Let $\mathbb R$ be the set of real numbers. We denote by $\mathcal F$ the set of all functions $f\colon\mathbb R\to\mathbb R$ such that $$f(x + f(y)) = f(x) + f(y)$$ for every $x,y\in\mathbb R$ Find all rational numbers $q$ such that for every function $f\in\mathcal F$, there exists some $z\in\mathbb R$ satisfying $f(z)=qz$.

Let $p$ be a prime. A [i]complete residue class modulo $p$[/i] is a set containing at least one element equivalent to $k \pmod{p}$ for all $k$. (a) Show that there exists an $n$ such that the $n$th row of Pascal's triangle forms a complete residue class modulo $p$. (b) Show that there exists an $n \le p^2$ such that the $n$th row of Pascal's triangle forms a complete residue class modulo $p$.

Consider the matrices $$A = \left(\begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right) \\ \mbox{ and } \\ B = \left(\begin{matrix} 1 & 0 \\ 2 & 1 \end{matrix}\right).$$ Let $k\geq 1$ an integer. Prove that for any nonzero $i_1,i_2,\dots,i_{k-1},j_1,j_2,\dots,j_k$ and any integers $i_0,i_k$ it holds that $$A^{i_0}B^{j_1}A^{i_1}B^{j_2}\cdots A^{i_{k-1}}B^{i_k}A^{i_k} \not = I.$$

Let $p,q$ be positive reals with sum 1. Show that for any $n$-tuple of reals $(y_1,y_2,...,y_n)$, there exists an $n$-tuple of reals $(x_1,x_2,...,x_n)$ satisfying $$p\cdot \max\{x_i,x_{i+1}\} + q\cdot \min\{x_i,x_{i+1}\} = y_i$$ for all $i=1,2,...,2017$, where $x_{2018}=x_1$.

When $10^{93}-93$ is expressed as a single whole number, the sum of the digits is $\text{(A)}\ 10 \qquad \text{(B)}\ 93 \qquad \text{(C)}\ 819 \qquad \text{(D)}\ 826 \qquad \text{(E)}\ 833$

On the sides of triangle $ ABC$ we consider points $ C_1,C_2 \in (AB), B_1,B_2 \in (AC), A_1,A_2 \in (BC)$ such that triangles $ A_1,B_1,C_1$ and $ A_2B_2C_2$ have a common centroid. Prove that sets $ [A_1,B_1]\cap [A_2B_2], [B_1C_1]\cap[B_2C_2], [C_1A_1]\cap [C_2A_2]$ are not empty.

Let $P(X) = a_n X^n + a_{n-1} X^{n-1} + \cdots + a_1 X + a_0$ be a polynomial with real coefficients such that $0 \leqslant a_i \leqslant a_0$ for $i = 1, 2, \ldots, n$. Prove that, if $P(X)^2 = b_{2n} X^{2n} + b_{2n-1} X^{2n-1} + \cdots + b_{n+1} X^{n+1} + \cdots + b_1 X + b_0$, then $4 b_{n+1} \leqslant P(1)^2$.

Let $ABC$ be a right triangle and isosceles, with $\angle C = 90^o$. Let $M$ be the midpoint of $AB$ and $N$ the midpoint of $AC$. Let $ P$ be such that $MNP$ is an equilateral triangle with $ P$ inside the quadrilateral $MBCN$. Calculate the measure of $\angle CAP$

Let $I$ denote the center of the circle inscribed in the right triangle $ABC$ with right angle at the vertex $A$. Next, denote by $M$ and $N$ the midpoints of the lines $AB$ and $BI$. Prove that the line $CI$ is tangent to the circumscribed circle of triangle $BMN$. (Patrik Bak, Josef Tkadlec)

Find all quadruples $(a,b,c,d)$ of positive integers satisfying that $a^2+b^2=c^2+d^2$ and such that $ac+bd$ divides $a^2+b^2$.

