and $B$. The tangents through $A$ and $B$ to $k$, different than $l$, intersect in point $C$. Find the geometric place of points $C$, when $A$ and $B$ change in such way so that $AP.BP$ is a constant.

The vertices of Durer's favorite regular decagon in clockwise order: $D_1, D_2, D_3, . . . , D_{10}$. What is the angle between the diagonals $D_1D_3$ and $D_2D_5$?

Two circles are externally tangent. Lines $\overline{PAB}$ and $\overline{PA'B'}$ are common tangents with $A$ and $A'$ on the smaller circle and $B$ and $B'$ on the larger circle. If $PA = AB = 4$, then the area of the smaller circle is [asy] size(250); defaultpen(fontsize(10pt)+linewidth(.8pt)); pair O=origin, Q=(0,-3sqrt(2)), P=(0,-6sqrt(2)), A=(-4/3,3.77-6sqrt(2)), B=(-8/3,7.54-6sqrt(2)), C=(4/3,3.77-6sqrt(2)), D=(8/3,7.54-6sqrt(2)); draw(Arc(O,2sqrt(2),0,360)); draw(Arc(Q,sqrt(2),0,360)); dot(A); dot(B); dot(C); dot(D); dot(P); draw(B--A--P--C--D); label("$A$",A,dir(A)); label("$B$",B,dir(B)); label("$A'$",C,dir(C)); label("$B'$",D,dir(D)); label("$P$",P,S);[/asy] $ \textbf{(A)}\ 1.44\pi\qquad\textbf{(B)}\ 2\pi\qquad\textbf{(C)}\ 2.56\pi\qquad\textbf{(D)}\ \sqrt{8}\pi\qquad\textbf{(E)}\ 4\pi $

There are $15$ lights on the ceiling of a room, numbered from $1$ to $15$. All lights are turned off. In another room, there are $15$ switches: a switch for lights $1$ and $2$, a switch for lights $2$ and $3$, a switch for lights $3$ en $4$, etcetera, including a sqitch for lights $15$ and $1$. When the switch for such a pair of lights is turned, both of the lights change their state (from on to off, or vice versa). The switches are put in a random order and all look identical. Raymond wants to find out which switch belongs which pair of lights. From the room with the switches, he cannot see the lights. He can, however, flip a number of switches, and then go to the other room to see which lights are turned on. He can do this multiple times. What is the minimum number of visits to the other room that he has to take to determine for each switch with certainty which pair of lights it corresponds to?

Find all real $a$ such that the equation $3^{\cos (2x)+1}-(a-5)3^{\cos^2(2x)}=7$ has a real root. [hide=Remark] This was the statement given at the contest, but there was actually a typo and the intended equation was $3^{\cos (2x)+1}-(a-5)3^{\cos^2(x)}=7$, which is much easier.

Prove that there are no distinct positive integers $x$ and $y$ such that $x^{2007} + y! = y^{2007} + x! $

let $m,n\in \mathbb{N}$ and $p(x),q(x),h(x)$ are polynomials with real Coefficients such that $p(x)$ is Descending. and for all $x\in \mathbb{R}$ $p(q(nx+m)+h(x))=n(q(p(x))+h(x))+m$ . prove that dont exist function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x\in \mathbb{R}$ $f(q(p(x))+h(x))=f(x)^{2}+1$

For all positive integers $n$, prove that there are integers $x, y$ relatively prime to $5$ such that $x^2 + y^2 = 5^n$.

On a plane several straight lines are drawn in such a way that each of them intersects exactly $15$ other lines. How many lines are drawn on the plane? Find all possibilities and justify your answer. Poland

Let $d(n)$ denote the number of positive divisors of the natural number $n$. Find all the natural numbers $n$ such that $$d(n) = \frac{n}{5}$$.

Suppose two functions $f(x)$ and $g(x)$ are defined for all $x$ with $2<x<4$ and satisfy: $2<f(x)<4,2<g(x)<4,f(g(x))=g(f(x))=x,f(x)\cdot g(x)=x^2$ for all $2<x<4$. Prove that $f(3)=g(3)$.

A plane cuts a sphere of radius $ 1$ into two pieces, one of which has three times the surface area of the other. What is the area of the disk that the sphere cuts out of the plane?

$find$ all non negative integers $m$, $n$ such that $mn-1$ divides $n^3-1$

Prove that if the numbers $ x_1 $ and $ x_2 $ are roots of the equation $ x^2 + px - 1 = 0 $, where $ p $ is an odd number, then for every natural $n$number $ x_1^n + x_2^n $ and $ x_1^{n+1} + x_2^{n+1} $ are integer and coprime.

Let $ABC$ be a triangle with incentre $I$ and circumcircle $\omega$. The incircle of the triangle $ABC$ touches the sides $BC$, $CA$ and $AB$ at $D$, $E$ and $F$, respectively. The circumcircle of triangle $ADI$ crosses $\omega$ again at $P$, and the lines $PE$ and $PF$ cross $\omega$ again at $X$and $Y$, respectively. Prove that the lines $AI$, $BX$ and $CY$ are concurrent.

Let $f(x)$ be a rational function with complex coefficients whose denominator does not have multiple roots. Let $u_0, u_1,... , u_n$ be the complex roots of $f$ and $w_1, w_2,..., w_m$ be the roots of $f'$. Suppose that $u_0$ is a simple root of $f$. Prove that \[ \sum_{k=1}^m \frac{1}{w_k - u_0} = 2\sum_{k = 1}^n\frac{1}{u_k - u_0}.\]

What is the area of the polygon whose vertices are the points of intersection of the curves $x^2+y^2=25$ and $(x-4)^2+9y^2=81$? ${{ \textbf{(A)}\ 24\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 37.5}\qquad\textbf{(E)}\ 42} $

Let $H = \{-2019,-2018, ...,-1, 0, 1, 2, ..., 2020\}$. Describe all functions $f : H \to H$ for which a) $x = f(x) - f(f(x))$ holds for every $x \in H$. b) $x = f(x) + f(f(x)) - f(f(f(x)))$ holds for every $x \in H$. c) $x = f(x) + 2f(f(x)) - 3f(f(f(x)))$ holds for every $x \in H$. PS. (a) + (b) for category E 1.5, (b) + (c) for category E+ 1.2

How $arc \sin(\cos(arc \sin x))$ and $arc \cos(\sin(arc \cos x))$ are related with each other?

Two points are located $10$ units apart, and a circle is drawn with radius $ r$ centered at one of the points. A tangent line to the circle is drawn from the other point. What value of $ r$ maximizes the area of the triangle formed by the two points and the point of tangency?

Let $ ABC$ be a triangle with incenter $ I$ and let $ \Gamma$ be a circle centered at $ I$, whose radius is greater than the inradius and does not pass through any vertex. Let $ X_{1}$ be the intersection point of $ \Gamma$ and line $ AB$, closer to $ B$; $ X_{2}$, $ X_{3}$ the points of intersection of $ \Gamma$ and line $ BC$, with $ X_{2}$ closer to $ B$; and let $ X_{4}$ be the point of intersection of $ \Gamma$ with line $ CA$ closer to $ C$. Let $ K$ be the intersection point of lines $ X_{1}X_{2}$ and $ X_{3}X_{4}$. Prove that $ AK$ bisects segment $ X_{2}X_{3}$.