We are given in the plane a line $\ell$ and three circles with centres $A,B,C$ such that they are all tangent to $\ell$ and pairwise externally tangent to each other. Prove that the triangle $ABC$ has an obtuse angle and find all possible values of this this angle. [i]Mircea Becheanu[/i]

Two ducks, Wat and Q, are taking a math test with $1022$ other ducklings. The test has $30$ questions, and the $n$th question is worth $n$ points. The ducks work independently on the test. Wat gets the $n$th problem correct with probability $\frac{1}{n^2}$ while Q gets the $n$th problem correct with probability $\frac{1}{n+1}$. Unfortunately, the remaining ducklings each answer all $30$ questions incorrectly. Just before turning in their test, the ducks and ducklings decide to share answers! On any question which Wat and Q have the same answer, the ducklings change their answers to agree with them. After this process, what is the expected value of the sum of all $1024$ scores? [i]Proposed by Evan Chen[/i]

Inside the acute-angled triangle $ ABC $ we consider the point $ P $ and its projections $ L, M, N $ to the sides $ BC, CA, AB $, respectively. Determine the point $ P $ for which the sum $ |BL|^2 + |CM|^2 + |AN|^2 $ is the smallest.

Prove that $ C_G\left( N_G(H) \right)\subset N_G\left( C_G(H) \right) , $ for any subgroup $ H $ of $ G, $ and characterize the groups $ G $ for which equality in this relation holds for all $ H\le G. $ [i]Here,[/i] $ C_G,N_G $ [i]are the centralizer, respectively, the normalizer of[/i] $ G. $

$S$ is the set of functions $f:\mathbb{N} \to \mathbb{R}$ that satisfy the following conditions: [b]I.[/b] $f(1) = 2$ [b]II.[/b] $f(n+1) \geq f(n) \geq \frac{n}{n + 1} f(2n)$ for $n = 1, 2, \ldots$ Find the smallest $M \in \mathbb{N}$ such that for any $f \in S$ and any $n \in \mathbb{N}, f(n) < M$.

Let $m,n \in Z, m\ne n, m \ne 0, n \ne 0$ . Find all $f: Z \to R$ such that $f(mx+ny)=mf(x)+nf(y)$ for all $x,y \in Z$ .

Let $n\geq 5$ an integer and consider a regular $n$-gon. Initially, Nacho is situated in one of the vertices of the $n$-gon, in which he puts a flag. He will start moving clockwise. First, he moves one position and puts another flag, then, two positions and puts another flag, etcetera, until he finally moves $n-1$ positions and puts a flag, in such a way that he puts $n$ flags in total. ¿For which values of $n$, Nacho will have put a flag in each of the $n$ vertices?

Determine the values of the real parameter $a$, such that the solutions of the system of inequalities $\begin{cases} \log_{\frac{1}{3}}{(3^{x}-6a)}+\frac{2}{\log_{a}{3}}<x-3\\ \log_{\frac{1}{3}}{(3^{x}-18)}>x-5\\ \end{cases}$ form an interval of length $\frac{1}{3}$.

Let $ABCD$ be a square of side length $6$. Points $E$ and $F$ are selected on rays $AB$ and $AD$ such that segments $EF$ and $BC$ intersect at a point $L$, $D$ lies between $A$ and $F$, and the area of $\triangle AEF$ is 36. Clio constructs triangle $PQR$ with $PQ=BL$, $QR=CL$ and $RP=DF$, and notices that the area of $\triangle PQR$ is $\sqrt{6}$. If the sum of all possible values of $DF$ is $\sqrt{m} + \sqrt{n}$ for positive integers $m \ge n$, compute $100m+n$. [i]Based on a proposal by Calvin Lee[/i]

You are given a square $n \times n$. The centers of some of some $m$ of its $1\times 1$ cells are marked. It turned out that there is no convex quadrilateral with vertices at these marked points. For each positive integer $n \geq 3$, find the largest value of $m$ for which it is possible. [i]Proposed by Oleksiy Masalitin, Fedir Yudin[/i]

Prove that there exists an ellipsoid touching all edges of an octahedron if and only if the octahedron's diagonals intersect. (Here an octahedron is a polyhedron consisting of eight triangular faces, twelve edges, and six vertices such that four faces meat at each vertex. The diagonals of an octahedron are the lines connecting pairs of vertices not connected by an edge).

Sebastian has a certain number of rectangles with areas that sum up to 3 and with side lengths all less than or equal to $1$. Demonstrate that with each of these rectangles it is possible to cover a square with side $1$ in such a way that the sides of the rectangles are parallel to the sides of the square. [b]Note:[/b] The rectangles can overlap and they can protrude over the sides of the square.

