A positive integer is called [i]monotone[/i] if has at least two digits and all its digits are nonzero and appear in a strictly increasing or strictly decreasing order. (a) Compute the sum of all monotone five-digit numbers. (b) Find the number of final zeros in the least common multiple of all monotone numbers (with any number of digits).

By the time a party is over, $ 28$ handshakes have occurred. If everyone shook everyone else's hand once, how many people attended the party?

We color one of the numbers $1,...,8$ with white or black according to the following rules: i) number $4$ gets colored white and one at lest of the following numbers gets colored black ii) if two numbers $a,b$ are colored in a different color and $a+b\le 8$, then number $a+b$ gets colored black. iii) if two numbers $a,b$ are colored in a different color and $a\cdot b\le 8$, then number $a\cdot b$ gets colored white. If by those rules, all numbers get colored, find the color of each number.

Let $n\geq 2$. Find all matrices $A\in\mathcal{M}_n(\mathbb{C})$ such that $$\text{rank}(A^2)+\text{rank}(B^2)\geq 2\text{rank}(AB),$$ for all $B\in\mathcal{M}_n(\mathbb{C})$. [i]Cristi Săvescu[/i]

Given is a polynomial $P(x)$ of degree $n>1$ with real coefficients. The equation $P(P(P(x)))=P(x)$ has $n^3$ distinct real roots. Prove that these roots could be split into two groups with equal arithmetic mean.

Consider a checkered $3m\times 3m$ square, where $m$ is an integer greater than $1.$ A frog sits on the lower left corner cell $S$ and wants to get to the upper right corner cell $F.$ The frog can hop from any cell to either the next cell to the right or the next cell upwards. Some cells can be [i]sticky[/i], and the frog gets trapped once it hops on such a cell. A set $X$ of cells is called [i]blocking[/i] if the frog cannot reach $F$ from $S$ when all the cells of $X$ are sticky. A blocking set is [i] minimal[/i] if it does not contain a smaller blocking set.[list=a][*]Prove that there exists a minimal blocking set containing at least $3m^2-3m$ cells. [*]Prove that every minimal blocking set containing at most $3m^2$ cells.

We are given a disk $\mathcal D$ of radius $1$ in the plane. (a) Prove that $\mathcal D$ cannot be covered with two disks of radii $r<1$. (b) Prove that, for some $r<1$, $\mathcal D$ can be covered with three disks of radius $r$. What is the smallest such $r$?

Define \[\begin{cases}d(n, 0)=d(n, n)=1&(n \ge 0),\\ md(n, m)=md(n-1, m)+(2n-m)d(n-1,m-1)&(0<m<n).\end{cases}\] Prove that $d(n, m)$ are integers for all $m, n \in \mathbb{N}$.

Let $n \geqslant 3$ be a positive integer and let $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ be a strictly increasing sequence of $n$ positive real numbers with sum equal to 2. Let $X$ be a subset of $\{1,2, \ldots, n\}$ such that the value of \[ \left|1-\sum_{i \in X} a_{i}\right| \] is minimised. Prove that there exists a strictly increasing sequence of $n$ positive real numbers $\left(b_{1}, b_{2}, \ldots, b_{n}\right)$ with sum equal to 2 such that \[ \sum_{i \in X} b_{i}=1. \]

Find all positive integers $n < 10^{100}$ for which simultaneously $n$ divides $2^n$, $n-1$ divides $2^n - 1$, and $n-2$ divides $2^n - 2$.

Let $A_1A_2A_3...A_n$ be a bicentric polygon with $n$ sides. Denote by $I_i$ the incenter of triangle $A_{i-1}A_iA_{i+1}, A_{i(i+1)}$ the intersection of $A_iA_{i+2}$ and $A_{i-1}A_{i+1},I_{i(i+1)}$ is the incenter of triangle $A_iA_{i(i+1)}A_{i+1}$ ($i = 1, n$). Prove that there exist $2n$ points $I_1, I_2, ..., I_n, I_{12}, I_{23}, ...., I_{n1}$ on the same circle. Nguyễn Văn Linh

Sides $\overline{AB}$ and $\overline{AC}$ of equilateral triangle $ABC$ are tangent to a circle at points $B$ and $C$ respectively. What fraction of the area of $\triangle ABC$ lies outside the circle? $ \textbf{(A) }\dfrac{4\sqrt{3}\pi}{27}-\frac{1}{3}\qquad \textbf{(B) } \frac{\sqrt{3}}{2}-\frac{\pi}{8}\qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) }\sqrt{3}-\frac{2\sqrt{3}\pi}{9}\qquad \textbf{(E) } \frac{4}{3}-\dfrac{4\sqrt{3}\pi}{27}$

The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).

