Let $ABC$ be an acute angled triangle and point $M$ chosen differently from $A,B,C$. Prove that $M$ is the orthocenter of triangle $ABC$ if and only if \[\frac{BC}{MA}\vec{MA}+\frac{CA}{MB}\vec{MB}+\frac{AB}{MC}\vec{MC}= \vec{0}\]

A directed line segment, starting point is $P(-1,1)$, finishing point is $Q(2,2)$. If line $l:x+my+m=0$ intersects $PQ$ at its extended line, then the range value of $m$ is________.

On a plane there is an invisible [color=#eee]rabbit[/color] (rabbit) hiding on a lattice point. We want to put $n$ hunters on some lattice points to catch the rabbit. In a turn each hunter can choose to shoot to left/right or top/bottom direction. On the $i$th turn there will be these steps in order 1. The rabbit jumps to an adjacent lattice point on the top, bottom, left, or right. 2. item Each hunter moves to an adjacent lattice point on the top, bottom, left or right (each hunter can move to different direction). Then they shoot a bullet which travels $\frac{334111214}{334111213}i$ units on the directions they chose. If a bullet hits the rabbit then it is caught. Find the smallest number $n$ such that the rabbit would definitely be caught in a finite number of turns. [i]Proposed by tob8y[/i]

On a circle there are $n$ red and $n$ blue arcs given in such a way that each red arc intersects each blue one. Prove that some point is contained by at least $n$ of the given coloured arcs.

There are $8436$ steel balls, each with radius $1$ centimeter, stacked in a tetrahedral pile, with one ball on top, $3$ balls in the second layer, $6$ in the third layer, $10$ in the fourth, and so on. Determine the height of the pile in centimeters.

Sir Alex plays the following game on a row of 9 cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations: [list=1] [*] Choose any number of the form $2^j$, where $j$ is a non-negative integer, and put it into an empty cell. [*] Choose two (not necessarily adjacent) cells with the same number in them; denote that number by $2^j$. Replace the number in one of the cells with $2^{j+1}$ and erase the number in the other cell. [/list] At the end of the game, one cell contains $2^n$, where $n$ is a given positive integer, while the other cells are empty. Determine the maximum number of moves that Sir Alex could have made, in terms of $n$. [i]Proposed by Warut Suksompong, Thailand[/i]

Consider infinite diagrams [asy] import graph; size(90); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; label("$n_{00} \ n_{01} \ n_{02} \ldots$", (1.14,1.38), SE*lsf); label("$n_{10} \ n_{11} \ n_{12} \ldots$", (1.2,1.8), SE*lsf); label("$n_{20} \ n_{21} \ n_{22} \ldots$", (1.2,2.2), SE*lsf); label("$\vdots \quad \vdots \qquad \vdots $", (1.32,2.72), SE*lsf); draw((1,1)--(3,1)); draw((1,1)--(1.02,2.62)); clip((-4.3,-10.94)--(-4.3,6.3)--(16.18,6.3)--(16.18,-10.94)--cycle); [/asy] where all but a finite number of the integers $n_{ij} , i = 0, 1, 2, \ldots, j = 0, 1, 2, \ldots ,$ are equal to $0$. Three elements of a diagram are called [i]adjacent[/i] if there are integers $i$ and $j$ with $i \geq 0$ and $j \geq 0$ such that the three elements are [b](i)[/b] $n_{ij}, n_{i,j+1}, n_{i,j+2},$ or [b](ii)[/b] $n_{ij}, n_{i+1,j}, n_{i+2,j} ,$ or [b](iii)[/b] $n_{i+2,j}, n_{i+1,j+1}, n_{i,j+2}.$ An elementary operation on a diagram is an operation by which three [i]adjacent[/i] elements $n_{ij}$ are changed into $n_{ij}'$ in such a way that $|n_{ij}-n_{ij}'|=1.$ Two diagrams are called equivalent if one of them can be changed into the other by a finite sequence of elementary operations. How many inequivalent diagrams exist?

Let $a,b,c$ be positive real numbers such that $ab+bc+ca\le 3abc$. Prove that \[\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\le \sqrt{2} (\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a})\]

Many Gothic cathedrals have windows with portions containing a ring of congruent circles that are circumscribed by a larger circle, In the figure shown, the number of smaller circles is four. What is the ratio of the sum of the areas of the four smaller circles to the area of the larger circle? [asy]unitsize(6mm); defaultpen(linewidth(.8pt)); draw(Circle((0,0),1+sqrt(2))); draw(Circle((sqrt(2),0),1)); draw(Circle((0,sqrt(2)),1)); draw(Circle((-sqrt(2),0),1)); draw(Circle((0,-sqrt(2)),1));[/asy]$ \textbf{(A)}\ 3\minus{}2\sqrt2 \qquad \textbf{(B)}\ 2\minus{}\sqrt2 \qquad \textbf{(C)}\ 4(3\minus{}2\sqrt2) \qquad \textbf{(D)}\ \frac12(3\minus{}\sqrt2)$ $ \textbf{(E)}\ 2\sqrt2\minus{}2$

