Let $ABC$ be an acute-angled triangle with $AC<BC.$ A circle passes through $A$ and $B$ and crosses the segments $AC$ and $BC$ again at $A_1$ and $B_1$ respectively. The circumcircles of $A_1B_1C$ and $ABC$ meet each other at points $P$ and $C.$ The segments $AB_1$ and $A_1B$ intersect at $S.$ Let $Q$ and $R$ be the reflections of $S$ in the lines $CA$ and $CB$ respectively. Prove that the points $P,$ $Q,$ $R,$ and $C$ are concyclic.

What is the smallest positive integer than can be expressed as the sum of nine consecutive integers, the sum of ten consecutive integers, and the sum of eleven consecutive integers?

Consider a finite sequence $a_1, a_2,...,a_n$ whose terms are natural numbers at most equal to $n$. Determine the maximum number of terms of such a sequence, if you know that every two of its neighboring terms are different and at the same time there is no quartet of terms in it such that $a_p = a_r \ne a_q = a_s$ for $p < q < r < s$.

How many real triples $(x,y,z)$ are there such that $x^4+y^4+z^4+1 = 4xyz$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ \text{Infinitely many} $

A sequence of real numbers $a_1, a_2, a_3, \dots$ satisfies $0 \leq a_1 \leq 1$ and $a_{n+1} = \tfrac{1 + \sqrt{a_n}}{2}$ for all positive integers $n$. If $a_1 + a_{2021} = 1$, then the product $a_1a_2a_3\cdots a_{2020}$ can be written in the form $m^k$, where $k$ is an integer and $m$ is a positive integer that is not divisible by any perfect square greater than $1$. Compute $m + k$.

Tina and Paul are playing a game on a square $S$. First, Tina selects a point $T$ inside $S$. Next, Paul selects a point $P$ inside $S$. Paul then colors blue all the points inside $S$ that are closer to $P$ than $T$ . Tina wins if the blue region thus produced is the interior of a triangle. Assuming that Paul is lazy and simply selects his point at random (and that Tina knows this), find, with proof, a point Tina can select to maximize her probability of winning, and compute this probability.

Let $M$, $N$ and $P$ be three non-collinear points. Construct using straight edge and compass a triangle for which $M$ and $N$ are the midpoints of two of its sides, and $P$ is its orthocenter.

Consider a $4 \times 4$ grid of squares. Aziraphale and Crowley play a game on this grid, alternating turns, with Aziraphale going first. On Aziraphale's turn, he may color any uncolored square red, and on Crowley's turn, he may color any uncolored square blue. The game ends when all the squares are colored, and Aziraphale's score is the area of the largest closed region that is entirely red. If Aziraphale wishes to maximize his score, Crowley wishes to minimize it, and both players play optimally, what will Aziraphale's score be?

In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height $0$. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn: [list] [*] The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller. [*] The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter. [/list] We say that a tree is [i]majestic[/i] if its height is at least $10^6$. Determine the largest $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.

Let $(a_n)_{n\ge 1}$ be a sequence given by $a_1 = 45$ and $$a_n = a^2_{n-1} + 15a_{n-1}$$ for $n > 1$. Prove that the sequence contains no perfect squares.

The sale ad read: "Buy three tires at the regular price and get the fourth tire for $\$3$." Sam paid $\$240$ for a set of four tires at the sale. What was the regular price of one tire? $\text{(A)}\ 59.25\text{ dollars} \qquad \text{(B)}\ 60\text{ dollars} \qquad \text{(C)}\ 70\text{ dollars} \qquad \text{(D)}\ 79\text{ dollars} \qquad \text{(E)}\ 80\text{ dollars}$

Let $ ABC$ be a triangle, and $ H$ its orthocenter. Let $ P$ be a point on the circumcircle of triangle $ ABC$ (distinct from the vertices $ A$, $ B$, $ C$), and let $ E$ be the foot of the altitude of triangle $ ABC$ from the vertex $ B$. Let the parallel to the line $ BP$ through the point $ A$ meet the parallel to the line $ AP$ through the point $ B$ at a point $ Q$. Let the parallel to the line $ CP$ through the point $ A$ meet the parallel to the line $ AP$ through the point $ C$ at a point $ R$. The lines $ HR$ and $ AQ$ intersect at some point $ X$. Prove that the lines $ EX$ and $ AP$ are parallel.

