Circle $\omega_1$ is centered at $O_1$ with radius $3$, and circle $\omega_2$ is centered at $O_2$ with radius $2$. Line $\ell$ is tangent to $\omega_1$ and $\omega_2$ at $X$, $Z$, respectively, and intersects segment $\overline{O_1O_2}$ at $Y$ . The circle through $O_1$, $X$, $Y$ has center $O_3$, and the circle through $O_2$, $Y$ , $Z$ has center $O_4$. Given that $O_1O_2 = 13$, find $O_3O_4$.

Find $(\log_2 x)^2$ if $\log_2 (\log_8 x) = \log_8 (\log_2 x)$.

Find all Arithmetic progressions $a_{1},a_{2},...$ of natural numbers for which there exists natural number $N>1$ such that for every $k\in \mathbb{N}$: $a_{1}a_{2}...a_{k}\mid a_{N+1}a_{N+2}...a_{N+k}$

Find all polynomials $f(x)=x^3+bx^2+cx+d$, where $b,c,d,$ are real numbers, such that $f(x^2-2)=-f(-x)f(x)$.

For positive integers $n$ and $k \geq 2$, define $E_k(n)$ as the greatest exponent $r$ such that $k^r$ divides $n!$. Prove that there are infinitely many $n$ such that $E_{10}(n) > E_9(n)$ and infinitely many $m$ such that $E_{10}(m) < E_9(m)$.

In a group of $2n$ students, each student has exactly $3$ friends within the group. The friendships are mutual and for each two students $A$ and $B$ which are not friends, there is a sequence $C_1, C_2, ..., C_r$ of students such that $A$ is a friend of $C_1$, $C_1$ is a friend of $C_2$, et cetera, and $C_r$ is a friend of $B$. Every student was asked to assess each of his three friendships with: "acquaintance", "friend" and "BFF". It turned out that each student either gave the same assessment to all of his friends or gave every assessment exactly once. We say that a pair of students is in conflict if they gave each other different assessments. Let $D$ be the set of all possible values of the total number of conflicts. Prove that $|D| \geq 3n$ with equality if and only if the group can be partitioned into two subsets such that each student is separated from all of his friends.

Given two monic polynomials $P(x)$ and $Q(x)$ with degrees 2016. $P(x)=Q(x)$ has no real root. [b]Prove that P(x)=Q(x+1) has at least one real root.[/b]

[b]p1.[/b] A straight line is painted in two colors. Prove that there are three points of the same color such that one of them is located exactly at the midpoint of the interval bounded by the other two. [b]p2.[/b] Find all positive integral solutions $x, y$ of the equation $xy = x + y + 3$. [b]p3.[/b] Can one cut a square into isosceles triangles with angle $80^o$ between equal sides? [b]p4.[/b] $20$ children are grouped into $10$ pairs: one boy and one girl in each pair. In each pair the boy is taller than the girl. Later they are divided into pairs in a different way. May it happen now that (a) in all pairs the girl is taller than the boy; (b) in $9$ pairs out of $10$ the girl is taller than the boy? [b]p5.[/b] Mr Mouse got to the cellar where he noticed three heads of cheese weighing $50$ grams, $80$ grams, and $120$ grams. Mr. Mouse is allowed to cut simultaneously $10$ grams from any two of the heads and eat them. He can repeat this procedure as many times as he wants. Can he make the weights of all three pieces equal? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Ruby has a non-negative integer $n$. In each second, Ruby replaces the number she has with the product of all its digits. Prove that Ruby will eventually have a single-digit number or $0$. (e.g. $86\rightarrow 8\times 6=48 \rightarrow 4 \times 8 =32 \rightarrow 3 \times 2=6$) [i]Proposed by Wong Jer Ren[/i]

Points $K$ and $N$ are the midpoints of sides $AC$ and $AB$ of triangle $ABC$. The inscribed circle $\omega$ of the triangle $AKN$ is tangent to $BC$. Find $BC$ if $AC + AB = n$. (Oleksii Karliuchenko)

Find all solutions of the following equation in integers $x,y: x^4+ y= x^3+ y^2$

Find all integers $a$ and $b$ satisfying \begin{align*} a^6 + 1 & \mid b^{11} - 2023b^3 + 40b, \qquad \text{and}\\ a^4 - 1 & \mid b^{10} - 2023b^2 - 41. \end{align*}

Given an acute triangle $ ABC$. The incircle of triangle $ ABC$ touches $ BC,CA,AB$ respectively at $ D,E,F$. The angle bisector of $ \angle A$ cuts $ DE$ and $ DF$ respectively at $ K$ and $ L$. Suppose $ AA_1$ is one of the altitudes of triangle $ ABC$, and $ M$ be the midpoint of $ BC$. (a) Prove that $ BK$ and $ CL$ are perpendicular with the angle bisector of $ \angle BAC$. (b) Show that $ A_1KML$ is a cyclic quadrilateral.

