Janson found $2025$ dogs on a circle. Janson wants to select some (possibly none) of the dogs to take home, such that no two selected dogs have exactly two dogs (whether selected or not) in between them. Let $S_{1}$ be the number of ways for him to do so. Ivan also found $2025$ cats on a circle. Ivan wants to select some (possibly none) of the cats to take home, such that no two selected cats have exactly five cats (whether selected or not) in between them. Let $S_{2}$ be the number of ways for him to do so. a) Prove that $S_{1}=S_{2}$. b) Prove that $S_{1}$ and $S_{2}$ are both perfect cubes.

Points $ A$ and $ B$ are on a circle of radius $ 5$ and $ AB\equal{}6$. Point $ C$ is the midpoint of the minor arc $ AB$. What is the length of the line segment $ AC$? $ \textbf{(A)}\ \sqrt{10} \qquad \textbf{(B)}\ \frac{7}{2} \qquad \textbf{(C)}\ \sqrt{14} \qquad \textbf{(D)}\ \sqrt{15} \qquad \textbf{(E)}\ 4$

Define the function $f:(0,1)\to (0,1)$ by \[\displaystyle f(x) = \left\{ \begin{array}{lr} x+\frac 12 & \text{if}\ \ x < \frac 12\\ x^2 & \text{if}\ \ x \ge \frac 12 \end{array} \right.\] Let $a$ and $b$ be two real numbers such that $0 < a < b < 1$. We define the sequences $a_n$ and $b_n$ by $a_0 = a, b_0 = b$, and $a_n = f( a_{n -1})$, $b_n = f (b_{n -1} )$ for $n > 0$. Show that there exists a positive integer $n$ such that \[(a_n - a_{n-1})(b_n-b_{n-1})<0.\] [i]Proposed by Denmark[/i]

Determine all three-digit numbers $N$ having the property that $N$ is divisible by 11, and $\dfrac{N}{11}$ is equal to the sum of the squares of the digits of $N$.

Let $a,b,c$ be positive real numbers, that satisfy $abc=1$. Prove the inequality: $$\frac{ab}{1+c}+\frac{bc}{1+a}+\frac{ca}{1+b} \geq \frac{27}{(a+b+c)(3+a+b+c)}$$

Let $n$ be a positive integer. Let $S$ be the set of $n^2$ cells in an $n\times n$ grid. Call a subset $T$ of $S$ a [b]double staircase [/b] if [list] [*] $T$ can be partitioned into $n$ horizontal nonoverlapping rectangles of dimensions $1 \times 1, 1 \times 2, ..., 1 \times n,$ and [*]$T$ can also be partitioned into $n$ vertical nonoverlapping rectangles of dimensions $1\times1, 2 \times 1, ..., n \times 1$. [/list] In terms of $n$, how many double staircases are there? (Rotations and reflections are considered distinct.) An example of a double staircase when $n = 3$ is shown below. [asy] unitsize(1cm); for (int i = 0; i <= 3; ++i) { draw((0,i)--(3,i),linewidth(0.2)); draw((i,0)--(i,3),linewidth(0.2)); } filldraw((0,0)--(1,0)--(1,1)--(0,1)--cycle, lightgray, linewidth(0.2)); filldraw((1,0)--(2,0)--(2,1)--(1,1)--cycle, lightgray, linewidth(0.2)); filldraw((2,0)--(3,0)--(3,1)--(2,1)--cycle, lightgray, linewidth(0.2)); filldraw((0,1)--(1,1)--(1,2)--(0,2)--cycle, lightgray, linewidth(0.2)); filldraw((1,1)--(2,1)--(2,2)--(1,2)--cycle, lightgray, linewidth(0.2)); filldraw((1,2)--(2,2)--(2,3)--(1,3)--cycle, lightgray, linewidth(0.2)); [/asy]

Let $\alpha \in \mathbb{C}\setminus \{0\}$ and $A \in \mathcal{M}_n(\mathbb{C})$, $A \neq O_n$, be such that $$A^2 + (A^*)^2 = \alpha A\cdot A^*,$$ where $A^* = (\bar A)^T.$ Prove that $\alpha \in \mathbb{R}$, $|\alpha| \le 2$. and $A\cdot A^* = A^*\cdot A.$

Let $P(x)$ a grade 3 polynomial such that: $$P(1)=1, P(2)=4, P(3)=9$$ Find the value of $P(10)+P(-6)$

If a polygon has both an inscribed circle and a circumscribed circle, then define the [i]halo[/i] of that polygon to be the region inside the circumcircle but outside the incircle. In particular, all regular polygons and all triangles have halos. (a) What is the area of the halo of a square with side length 2? (b) What is the area of the halo of a 3-4-5 right triangle? (c) What is the area of the halo of a regular 2016-gon with side length 2?