Two logs of length 10 are laying on the ground touching each other. Their radii are 3 and 1, and the smaller log is fastened to the ground. The bigger log rolls over the smaller log without slipping, and stops as soon as it touches the ground again. The volume of the set of points swept out by the larger log as it rolls over the smaller one can be expressed as $n \pi$, where $n$ is an integer. Find $n$.

Let $q{}$ be an arbitrary polynomial with complex coefficients which is not identically $0$ and $\Gamma_q =\{z : |q(z)| = 1\}$ be its contour line. Prove that for every point $z_0\in\Gamma_q$ there is a polynomial $p{}$ for which $|p(z_0)| = 1$ and $|p(z)|<1$ for any $z\in\Gamma_q\setminus\{z_0\}.$

A circle $\omega$ and two points $A, B$ of this circle are given. Let $C$ be an arbitrary point on one of arcs $AB$ of $\omega$; $CL$ be the bisector of triangle $ABC$; the circle $BCL$ meet $AC$ at point $E$; and $CL$ meet $BE$ at point $F$. Find the locus of circumcenters of triangles $AFC$.

The base of a triangle is $80$, and one side of the base angle is $60^\circ$. The sum of the lengths of the other two sides is $90$. The shortest side is: $ \textbf{(A)}\ 45 \qquad\textbf{(B)}\ 40\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 17\qquad\textbf{(E)}\ 12 $

Let $A_1, A_2, ... , A_k$ be the subsets of $\left\{1,2,3,...,n\right\}$ such that for all $1\leq i,j\leq k$:$A_i\cap A_j \neq \varnothing$. Prove that there are $n$ distinct positive integers $x_1,x_2,...,x_n$ such that for each $1\leq j\leq k$: $$lcm_{i \in A_j}\left\{x_i\right\}>lcm_{i \notin A_j}\left\{x_i\right\}$$ [i]Proposed by Morteza Saghafian, Mahyar Sefidgaran[/i]

The letters $P$, $Q$, and $R$ are entered in a $20\times 20$ grid according to the pattern shown below. How many $P$s, $Q$s, and $R$s will appear in the completed table? [asy] usepackage("mathdots"); size(5cm); draw((0,0)--(6,0),linewidth(1.5)+mediumgray); draw((0,1)--(6,1),linewidth(1.5)+mediumgray); draw((0,2)--(6,2),linewidth(1.5)+mediumgray); draw((0,3)--(6,3),linewidth(1.5)+mediumgray); draw((0,4)--(6,4),linewidth(1.5)+mediumgray); draw((0,5)--(6,5),linewidth(1.5)+mediumgray); draw((0,0)--(0,6),linewidth(1.5)+mediumgray); draw((1,0)--(1,6),linewidth(1.5)+mediumgray); draw((2,0)--(2,6),linewidth(1.5)+mediumgray); draw((3,0)--(3,6),linewidth(1.5)+mediumgray); draw((4,0)--(4,6),linewidth(1.5)+mediumgray); draw((5,0)--(5,6),linewidth(1.5)+mediumgray); label(scale(.9)*"\textsf{P}", (.5,.5)); label(scale(.9)*"\textsf{Q}", (.5,1.5)); label(scale(.9)*"\textsf{R}", (.5,2.5)); label(scale(.9)*"\textsf{P}", (.5,3.5)); label(scale(.9)*"\textsf{Q}", (.5,4.5)); label("$\vdots$", (.5,5.6)); label(scale(.9)*"\textsf{Q}", (1.5,.5)); label(scale(.9)*"\textsf{R}", (1.5,1.5)); label(scale(.9)*"\textsf{P}", (1.5,2.5)); label(scale(.9)*"\textsf{Q}", (1.5,3.5)); label(scale(.9)*"\textsf{R}", (1.5,4.5)); label("$\vdots$", (1.5,5.6)); label(scale(.9)*"\textsf{R}", (2.5,.5)); label(scale(.9)*"\textsf{P}", (2.5,1.5)); label(scale(.9)*"\textsf{Q}", (2.5,2.5)); label(scale(.9)*"\textsf{R}", (2.5,3.5)); label(scale(.9)*"\textsf{P}", (2.5,4.5)); label("$\vdots$", (2.5,5.6)); label(scale(.9)*"\textsf{P}", (3.5,.5)); label(scale(.9)*"\textsf{Q}", (3.5,1.5)); label(scale(.9)*"\textsf{R}", (3.5,2.5)); label(scale(.9)*"\textsf{P}", (3.5,3.5)); label(scale(.9)*"\textsf{Q}", (3.5,4.5)); label("$\vdots$", (3.5,5.6)); label(scale(.9)*"\textsf{Q}", (4.5,.5)); label(scale(.9)*"\textsf{R}", (4.5,1.5)); label(scale(.9)*"\textsf{P}", (4.5,2.5)); label(scale(.9)*"\textsf{Q}", (4.5,3.5)); label(scale(.9)*"\textsf{R}", (4.5,4.5)); label("$\vdots$", (4.5,5.6)); label(scale(.9)*"$\dots$", (5.5,.5)); label(scale(.9)*"$\dots$", (5.5,1.5)); label(scale(.9)*"$\dots$", (5.5,2.5)); label(scale(.9)*"$\dots$", (5.5,3.5)); label(scale(.9)*"$\dots$", (5.5,4.5)); label(scale(.9)*"$\iddots$", (5.5,5.6)); [/asy] $\textbf{(A)}~132~\text{Ps}, 134~\text{Qs}, 134~\text{Rs}\qquad\textbf{(B)}~133~\text{Ps}, 133~\text{Qs}, 134~\text{Rs}\qquad\textbf{(C)}~133~\text{Ps}, 134~\text{Qs}, 133~\text{Rs}$\\ $\textbf{(D)}~134~\text{Ps}, 132~\text{Qs}, 134~\text{Rs}\qquad\textbf{(E)}~134~\text{Ps}, 133~\text{Qs}, 133~\text{Rs}\qquad$