For a positive integer $n$, $S_n$ is the set of positive integer $n$-tuples $(a_1,a_2, \cdots ,a_n)$ which satisfies the following. (i). $a_1=1$. (ii). $a_{i+1} \le a_i+1$. For $k \le n$, define $N_k$ as the number of $n$-tuples $(a_1, a_2, \cdots a_n) \in S_n$ such that $a_k=1, a_{k+1}=2$. Find the sum $N_1 + N_2+ \cdots N_{k-1}$.

Let $ABC$ arbitrary triangle ($AB \neq BC \neq AC \neq AB$) And O,I,H it's circum-center, incenter and ortocenter (point of intersection altitudes). Prove, that 1) $\angle OIH > 90^0$(2 points) 2)$\angle OIH >135^0$(7 points) balls for 1) and 2) not additive.

A set of points in the plane is called [i]sane[/i] if no three points are collinear and the angle between any three distinct points is a rational number of degrees. (a) Does there exist a countably infinite sane set $\mathcal{P}$? (b) Does there exist an uncountably infinite sane set $\mathcal{Q}$? [i]Proposed by Tony Wang[/i]

Find all positive integers $n$, such that the number \[ \frac{n^{2021}+101}{n^2+n+1}\] is an integer.

Prove that $$\sum\limits_{n = 1}^{\infty}\frac{1}{\sqrt{n}\left(n+1\right)} &lt; 2.$$ Proposed by Ivan Krijan, University of Zagreb

Evaluate $\lim_{x \to +\infty}\sqrt{x^2+x+1}\frac{x-ln(e^x+x)}{x}$.

Calculate the integral \[\int_{-\infty}^{+\infty}e^{ikx}\frac{1-e^x}{1+e^x}dx\]

Let $s_1, s_2, s_3$ be the respective sums of $n$, $2n$, $3n$ terms of the same arithmetic progression with $a$ as the first term and $d$ as the common difference. Let $R=s_3-s_2-s_1$. Then $R$ is dependent on: $ \textbf{(A)}\ a \text{ } \text{and} \text{ } d\qquad\textbf{(B)}\ d \text{ } \text{and} \text{ } n\qquad\textbf{(C)}\ a \text{ } \text{and} \text{ } n\qquad\textbf{(D)}\ a, d, \text{ } \text{and} \text{ } n\qquad$ $\textbf{(E)}\ \text{neither} \text{ } a \text{ } \text{nor} \text{ } d \text{ } \text{nor} \text{ } n $

Let $S$ be a finite set and $P$ the set of all subsets of $S$. Show that one can label the elements of $P$ as $A_i$ such that (1) $A_1 =\emptyset$. (2) For each $n\geq1 $ we either have $A_{n-1}\subset A_{n}$ and $|A_{n} \setminus A_{n-1}|=1$ or $A_{n}\subset A_{n-1}$ and $|A_{n-1} \setminus A_{n}|=1.$

Given an isocleos triangle $ABC$ with equal sides $AB=AC$ and incenter $I$.Let $\Gamma_1$be the circle centered at $A$ with radius $AB$,$\Gamma_2$ be the circle centered at $I$ with radius $BI$.A circle $\Gamma_3$ passing through $B,I$ intersects $\Gamma_1$,$\Gamma_2$ again at $P,Q$ (different from $B$) respectively.Let $R$ be the intersection of $PI$ and $BQ$.Show that $BR \perp CR$.

$$\int\frac{\ln^2(x)}{x}\,dx$$ [i]Proposed by Connor Gordon[/i]

Circle $B$, which has radius 2008, is tangent to horizontal line $A$ at point $P$. Circle $C_1$ has radius 1 and is tangent both to circle $B$ and to line $A$ at a point to the right of point $P$. Circle $C_2$ has radius larger than 1 and is tangent to line $A$ and both circles B and $C_1$. For $n>1$, circle $C_n$ is tangent to line $A$ and both circles $B$ and $C_{n-1}$. Find the largest value of n such that this sequence of circles can be constructed through circle $C_n$ where the n circles are all tangent to line $A$ at points to the right of $P$. [asy] size(300); draw((-10,0)--(10,0)); draw(arc((0,10),10,210,330)); label("$P$",(0,0),S); pair C=(0,10),X=(12,3); for(int kk=0;kk<6;++kk) { pair Y=(X.x-X.y,X.y); for(int k=0;k<20;++k) Y+=(abs(Y-X)-X.y-Y.y,abs(Y-C)-10-Y.y)/3; draw(circle(Y,Y.y)); X=Y; }[/asy]