Denote by $\mathbb Z^2$ the set of all points $(x,y)$ in the plane with integer coordinates.  For each integer $n\geq 0$, let $P_n$ be the subset of $\mathbb Z^2$ consisting of the point $(0,0)$ together with all points $(x,y)$ such that $x^2+y^2=2^k$ for some integer $k\leq n$.  Determine, as a function of $n$, the number of four-point subsets of $P_n$ whose elements are the vertices of a square.

On each side of a triangle, $5$ points are chosen (other than the vertices of the triangle) and these $15$ points are colored red. How many ways are there to choose four red points such that they form the vertices of a quadrilateral?

Let $N$ be the number of ways to draw 22 straight edges between 10 labeled points, of which no three are collinear, such that no triangle with vertices among these 10 points is created, and there is at most one edge between any two labeled points. Compute $\dfrac{N}{9!}$.

$ABCDEF$ is a regular hexagon. A frog sarts jumping at $A$, each time it can jump to one of the two adjacent points. If the frog jump to $D$ in no more than five times, it stops. After five jumpings, if the frog hasn't jumped to $D$ yet, it will stop as well. Then the number of different ways to jump is________.

We say distinct positive integers  $a_1,a_2,\ldots ,a_n $ are "good" if their sum is equal to the sum of all pairwise $\gcd $'s among them. Prove that there are infinitely many $n$ s such that $n$ good numbers exist. [i]Proposed by Morteza Saghafian[/i]

Let $a, b$ be non-zero integers. Let $m(a, b)$ be the smallest value of $\cos ax + \cos bx$ (for real $x$). Show that for some $r$, $m(a, b) \le r < 0$ for all $a, b$.

The point $O$ inside the convex polygon makes isosceles triangle with all the pairs of its vertices. Prove that $O$ is the centre of the circumscribed circle. [u]other formulation:[/u] $P$ is a convex polygon and $X$ is an interior point such that for every pair of vertices $A, B$, the triangle $XAB$ is isosceles. Prove that all the vertices of $P$ lie on a circle with center $X$.

Let $ABCD$ be a quadrilateral inscribed in a circle $\Gamma$ such that $AB=BC=CD$. Let $M$ and $N$ be the midpoints of $AD$ and $AB$ respectively. The line $CM$ meets $\Gamma$ again at $E$. Prove that the tangent at $E$ to $\Gamma$, the line $AD$ and the line $CN$ are concurrent.

A bored student walks down a hall that contains a row of closed lockers, numbered 1 to 1024. He opens the locker numbered 1, and then alternates between skipping and opening each closed locker thereafter. When he reaches the end of the hall, the student turns around and starts back. He opens the first closed locker he encounters, and then alternates between skipping and opening each closed locker thereafter. The student continues wandering back and forth in this manner until every locker is open. What is the number of the last locker he opens?

A scalene triangle $ABC$ is inscribed within circle $\omega$. The tangent to the circle at point $C$ intersects line $AB$ at point $D$. Let $I$ be the center of the circle inscribed within $\triangle ABC$. Lines $AI$ and $BI$ intersect the bisector of $\angle CDB$ in points $Q$ and $P$, respectively. Let $M$ be the midpoint of $QP$. Prove that $MI$ passes through the middle of arc $ACB$ of circle $\omega$.

A sequence $a_0,a_1,a_2,\cdots,a_n,\cdots$ satisfies that $a_0=3$, and $(3-a_{n-1})(6+a_n)=18$, then the value of $\sum_{i=0}^{n}\frac{1}{a_i}$ is________.

$BE$ is the altitude of acute triangle $ABC$. The line $\ell$ touches the circumscribed circle of the triangle $ABC$ at point $B$. A perpendicular $CF$ is drawn from $C$ on line $\ell$. Prove that the lines $EF$ and $AB$ are parallel.

On a line $\ell$ there exists $3$ points $A, B$, and $C$ where $B$ is located between $A$ and $C$. Let $\Gamma_1, \Gamma_2, \Gamma_3$ be circles with $AC, AB$, and $BC$ as diameter respectively; $BD$ is a segment, perpendicular to $\ell$ with $D$ on $\Gamma_1$. Circles $\Gamma_4, \Gamma_5, \Gamma_6$ and $\Gamma_7$ satisfies the following conditions: $\bullet$ $\Gamma_4$ touches $\Gamma_1, \Gamma_2$, and$ BD$. $\bullet$ $\Gamma_5$ touches $\Gamma_1, \Gamma_3$, and $BD$. $\bullet$ $\Gamma_6$ touches $\Gamma_1$ internally, and touches $\Gamma_2$ and $\Gamma_3$ externally. $\bullet$ $\Gamma_7$ passes through $B$ and the tangent points of $\Gamma_2$ with $\Gamma_6$, and $\Gamma_3$ with $\Gamma_6$. Show that the circles $\Gamma_4, \Gamma_5$, and $\Gamma_7$ are congruent.

Prove that there exists a positive integer $n$, such that the number $k^2+k+n$ does not have a prime divisor less than $2008$ for any integer $k$.

Let $p=2017$ be a prime. Suppose that the number of ways to place $p$ indistinguishable red marbles, $p$ indistinguishable green marbles, and $p$ indistinguishable blue marbles around a circle such that no red marble is next to a green marble and no blue marble is next to a blue marble is $N$. (Rotations and reflections of the same configuration are considered distinct.) Given that $N=p^m\cdot n$, where $m$ is a nonnegative integer and $n$ is not divisible by $p$, and $r$ is the remainder of $n$ when divided by $p$, compute $pm+r$. [i]Proposed by Yannick Yao[/i]

Let $ABC$ be a triangle with incenter $I$ and let $X$, $Y$ and $Z$ be the incenters of the triangles $BIC$, $CIA$ and $AIB$, respectively. Let the triangle $XYZ$ be equilateral. Prove that $ABC$ is equilateral too. [i]Proposed by Mirsaleh Bahavarnia, Iran[/i]