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for every $x,y\in\mathbb{R}$ the following equality holds: $$f(xf(y)+x^2+y)=f(x)f(y)+xf(x)+f(y).$$

Ana and Beto play on a grid of $2022 \times 2022$. Ana colors the sides of some squares on the board red, so that no square has two red sides that share a vertex. Next, Bob must color a blue path that connects two of the four corners of the board, following the sides of the squares and not using any red segments. If Beto succeeds, he is the winner, otherwise Ana wins. Who has a winning strategy?

Let $n=x-y^{x-y}$. Find $n$ when $x=2$ and $y=-2$. $\textbf{(A)}\ -14 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 256$

Given a circle with center $O$. Points $A$ and $B$ lie on the circle such that triangle $OBA$ is equilateral. Let $C$ be a point outside the circle with $\angle ACB = 45^{\circ}$. Line $CA$ intersects the circle at point $D$, and the line $CB$ intersects the circle at point $E$. Find $\angle DBE$.

A dot is marked at each vertex of a triangle $ABC$. Then, $2$, $3$, and $7$ more dots are marked on the sides $AB$, $BC$, and $CA$, respectively. How many triangles have their vertices at these dots?

Jeffrey stands on a straight horizontal bridge that measures $20000$ meters across. He wishes to place a pole vertically at the center of the bridge so that the sum of the distances from the top of the pole to the two ends of the bridge is $20001$ meters. To the nearest meter, how long of a pole does Jeffrey need?

Prove that any convex polyhedron with $10n$ faces, has at least $n$ faces with the same number of sides. (A Kanel)

we named a $n*n$ table $selfish$ if we number the row and column with $0,1,2,3,...,n-1$.(from left to right an from up to down) for every {$ i,j\in{0,1,2,...,n-1}$} the number of cell $(i,j)$ is equal to the number of number $i$ in the row $j$. for example we have such table for $n=5$ 1 0 3 3 4 1 3 2 1 1 0 1 0 1 0 2 1 0 0 0 1 0 0 0 0 prove that for $n>5$ there is no $selfish$ table

Sixteen real numbers are arranged in a magic square of side $4$ so that the sum of numbers in each row, column or main diagonal equals $k$. Prove that the sum of the numbers in the four corners of the square is also $k$.