Alex the Kat plays the following game. First, he writes the number $27000$ on a blackboard. Each minute, he erases the number on the blackboard and replaces it with a number chosen uniformly randomly from its positive divisors, including itself. Find the probability that, after $2019$ minutes, the number on the blackboard is $1$.

Let $ABIH$,$BDEC$ and $ACFG$ be arbitrary rectangles constructed (externally) on the sides of triangle $ABC$.Choose point $S$ outside rectangle $ABIH$ (on the opposite side as triangle $ABC$) such that $\angle SHI=\angle FAC$ and $\angle HIS=\angle EBC$.Prove that the lines $FI,EH$ and $CS$ are concurrent(i.e., the three lines intersect in one point).

Consider $\triangle ABC$. Choose a point $M$ on side $BC$ and let $O$ be the center of the circle passing through the vertices of $\triangle ABM$. Let $k$ be the circle that passes through $A$ and $M$ and whose center lies on $BC$. Let line $MO$ intersect $K$ again in point $K$. Prove that the line $BK$ is the same for any point $M$ on segment $BC$, so long as all of these constructions are well-defined. [i]Proposed by Evan Chen[/i]

$5.$Let x be a real number with $0<x<1$ and let $0.c_1c_2c_3...$ be the decimal expansion of x.Denote by $B(x)$ the set of all subsequences of $c_1c_2c_3$ that consist of 6 consecutive digits. For instance , $B(\frac{1}{22})={045454,454545,545454}$ Find the minimum number of elements of $B(x)$ as $x$ varies among all irrational numbers with $0<x<1$

Consider some graph $G$ with $2019$ nodes. Let's define [i]inverting[/i] a vertex $v$ the following process: for every other vertex $u$, if there was an edge between $v$ and $u$, it is deleted, and if there wasn't, it is added. We want to minimize the number of edges in the graph by several [i]invertings[/i] (we are allowed to invert the same vertex several times). Find the smallest number $M$ such that we can always make the number of edges in the graph not larger than $M$, for any initial choice of $G$. [i]Proposed by Arsenii Nikolaev, Anton Trygub (Ukraine)[/i]

Find all four-digit numbers such that when decomposed into prime factors, each number has the sum of its prime factors equal to the sum of the exponents.

Determine whether there exist $100$ distinct lines in the plane having exactly $1985$ distinct points of intersection

Let $\theta_i \in \left ( 0,\frac{\pi}{4} \right ]$ for $i=1,2,3,4$. Prove that: $\tan \theta _1 \tan \theta _2 \tan \theta _3 \tan \theta _4 \le (\frac{\sin^8 \theta _1+\sin^8 \theta _2+\sin^8 \theta _3+\sin^8 \theta _4}{\cos^8 \theta _1+\cos^8 \theta _2+\cos^8 \theta _3+\cos^8 \theta _4})^\frac{1}{2}$ [hide=edit]@below, fixed now. There were some problems (weird characters) so aops couldn't send it.[/hide]

Let be given $ n \ge 2$ lines on a plane, not all concurrent and no two parallel. Prove that there is a point which belongs to exactly two of the given lines.

Given are four circles $\Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4$. Circles $\Gamma_1$ and $\Gamma_2$ are externally tangent at point $A$. Circles $\Gamma_2$ and $\Gamma_3$ are externally tangent at point $B$. Circles $\Gamma_3$ and $\Gamma_4$ are externally tangent at point $C$. Circles $\Gamma_4$ and $\Gamma_1$ are externally tangent at point $D$. Prove that $ABCD$ is cyclic.

Write down $1, 2, 3, ... , 2015$ in a row on a whiteboard. Every minute, select a pair of adjacent numbers at random, erase them, and insert their sum where you selected the numbers. (For instance, selecting $3$ and $4$ from $1, 2, 3, 4, 5$ would result in $1, 2, 7, 5$.) Repeat this process until you have two numbers remaining. What is the probability that the smaller number is less than or equal to $2015$?

Compute $\sqrt{(31)(30)(29)(28)+1}$.

Determine whether it's possible to cover a $K_{2012}$ with a) 1000 $K_{1006}$'s; b) 1000 $K_{1006,1006}$'s. [i]David Yang.[/i]

(a) Find out, whether there exists a differentiable function $f: R \longrightarrow R$, not equaling $0$ for all $x \in R$, satisfying the conditions $2f(f(x)) = f(x) \geq 0$ for all $x \in R$. (b) Find out, whether there exists a differentiable function $f: R \longrightarrow R$, not equaling $0$ for all $x \in R$, satisfying the conditions $-1 \leq 2f(f(x)) = f(x) \leq 1$ for all $x \in R$.

Victor writes down all $7-$digit numbers using the digits $1, 2, 3, 4, 5, 6,$ and $7$ exactly once. Prove that there are no two numbers among them where one is a multiple of the other.

Find the value of $a$ satisfying \begin{align*} a+b&=3\\ b+c&=11\\ c+a&=61 \end{align*}