For each integer $C>1$ decide whether there exist pairwise distinct positive integers $a_1,a_2,a_3,...$ such that for every $k\ge 1$, $a_{k+1}^k$ divides $C^ka_1a_2...a_k$. [i]Proposed by Daniel Liu

(1) Prove that, on the complex plane, the area of the convex hull of all complex roots of $z^{20}+63z+22=0$ is greater than $\pi$. (2) Let $a_1,a_2,\ldots,a_n$ be complex numbers with sum $1$, and $k_1<k_2<\cdots<k_n$ be odd positive integers. Let $\omega$ be a complex number with norm at least $1$. Prove that the equation \[ a_1 z^{k_1}+a_2 z^{k_2}+\cdots+a_n z^{k_n}=w \] has at least one complex root with norm at most $3n|\omega|$.

Determine all $n \in \mathbb{N}$ for which $n^{10} + n^5 + 1$ is prime.

Let $A,B$ be adjacent vertices of a regular $n$-gon ($n\ge5$) with center $O$. A triangle $XYZ$, which is congruent to and initially coincides with $OAB$, moves in the plane in such a way that $Y$ and $Z$ each trace out the whole boundary of the polygon, with $X$ remaining inside the polygon. Find the locus of $X$.

Let $ABC$ be a triangle with circumcircle $c$ and circumcenter $O$, and let $D$ be a point on the side $BC$ different from the vertices and the midpoint of $BC$. Let $K$ be the point where the circumcircle $c_1$ of the triangle $BOD$ intersects $c$ for the second time and let $Z$ be the point where $c_1$ meets the line $AB$. Let $M$ be the point where the circumcircle $c_2$ of the triangle $COD$ intersects $c$ for the second time and let $E$ be the point where $c_2$ meets the line $AC$. Finally let $N$ be the point where the circumcircle $c_3$ of the triangle $AEZ$ meets $c$ again. Prove that the triangles $ABC$ and $NKM$ are congruent.

Suppose that every point in the plane is coloured either black or white. Must there be an equilateral triangle such that all of its vertices are the same colour?

For which positive integer $n$ is the quantity $\frac{n}{3} + \frac{40}{n}$ minimized? [i]Proposed by Eugene Chen[/i]

Lucy starts by writing $s$ integer-valued $2022$-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\mathbf{v}=(v_1,\ldots,v_{2022})$ and $\mathbf{w}=(w_1,\ldots,w_{2022})$ that she has already written, and apply one of the following operations to obtain a new tuple: \begin{align*} \mathbf{v}+\mathbf{w}&=(v_1+w_1,\ldots,v_{2022}+w_{2022}) \\ \mathbf{v} \lor \mathbf{w}&=(\max(v_1,w_1),\ldots,\max(v_{2022},w_{2022})) \end{align*} and then write this tuple on the blackboard. It turns out that, in this way, Lucy can write any integer-valued $2022$-tuple on the blackboard after finitely many steps. What is the smallest possible number $s$ of tuples that she initially wrote?

Let $CH$ be the altitude of triangle ABC, $O$ be the center of the circle circumscribed around it. Point $T$ is the projection of point $C$ on the line $TO$. Prove that the line $TH$ bisects the side $BC$.

Find all pairs of positive integers $(m, n)$ such that $m^2-mn+n^2+1$ divides both numbers $3^{m+n}+(m+n)!$ and $3^{m^3+n^3}+m+n$. [i]Proposed by Dorlir Ahmeti[/i]

Let $n \in \mathbb{N}$, $n \geq 2$, $a_1, a_2, \cdots , a_n \in \mathbb{R}$ and $a_n = max \{a_1, a_2,\cdots , a_n\}$ [list=1] [*]If $t_k$, $t'_k \in \mathbb{R}$, $k \in \{1, 2,\cdots , n\}$ , $t_k \leq t'_k$, for any $k \in \{1, 2, \cdots, n - 1\}$ and $$\sum \limits_{k=1}^nt_k=\sum \limits_{k=1}^nt'_k$$Prove that $$\sum \limits_{k=1}^nt_ka_k\geq \sum \limits_{k=1}^nt'_ka_k$$ [*] If $b_k$, $c_k \in \mathbb{R}^*_+$, $k \in \{1, 2,\cdots , n\}$ , $b_k \leq c_k$ for any $k \in \{1, 2,\cdots, k - 1\}$ and $$b_1b_2\cdots b_n=c_1c_2\cdots c_n$$Prove that $$\prod \limits_{k=1}^n b_k^{a_k}\geq \prod \limits_{k=1}^nc_k^{a_k}$$ [/list]

Let $\Gamma_1$ and $\Gamma_2$ be two circles intersecting at $P$ and $Q$. The common tangent, closer to $P$, of $\Gamma_1$ and $\Gamma_2$ touches $\Gamma_1$ at $A$ and $\Gamma_2$ at $B$. The tangent of $\Gamma_1$ at $P$ meets $\Gamma_2$ at $C$, which is different from $P$, and the extension of $AP$ meets $BC$ at $R$. Prove that the circumcircle of triangle $PQR$ is tangent to $BP$ and $BR$.