There are three empty jugs on a table. Winnie the Pooh, Rabbit, and Piglet put walnuts in the jugs one by one. They play successively, with the initial determined by a draw. Thereby Winnie the Pooh plays either in the first or second jug, Rabbit in the second or third, and Piglet in the first or third. The player after whose move there are exactly 1999 walnuts loses the games. Show that Winnie the Pooh and Piglet can cooperate so as to make Rabbit lose.

On a social network, no user has more than ten friends ( the state "friendship" is symmetrical). The network is connected: if, upon learning interesting news a user starts sending it to its friends, and these friends to their own friends and so on, then at the end, all users hear about the news. Prove that the network administration can divide users into groups so that the following conditions are met: [list] [*] each user is in exactly one group [*] each group is connected in the above sense [*] one of the groups contains from $1$ to $100$ members and the remaining from $100$ to $900$. [/list]

Prove that the equation $$\sqrt{2-x^2}+\sqrt[3]{3-x^3}=0$$ has no real solutions.

Find all functions $f: (0, \infty) \to (0, \infty)$ such that \begin{align*} f(y(f(x))^3 + x) = x^3f(y) + f(x) \end{align*} for all $x, y>0$. [i]Proposed by Jason Prodromidis, Greece[/i]

Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\omega$ whose diagonals meet at $F$. Lines $AB$ and $CD$ meet at $E$. Segment $EF$ intersects $\omega$ at $X$. Lines $BX$ and $CD$ meet at $M$, and lines $CX$ and $AB$ meet at $N$. Prove that $MN$ and $BC$ concur with the tangent to $\omega$ at $X$. [i]Proposed by Allen Liu[/i]

Given real numbers $a,b,c,d$ such that $\sin{a}+b >\sin{c}+d, a+\sin{b}>c+\sin{d}$, prove that $a+b>c+d$

Let be a function $ f:(0,\infty )\longrightarrow\mathbb{R} . $ [b]a)[/b] Show that if $ f $ is differentiable and $ \lim_{x\to \infty } xf'(x)=1, $ then $ \lim_{x\to\infty } f(x)=\infty .$ [b]b)[/b] Prove that if $ f $ is twice differentiable and $ f''+5f'+6f $ has limit at plus infinity, then: $$ \lim_{x\to\infty } f(x)=\frac{1}{6}\lim_{x\to\infty } \left( f''(x)+5f'(x)+6f(x)\right) $$ [i]Dorel Duca[/i] and [i]Dorian Popa[/i]

Find all pairs of positive integers $ x,y$ satisfying the equation \[ y^x \equal{} x^{50}\]

In the Cartesian plane is given a polygon $P$ whose vertices have integer coordinates and with sides parallel to the coordinate axes. Show that if the length of each edge of $P$ is an odd integer, then the surface of P cannot be partitioned into $2\times 1$ rectangles.

Determine all real polynomials $P$ of degree at most $22$ for which $$kP (k + 1) - (k + 1)P (k) = k^2 + k + 1$$ for all $k \in \{1, 2, 3, . . . , 21, 22\}$.

a) The incircle of a triangle $ABC$ touches $AC$ and $AB$ at points $B_0$ and $C_0$ respectively. The bisectors of angles $B$ and $C$ meet the perpendicular bisector to the bisector $AL$ in points $Q$ and $P$ respectively. Prove that the lines $PC_0, QB_0$ and $BC$ concur. b) Let $AL$ be the bisector of a triangle $ABC$. Points $O_1$ and $O_2$ are the circumcenters of triangles $ABL$ and $ACL$ respectively. Points $B_1$ and $C_1$ are the projections of $C$ and $B$ to the bisectors of angles $B$ and $C$ respectively. Prove that the lines $O_1C_1, O_2B_1,$ and $BC$ concur. c) Prove that the two points obtained in pp. a) and b) coincide.

We call the $main$ $divisors$ of a composite number $n$ the two largest of its natural divisors other than $n$. Composite numbers $a$ and $b$ are such that the main divisors of $a$ and $b$ coincide. Prove that $a=b$.

Let $a,b,c \in \mathbb {R}^{+}$ and $abc=1$ prove that: $\frac {a+b}{(a+b+1)^2}+\frac {b+c}{(b+c+1)^2}+\frac {c+a}{(c+a+1)^2} \geq \frac {2}{a+b+c}$