There are $169$ non-zero digits written around a circle. Prove that they can be split into $14$ non-empty blocks of consecutive digits so that among the $14$ natural numbers formed by the digits in those blocks, at least $13$ of them are divisible by $13$ (the digits in each block are read in clockwise direction).

Solve in positive integers the equation $10^{a}+2^{b}-3^{c}=1997$.

Let $j$ and $k$ be integers. Prove that the inequality $$\lfloor(j+k)\alpha\rfloor+\lfloor(j+k)\beta\rfloor\ge\lfloor j\alpha\rfloor+\lfloor j\beta\rfloor+\lfloor k(\alpha+\beta)\rfloor$$holds for all real numbers $\alpha,\beta$ if and only if $j=k$.

Determine the set of integers $n$ for which $n^2+19n+92$ is a square.

Prove that there are infinitely many integers $k\geqslant 2024$ for which there exists a set $\{a_1,\ldots,a_k\}$ with the following properties:[list] [*]$a_1{}$ is a positive integer and $a_{i+1}=a_i+1$ for all $1\leqslant i<k,$ and [*]$2(a_1\cdots a_{k-2}-1)^2$ is divisible by $2(a_1+\cdots+a_k)+a_1-a_1^2.$ [/list][i](Proposed by Prajit Adhikari, Nepal)[/i]

Let $AB$ be a segment of a plane. Is it possible to paint the plane in $2009$ colors in such a way that both of the following conditions are satisfied? 1) Every two points of the same color can be connected by a polygonal line. 2) For any point $C$ of $AB$, every $n \in N$ and every $k\in \{1,2,3,...,2009\}$ , there exists a point $D$, painted in $k$-th color such that the length of $CD$ is less than $0,0...01$, where all the zeros after the decimal point are exactly $n$.

A $6 \times 6$ board is given such that each unit square is either red or green. It is known that there are no $4$ adjacent unit squares of the same color in a horizontal, vertical, or diagonal line. A $2 \times 2$ subsquare of the board is [i]chesslike[/i] if it has one red and one green diagonal. Find the maximal possible number of chesslike squares on the board. [i]Proposed by Nikola Velov[/i]

Given points $D$ and $E$ on the legs $[CA]$ and $[CB]$, respectively, of the isosceles right triangle. $|CD| = |CE|$. The extensions of the perpendiculars from $D$ and $C$ to the line $AE$ cross the hypotenuse $AB$ in the points $K$ and $L$. Prove that $|KL| = |LB|$

For every positive integer $n$, let \[h(n) = 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}.\] For example, $h(1) = 1$, $h(2) = 1+\frac{1}{2}$, $h(3) = 1+\frac{1}{2}+\frac{1}{3}$. Prove that for $n=2,3,4,\ldots$ \[n+h(1)+h(2)+h(3)+\cdots+h(n-1) = nh(n).\]

Find all positive integers $n$ for which there exist non-negative integers $a_1, a_2, \ldots, a_n$ such that \[ \frac{1}{2^{a_1}} + \frac{1}{2^{a_2}} + \cdots + \frac{1}{2^{a_n}} = \frac{1}{3^{a_1}} + \frac{2}{3^{a_2}} + \cdots + \frac{n}{3^{a_n}} = 1. \] [i]Proposed by Dusan Djukic, Serbia[/i]

Let $x_1, \ldots , x_{100}$ be nonnegative real numbers such that $x_i + x_{i+1} + x_{i+2} \leq 1$ for all $i = 1, \ldots , 100$ (we put $x_{101 } = x_1, x_{102} = x_2).$ Find the maximal possible value of the sum $S = \sum^{100}_{i=1} x_i x_{i+2}.$ [i]Proposed by Sergei Berlov, Ilya Bogdanov, Russia[/i]

Find all nonnegative real numbers $ x$ for which $ \sqrt[3]{13\plus{}\sqrt{x}}\plus{}\sqrt[3]{13\minus{}\sqrt{x}}$ is an integer.

A regular 1001-gon is drawn on a board, the vertiecs of which are numbered with $1,2,\ldots,1001.$ Is it possible to label the vertices of a cardboard 1001-gon with the numbers $1,2,\ldots,1001$ such that for any overlap between the two 1001-gons, there are two vertices with the same number one over the other? Note that the cardboard polygon can be inverted.

Solve the system of equations \[ |a_1-a_2|x_2+|a_1-a_3|x_3+|a_1-a_4|x_4=1 \] \[ |a_2-a_1|x_1+|a_2-a_3|x_3+|a_2-a_4|x_4=1 \] \[ |a_3-a_1|x_1+|a_3-a_2|x_2+|a_3-a_4|x_4=1 \] \[ |a_4-a_1|x_1+|a_4-a_2|x_2+|a_4-a_3|x_3=1 \] where $a_1, a_2, a_3, a_4$ are four different real numbers.