Find all positive integers $n$ and $m$ such that $$\dbinom{n}{1} + \dbinom{n}{3} = 2^m.$$

$\textbf{N1.}$ Find all nonnegative integers $a,b,c$ such that \begin{align*} a^2+b^2+c^2-ab-bc-ca = a+b+c \end{align*} [i]Proposed by usjl[/i]

Let $ x$, $ y$, $ z$ be real numbers such that $ 0 < x,y,z < 1$ and $ xyz \equal{} (1 \minus{} x)(1 \minus{} y)(1 \minus{} z)$. Show that at least one of the numbers $ (1 \minus{} x)y,(1 \minus{} y)z,(1 \minus{} z)x$ is greater than or equal to $ \frac {1}{4}$

Let $H$ be the orthocenter of triangle $ABC$, $X$ be an arbitrary point. A circle with a diameter of $XH$ intersects lines $AH, BH, CH$ at points $A_1, B_1, C_1$ for the second time, and lines $AX BX, CX$ at points $A_2, B_2, C_2$. Prove that lines A$_1A_2, B_1B_2, C_1C_2$ intersect at one point.

A line segment $B_0B_n$ is divided into $n$ equal parts at points $B_1,B_2,...,B_{n-1} $ and $A$ is a point such that $\angle B_0AB_n$ is a right angle. Prove that : $$\sum_{k=0}^{n} |AB_k|^{2} = \sum_{k=0}^{n} |B_0B_k|^2$$

Find all the functions $f:(0,+\infty) \to R $ that satisfy the equation $$f(x)f(y)+f\big(\frac{\lambda}{x})f(\frac{\lambda}{y})=2f(xy)$$ for all pairs of $x,y$ real and positive numbers, where $\lambda$ is a positive real number such that $f(\lambda )=1$

A row of 2021 balls is given. Pasha and Vova play a game, taking turns to perform moves; Pasha begins. On each turn a boy should paint a non-painted ball in one of the three available colors: red, yellow, or green (initially all balls are non-painted). When all the balls are colored, Pasha wins, if there are three consecutive balls of different colors; otherwise Vova wins. Who has a winning strategy?

The city is a rectangle divided onto squares by $m$ streets coming from the West to the East and $n$ streets coming from the North to the South. There are militioners (policemen) on the streets but not on the crossroads. They watch the certain automobile, moving along the closed route, marking the time and the direction of its movement. Its trace is not known in advance, but they know, that it will not pass over the same segment of the way twice. What is the minimal number of the militioners providing the unique determination of the route according to their reports?

Find the locus of points inside a trihedral angle such that the sum of their distances from the faces of the trihedral angle has a fixed positive value $a$.

Prove that if the integers $ a $ and $ b $ satisfy the equation $$ 2a^2 + a = 3b^2 + b,$$ then the numbers $ a - b $ and $ 2a + 2b + 1 $ are squares of integers.

The incircle of the triangle $ABC$ touches the sides $AC$ and $BC$ at points $P$ and $Q$ respectively. $N$ and $M$ are the midpoints of $AC$ and $BC$ respectively. Let $X=AM\cap BP, Y=BN\cap AQ$. Given $C,X,Y$ are collinear, prove that $CX$ is the angle bisector of the angle $ACB$. I. Gorodnin

Solve the system of equations $$ \begin{array}{l}<br /> x^2y^2 + x^2z^2 = axyz\\<br /> y^2z^2 + y^2x^2 = bxyz\\<br /> z^2x^2 + z^2y^2 = cxyz.<br /> \end{array}$$

Doug can paint a room in $ 5$ hours. Dave can paint the same room in $ 7$ hours. Doug and Dave paint the room together and take a one-hour break for lunch. Let $ t$ be the total time, in hours, required for them to complete the job working together, including lunch. Which of the following equations is satisfied by $ t$? $ \textbf{(A)}\ \left(\frac{1}{5}\plus{}\frac{1}{7}\right)(t\plus{}1)\equal{}1 \qquad \textbf{(B)}\ \left(\frac{1}{5}\plus{}\frac{1}{7}\right)t\plus{}1\equal{}1 \qquad \textbf{(C)}\ \left(\frac{1}{5}\plus{}\frac{1}{7}\right)t\equal{}1 \\ \textbf{(D)}\ \left(\frac{1}{5}\plus{}\frac{1}{7}\right)(t\minus{}1)\equal{}1 \qquad \textbf{(E)}\ (5\plus{}7)t\equal{}1$