Find the smallest positive integer $n$ such that $\underbrace{2^{2^{...^{2}}}}_{n}> 3^{3^{3^3}}$. (The notation $\underbrace{2^{2^{...^{2}}}}_{n}$ is used to denote a power tower with $n$ $2$’s. For example, $\underbrace{2^{2^{...^{2}}}}_{n}$ with $n = 4$ would equal $2^{2^{2^2}}$.)

In triangle $ABC,$ $AB=BC,$ and let $I$ be the incentre of $\triangle ABC.$ $M$ is the midpoint of segment $BI.$ $P$ lies on segment $AC,$ such that $AP=3PC.$ $H$ lies on line $PI,$ such that $MH\perp PH.$ $Q$ is the midpoint of the arc $AB$ of the circumcircle of $\triangle ABC$. Prove that $BH\perp QH.$

For positive integer numbers $a$, $b$ and $c$ it is known that $a^2+b^2+c^2$ and $a^3+b^3+c^3$ are both divisible by $a+b+c$. In addition, $gcd(a+b+c, 6) = 1$. Prove that $a^5+b^5+c^5$ is divisible by $(a+b+c)^2$. [i] A. Antropov [/i]

The princess wishes to have a bracelet with $r$ rubies and $s$ emeralds arranged in such order that there exist two jewels on the bracelet such that starting with these and enumerating the jewels in the same direction she would obtain identical sequences of jewels. Prove that it is possible to fulfill the princess’s wish if and only if $r$ and $s$ have a common divisor.

We are given the set $$M = \{-2^{2022}, -2^{2021}, . . . , -2^{2}, -2, -1, 1, 2, 2^2, . . . , 2^{2021}, 2^{2022}\}.$$ Let $T$ be a subset of $M$, such that neighbouring numbers have the same difference when the elements are ordered by size. (a) Determine the maximum number of elements that such a set $T$ can contain. (b) Determine all sets $T$ with the maximum number of elements. [i](Walther Janous)[/i]

Consider the isosceles right triangle $ABC$ with $\angle BAC = 90^\circ$. Let $\ell$ be the line passing through $B$ and the midpoint of side $AC$. Let $\Gamma$ be the circumference with diameter $AB$. The line $\ell$ and the circumference $\Gamma$ meet at point $P$, different from $B$. Show that the circumference passing through $A,\ C$ and $P$ is tangent to line $BC$ at $C$.

Suppose $\, q_{0}, \, q_{1}, \, q_{2}, \ldots \; \,$ is an infinite sequence of integers satisfying the following two conditions: (i) $\, m-n \,$ divides $\, q_{m}-q_{n}\,$ for $\, m > n \geq 0,$ (ii) there is a polynomial $\, P \,$ such that $\, |q_{n}| < P(n) \,$ for all $\, n$ Prove that there is a polynomial $\, Q \,$ such that $\, q_{n}= Q(n) \,$ for all $\, n$.

Find all surjective functions $f: \mathbb R \to \mathbb R$ such that for every $x,y\in \mathbb R,$ we have \[f(x+f(x)+2f(y))=f(2x)+f(2y).\]

Let $ABC$ be a triangle with $E$ being the centre of its Euler circle. Through $E$, construct the lines $PS, MQ, NR$ parallel to $BC,CA,AB$ ($R,Q$ are on the line $BC, N, P$ on the line $AC,M, S$ on the line $AB$). Prove that the four Euler lines of triangles $ABC,AMN,BSR,CPQ$ are concurrent. Nguyễn Văn Linh

Let $U=\{1,2,\ldots ,n\}$, where $n\geq 3$. A subset $S$ of $U$ is said to be [i]split[/i] by an arrangement of the elements of $U$ if an element not in $S$ occurs in the arrangement somewhere between two elements of $S$. For example, 13542 splits $\{1,2,3\}$ but not $\{3,4,5\}$. Prove that for any $n-2$ subsets of $U$, each containing at least 2 and at most $n-1$ elements, there is an arrangement of the elements of $U$ which splits all of them.

Points $X$ and $Y$ are chosen inside an acute triangle $ABC$ so that: $$\angle AXB = \angle CYB = 180^\circ - \angle ABC, \text{ } \angle ABX = \angle CBY$$ Show that the points $X$ and $Y$ are equidistant from the center of the circumscribed circle of $\triangle ABC$. [i]Proposed by Anton Trygub[/i]

Prove that for any prime $p,$ there exists a positive integer $n$ such that \[1^n+2^{n-1}+3^{n-2}+\cdots+n^1\equiv 2020\pmod{p}.\] [i]Robin Son[/i]