Let $p$ be an odd prime number. For every integer $a,$ define the number $S_a = \sum^{p-1}_{j=1} \frac{a^j}{j}.$ Let $m,n \in \mathbb{Z},$ such that $S_3 + S_4 - 3S_2 = \frac{m}{n}.$ Prove that $p$ divides $m.$ [i]Proposed by Romeo Meštrović, Montenegro[/i]

In convex quadrilateral $ABCD, \angle BAC=\angle CAD.$ $E$ lies on segment $CD$, and $BE$ and $AC$ intersect at $F,$ $DF$ and $BC$ intersect at $G.$ Prove that $\angle GAC=\angle EAC.$

Let $ABCD$ be a cyclic quadrilateral with center $O$ such that $BD$ bisects $AC.$ Suppose that the angle bisector of $\angle ABC$ intersects the angle bisector of $\angle ADC$ at a single point $X$ different than $B$ and $D.$ Prove that the line passing through the circumcenters of triangles $XAC$ and $XBD$ bisects the segment $OX.$ [i]Proposed by Viktor Ahmeti and Leart Ajvazaj, Kosovo[/i]

Suppose that $ V$ is a locally compact topological space that admits no countable covering with compact sets. Let $ \textbf{C}$ denote the set of all compact subsets of the space $ V$ and $ \textbf{U}$ the set of open subsets that are not contained in any compact set. Let $ f$ be a function from $ \textbf{U}$ to $ \textbf{C}$ such that $ f(U)\subseteq U$ for all $ U \in \textbf{U}$. Prove that either (i) there exists a nonempty compact set $ C$ such that $ f(U)$ is not a proper subset of $ C$ whenever $ C \subseteq U \in \textbf{U}$, (ii) or for some compact set $ C$, the set \[ f^{-1}(C)= \bigcup \{U \in \textbf{U}\;: \ \;f(U)\subseteq C\ \}\] is an element of $ \textbf{U}$, that is, $ f^{-1}(C)$ is not contained in any compact set. [i]A. Mate[/i]

For a natural number $n,$ let $a_n$ be the sum of all products $xy$ over all integers $x$ and $y$ with $1 \leq x < y \leq n.$ For example, $a_3 = 1\cdot2 + 2\cdot3 + 1\cdot3 = 11.$ Determine the smallest $n \in \mathbb{N}$ such that $n > 1$ and $a_n$ is a multiple of $2020.$

There is $1$ integer in between $300$ and $400$ (base $10$) inclusive such that its last digit is $7$ when written in bases $8$, $10$, and $12$. Find this integer, in base $10$. [i]2015 CCA Math Bonanza Individual Round #9[/i]

Let $p$ be a prime number. Determine the largest possible $n$ such that the following holds: it is possible to fill an $n\times n$ table with integers $a_{ik}$ in the $i$th row and $k$th column, for $1\le i,k\le n$, such that for any quadruple $i,j,k,l$ with $1\le i<j\le n$ and $1\le k<l\le n$, the number $a_{ik}a_{jl}-a_{il}a_{jk}$ is not divisible by $p$. [i]Proposed by oneplusone[/i]

In some computer languages (such as APL), when there are no parentheses in an algebraic expression, the operations are grouped from left to right. Thus, $ a \times b \minus{} c$ in such languages means the same as $ a(b\minus{}c)$ in ordinary algebraic notation. If $ a \div b \minus{} c \plus{} d$ is evaluated in such a language, the result in ordinary algebraic notation would be $ \textbf{(A)}\ \frac{a}{b} \minus{} c \plus{} d \qquad \textbf{(B)}\ \frac{a}{b} \minus{} c \minus{} d \qquad \textbf{(C)}\ \frac{d \plus{} c \minus{} b}{a} \qquad \textbf{(D)}\ \frac{a}{b \minus{} c \plus{} d} \qquad \textbf{(E)}\ \frac{a}{b\minus{}c\minus{}d}$

Let $ABC$ be a triangle with incenter $I$, incircle $\gamma$ and circumcircle $\Gamma$. Let $M$, $N$, $P$ be the midpoints of sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ and let $E$, $F$ be the tangency points of $\gamma$ with $\overline{CA}$ and $\overline{AB}$, respectively. Let $U$, $V$ be the intersections of line $EF$ with line $MN$ and line $MP$, respectively, and let $X$ be the midpoint of arc $\widehat{BAC}$ of $\Gamma$. Given that $AB = 5$, $AC = 8$, and $\angle A = 60^{\circ}$, compute the area of triangle $XUV$.

Consider the set of grid points $(m,n)$ in the plane, $m,n$ integers. Let $\sigma$ be a finite subset and define \[S(\sigma)=\sum_{(m,n)\in\sigma}(100-|m|-|n|) \] Find the maximum of $S$, taken over the set of all such subsets $\sigma$.

Compute the smallest positive integer $x$ which satisfies $x^2-8x+1\equiv0\pmod{22}$ and $x^2-22x+1\equiv0\pmod{8}$.