Prove that if $A{}$ is an $n\times n$ matrix with complex entries such that $A+A^*=A^2A^*$ then $A=A^*$. (Here, we denote by $M^*$ the conjugate transpose $\overline{M}^t$ of the matrix $M{}$).

The sum of $n$ terms of an arithmetic progression is $153$, and the common difference is $2$. If the first interm is an integer, and $n>1$, then the number of possible values for $n$ is: $ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6 $

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ a continuous function such that for any $x\in \mathbb{R}$, the limit $\lim_{h\to 0} \left|\frac{f(x+h)-f(x)}{h}\right|$ exists and it is finite. Prove that in any real point, $f$ is differentiable or it has finite one-side derivates, of the same modul, but different signs.

For $n \geq 3$ find the eigenvalues (with their multiplicities) of the $n \times n$ matrix $$\begin{bmatrix} 1 & 0 & 1 & 0 & 0 & 0 & \dots & \dots & 0 & 0\\ 0 & 2 & 0 & 1 & 0 & 0 & \dots & \dots & 0 & 0\\ 1 & 0 & 2 & 0 & 1 & 0 & \dots & \dots & 0 & 0\\ 0 & 1 & 0 & 2 & 0 & 1 & \dots & \dots & 0 & 0\\ 0 & 0 & 1 & 0 & 2 & 0 & \dots & \dots & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 2 & \dots & \dots & 0 & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & & \vdots & \vdots\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & & \ddots & \vdots & \vdots\\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & \dots & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & \dots & 0 & 1 \end{bmatrix}$$

A $2^{2014} + 1$ by $2^{2014} + 1$ grid has some black squares filled. The filled black squares form one or more snakes on the plane, each of whose heads splits at some points but never comes back together. In other words, for every positive integer $n$ greater than $2$, there do not exist pairwise distinct black squares $s_1$, $s_2$, \dots, $s_n$ such that $s_i$ and $s_{i+1}$ share an edge for $i=1,2, \dots, n$ (here $s_{n+1}=s_1$). What is the maximum possible number of filled black squares? [i]Proposed by David Yang[/i]

Find the greatest prime factor of $20!+20!+21!$.

A line with negative slope passing through the point $(18,8)$ intersects the $x$ and $y$ axes at $(a,0)$ and $(0,b)$, respectively. What is the smallest possible value of $a+b$?

Let $A$ be the sequence of zeroes and ones (binary sequence). The sequence can be modified by the following operation: we may pick a block or a contiguous subsequence where there are an unequal number of zeroes and ones, and then flip their order within the block (so block $a_1, a_2, \ldots, a_r$ becomes $a_r, a_{r-1}, \ldots, a_1$). As an example, let $A$ be the sequence $1,1,0,0,1$. We can pick block $1,0,0$ and flip it, so the sequence $1,\boxed{1,0,0},1$ becomes $1,\boxed{0,0,1},1$. However, we cannot pick block $1,1,0,0$ and flip their order since they contain the same number of $1$s and $0$s. Two sequences $A$ and $B$ are called [i]related[/i] if $A$ can be transformed into $B$ using a finite number the operation mentioned above. Determine the largest natural number $n$ for which there exists $n$ different sequences $A_1, A_2, \ldots, A_n$ where each sequence consists of 2022 digits, and for every index $i \neq j$, the sequence $A_i$ is not related to $A_j$.

There are $41$ letters on a circle, each letter is $A$ or $B$. It is allowed to replace $ABA$ by $B$ and conversely, as well as to replace $BAB$ by $A$ and conversely. Is it necessarily true that it is possible to obtain a circle containing a single letter repeating these operations? Maxim Didin

Find all $ x,y,z\in\mathbb{N}_{0}$ such that $ 7^{x} \plus{} 1 \equal{} 3^{y} \plus{} 5^{z}$. [i]Alternative formulation:[/i] Solve the equation $ 1\plus{}7^{x}\equal{}3^{y}\plus{}5^{z}$ in nonnegative integers $ x$, $ y$, $ z$.

If $f(x)$ is a monic quartic polynomial such that $f(-1)=-1$, $f(2)=-4$, $f(-3)=-9$, and $f(4)=-16$, find $f(1)$.

Sixteen chairs are arranged in a row. Eight people each select a chair in which to sit so that no person sits next to two other people. Let $N$ be the number of subsets of $16$ chairs that could be selected. Find the remainder when $N$ is divided by $1000$.

Given a triangle $ABC$, in which the angle $B$ is three times the angle $C$. On the side $AC$, point $D$ is chosen such that the angle $BDC$ is twice the angle $C$. Prove that $BD + BA = AC$.

For each positive integer $n\leq 49$ we define the numbers $a_n = 3n+\sqrt{n^2-1}$ and $b_n=2(\sqrt{n^2+n}+\sqrt{n^2-n})$. Prove that there exist two integers $A,B$ such that \[ \sqrt{a_1-b_1}+\sqrt{a_2-b_2} + \cdots + \sqrt{a_{49}-b_{49}} = A+B\sqrt2. \]

Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$. [i]Proposed by Hojoo Lee, Korea[/i]

Let $\gamma$ be the incircle of $\triangle ABC$ (i.e. the circle inscribed in $\triangle ABC$) and $I$ be the center of $\gamma$. Let $D$, $E$ and $F$ be the feet of the perpendiculars from $I$ to $BC$, $CA$, and $AB$ respectively. Let $D'$ be the point on $\gamma$ such that $DD'$ is a diameter of $\gamma$. Suppose the tangent to $\gamma$ through $D$ intersects the line $EF$ at $P$. Suppose the tangent to $\gamma$ through $D'$ intersects the line $EF$ at $Q$. Prove that $\angle PIQ + \angle DAD' = 180^{\circ}$.