Let point $A$ be one of the intersections of circles $c$ and $k$. Let $X_1$ and $X_2$ be arbitrary points on circle $c$. Let $Y_i$ denote the intersection of line $AX_i$ and circle $k$ for $i=1,2$. Let $P_1$, $P_2$ and $P_3$ be arbitrary points on circle $k$, and let $O$ denote the center of circle $k$. Let $K_{ij}$ denote the center of circle $(X_iY_iP_j)$ for $i=1,2$ and $j=1,2,3$. Let $L_j$ denote the center of circle $(OK_{1j}K_{2j})$ for $j=1,2,3$. Prove that points $L_1$, $L_2$ and $L_3$ are collinear. Proposed by [i]Vilmos Molnár-Szabó[/i], Budapest

Let $n,k \in \mathbb{Z}^{+}$ with $k \leq n$ and let $S$ be a set containing $n$ distinct real numbers. Let $T$ be a set of all real numbers of the form $x_1 + x_2 + \ldots + x_k$ where $x_1, x_2, \ldots, x_k$ are distinct elements of $S.$ Prove that $T$ contains at least $k(n-k)+1$ distinct elements.

Let $ABC$ be a triangle with $AB < AC$ and let $D{}$ be the other intersection point of the angle bisector of $\angle A$ with the circumcircle of the triangle $ABC$. Let $E{}$ and $F{}$ be points on the sides $AB$ and $AC$ respectively, such that $AE = AF$ and let $P{}$ be the point of intersection of $AD$ and $EF$. Let $M{}$ be the midpoint of $BC{}$. Prove that $AM$ and the circumcircles of the triangles $AEF$ and $PMD$ pass through a common point.

Let $ a,b,c\in\mathbb{R}_+^*, $ and $ f:[0,a]\longrightarrow [0,b] $ bijective and non-decreasing. Prove that: $$ \frac{1}{b}\int_0^a f^2 (x)dx +\frac{1}{a}\int_0^b \left( f^{-1} (x)\right)^2dx\le ab. $$

Find all real polynomials $ f$ with $ x,y \in \mathbb{R}$ such that \[ 2 y f(x \plus{} y) \plus{} (x \minus{} y)(f(x) \plus{} f(y)) \geq 0. \]

Each of the quadratic polynomials $P(x), Q(x)$ and $P(x)+Q(x)$ with real coefficients has a repeated root. Is it guaranteed that those roots coincide? [i]Boris Frenkin[/i]

If $x^2\cos\theta-x(1-x)+(1-x)^2\sin\theta>0$ for all $x\in[0,1]$, find the range value of $\theta$.

Let $ABC$ be an equilateral triangle. From the vertex $A$ we draw a ray towards the interior of the triangle such that the ray reaches one of the sides of the triangle. When the ray reaches a side, it then bounces off following the law of reflection, that is, if it arrives with a directed angle $\alpha$, it leaves with a directed angle $180^{\circ}-\alpha$. After $n$ bounces, the ray returns to $A$ without ever landing on any of the other two vertices. Find all possible values of $n$.

Nick has a $3\times3$ grid and wants to color each square in the grid one of three colors such that no two squares that are adjacent horizontally or vertically are the same color. Compute the number of distinct grids that Nick can create.

Let $ABCD$ be a cyclic quadrilateral. The diagonals $AC$ and $BD$ meet at $P$, and $DA $ and $CB$ meet at $Q$. Suppose $PQ$ is perpendicular to $AC$. Let $E$ be the midpoint of $AB$. Prove that $PE$ is perpendicular to $BC$.

Rick has $7$ books on his shelf: three identical red books, two identical blue books, a yellow book, and a green book. Dave accidentally knocks over the shelf and has to put the books back on in the same order. He knows that none of the red books were next to each other and that the yellow book was one of the first four books on the shelf, counting from the left. If Dave puts back the books according to the rules, but otherwise randomly, what is the probability that he puts the books back correctly?

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy \[f(xf(y) + f(x)) = f(x)f(y) + 2f(x) + f(y) - 1,\] for every $x, y \in \mathbb{R}$, and $f(kx) > kf(x)$ for every $x \in \mathbb{R}$ and $k \in \mathbb{R}$, such that $k > 1$.

$AL$ is internal bisector of scalene $\triangle ABC$ ($L \in BC$). $K$ is chosen on segment $AL$. Point $P$ lies on the same side with respect to line $BC$ as point $A$ such that $\angle BPL = \angle CKL$ and $\angle CPL = \angle BKL$. $M$ is midpoint of segment $KP$, and $D$ is foot of perpendicular from $K$ on $BC$. Prove that $\angle AMD = 180^\circ - |\angle ABC - \angle ACB|$. [i]Proposed by Mykhailo Shtandenko and Fedir Yudin[/i]