A quadrilateral is called $\textit{innovative}$ if its diagonals divide it into $4$ triangles, having the same sets of angle measures. Find the angle measures of an $\textit{innovative}$ quadrilateral, given that one of its angles has measure $13^{\circ}$.

Find all pairs $(a, b)$ of coprime positive integers, such that $a<b$ and $$b \mid (n+2)a^{n+1002}-(n+1)a^{n+1001}-na^{n+1000}$$for all positive integers $n$.

In every cell of a board $9 \times 9$ is written an integer. For any $k$ numbers in the same row (column), their sum is also in the same row (column). Find the smallest possible number of zeroes in the board for $a)$ $k=5;$ $b)$ $k=8.$

Given is an obtuse isosceles triangle $ABC$ with $CA=CB$ and circumcenter $O$. The point $P$ on $AB$ is such that $AP<\frac{AB} {2}$ and $Q$ on $AB$ is such that $BQ=AP$. The circle with diameter $CQ$ meets $(ABC)$ at $E$ and the lines $CE, AB$ meet at $F$. If $N$ is the midpoint of $CP$ and $ON, AB$ meet at $D$, show that $ODCF$ is cyclic.

Let $p, q$ be coprime integers, such that $|\frac{p} {q}| \leq 1$. For which $p, q$, there exist even integers $b_1, b_2, \ldots, b_n$, such that $$\frac{p} {q}=\frac{1}{b_1+\frac{1}{b_2+\frac{1}{b_3+\ldots}}}? $$

Given is an acute triangle $ABC$ with circumcenter $O$. The point $P$ on $BC$ such that $BP<\frac{BC} {2}$ and the point $Q$ is on $BC$, such that $CQ=BP$. The line $AO$ meets $BC$ at $D$ and $N$ is the midpoint of $AP$. The circumcircle of $(ODQ)$ meets $(BOC)$ at $E$. The lines $NO, OE$ meet $BC$ at $K, F$. Show that $AOKF$ is cyclic.

Find all positive integers $k$, so that there exists a polynomial $f(x)$ with rational coefficients, such that for all sufficiently large $n$, $$f(n)=\text{lcm}(n+1, n+2, \ldots, n+k).$$

In every cell of a board $101 \times 101$ is written a positive integer. For any choice of $101$ cells from different rows and columns, their sum is divisible by $101$. Show that the number of ways to choose a cell from each row of the board, so that the total sum of the numbers in the chosen cells is divisible by $101$, is divisible by $101$.

A quadruplet of distinct positive integers $(a, b, c, d)$ is called $k$-good if the following conditions hold: 1. Among $a, b, c, d$, no three form an arithmetic progression. 2. Among $a+b, a+c, a+d, b+c, b+d, c+d$, there are $k$ of them, forming an arithmetic progression. $a)$ Find a $4$-good quadruplet. $b)$ What is the maximal $k$, such that there is a $k$-good quadruplet?

The points $A_1, B_1, C_1$ are chosen on the sides $BC, CA, AB$ of a triangle $ABC$ so that $BA_1=BC_1$ and $CA_1=CB_1$. The lines $C_1A_1$ and $A_1B_1$ meet the line through $A$, parallel to $BC$, at $P, Q$. Let the circumcircles of the triangles $APC_1$ and $AQB_1$ meet at $R$. Given that $R$ lies on $AA_1$, show that $R$ lies on the incircle of $ABC$.

Find the smallest possible number of divisors a positive integer $n$ may have, which satisfies the following conditions: 1. $24 \mid n+1$; 2. The sum of the squares of all divisors of $n$ is divisible by $48$ ($1$ and $n$ are included).

Let $G$ be a complete bipartite graph with partition sets $A$ and $B$ of sizes $km$ and $kn$, respectively. The edges of $G$ are colored in $k$ colors. Prove that there exists a monochromatic connected component with at least $m+n$ vertices (which means that there exists a color and a set of vertices, such that between any two of them, there is a path consisting of edges only in that color).

Let $x_0, x_1, \ldots$ be a sequence of real numbers such that $x_0=1$ and $x_{n+1}=\sin(x_n)+\frac{\pi} {2}-1$ for all $n \geq 0$. Show that the sequence converges and find its limit.

Given is an acute triangle $ABC$ with incenter $I$ and the incircle touches $BC, CA, AB$ at $D, E, F$. The circle with center $C$ and radius $CE$ meets $EF$ for the second time at $K$. If $X$ is the $C$-excircle touchpoint with $AB$, show that $CX, KD, IF$ concur.

Solve in positive integers the equation $$m^{\frac{1}{n}}+n^{\frac{1}{m}}=2+\frac{2}{mn(m+n)^{\frac{1}{m}+\frac{1}{n}}}.$$

A set of points in the plane is called $\textit{good}$ if the distance between any two points in it is at most $1$. Let $f(n, d)$ be the largest positive integer such that in any $\textit{good}$ set of $3n$ points, there is a circle of diameter $d$, which contains at least $f(n, d)$ points. Prove that there exists a positive real $\epsilon$, such that for all $d \in (1-\epsilon, 1)$, the value of $f(n, d)$ does not depend on $d$ and find that value as a function of $n$.