$2020$ magicians are divided into groups of $2$ for the Lexington Magic Tournament. After every $5$ days, which is the duration of one match, teams are rearranged so no $2$ people are ever on the same team. If the longest tournament is $n$ days long, what is the value of $n?$ [i]Proposed by Ephram Chun[/i]

a) A certain committee has gathered $40$ times. There were $10$ members on every meeting. Not a single couple has met on the meetings twice. Prove that there were no less then $60$ members in the committee. b) Prove that you can not construct more then $30$ subcommittees of $5$ members from the committee of $25$ members, with no couple of subcommittees having more than one common member.

Let $ABC$ be a triangle, not right-angled, with positive integer angle measures (in degrees) and circumcenter $O$. Say that a triangle $ABC$ is [i]good[/i] if the following three conditions hold: (a) There exists a point $P\neq A$ on side $AB$ such that the circumcircle of $\triangle POA$ is tangent to $BO$. (b) There exists a point $Q\neq A$ on side $AC$ such that the circumcircle of $\triangle QOA$ is tangent to $CO$. (c) The perimeter of $\triangle APQ$ is at least $AB+AC$. Determine the number of ordered triples $(\angle A, \angle B,\angle C)$ for which $\triangle ABC$ is good. [i]Proposed by Vincent Huang[/i]

The positive integers $x$ and $y$ are such that $x^{2022}+x+y^2$ is divisible by $xy$. a) Give an example of such integers $x$ and $y$, with $x>y$. b) Prove that $x$ is a perfect square.

Find the least natural number $n$, such that the following inequality holds:$\sqrt{\dfrac{n-2011}{2012}}-\sqrt{\dfrac{n-2012}{2011}}<\sqrt[3]{\dfrac{n-2013}{2011}}-\sqrt[3]{\dfrac{n-2011}{2013}}$.

Examine if we can place $9$ convex $6$-angled polygons the one next to the other (with common only one side or part of her) to construct a convex $39$-angled polygon.

$M_n=\{\overline{0.a_1a_2\cdots a_n}|a_i\in{0,1},i=1,2,\cdots,n,a_n=1\}$. $T_n=|M_n|,S_n=\sum_{x\in M_n}x$, then $\lim_{n\to\infty}\frac{S_n}{T_n}=$________.

Find all real numbers $p$ such that the equation $$\sqrt{x^2-5p^2}=px-1$$ has a root $x=3$. Then, solve the equation for the determined values of $p$.

On the sides of the convex hexagon $ABCDEF$ into the outer side were built equilateral triangles $ABC_1$, $BCD_1$, $CDE_1$, $DEF_1$, $EFA_1$ and $FAB_1$. The triangle $B_1D_1F_1$ is equilateral too. Prove that, the triangle $A_1C_1E_1$ is also equilateral.

A positive integer $n$ is [i] tripariable [/i] if it is possible to partition the set $\{1, 2, \dots, n\}$ into disjoint pairs such that the sum of two elements in each pair is a power of $3$. For example $6$ is tripariable because $\{1, 2, \dots, n\}=\{1,2\}\cup\{3,6\}\cup\{4,5\}$ and $$1+2=3^1,\quad 3+6 = 3^2\quad\text{and}\quad4+5=3^2$$ are all powers of 3. How many positive integers less than or equal to 2024 are tripariable?

Four ambassadors and one advisor for each of them are to be seated at a round table with $12$ chairs numbered in order from $1$ to $12$. Each ambassador must sit in an even-numbered chair. Each advisor must sit in a chair adjacent to his or her ambassador. There are $N$ ways for the $8$ people to be seated at the table under these conditions. Find the remainder when $N$ is divided by $1000$.

Given a real number $x>1$, prove that there exists a real number $y >0$ such that \[\lim_{n \to \infty} \underbrace{\sqrt{y+\sqrt {y + \cdots+\sqrt y}}}_{n \text{ roots}}=x.\]

Let $ABCD$ be a square with side length $8$. Let $M$ be the midpoint of $BC$ and let $\omega$ be the circle passing through $M, A$, and $D$. Let $O$ be the center of $\omega, X$ be the intersection point (besides A) of $\omega$ with $AB$, and $Y$ be the intersection point of $OX$ and $AM$. If the length of $OY$ can be written in simplest form as $\frac{m}{n}$ , compute $m + n$.

Given two heaps of checkers. the bigger contains $m$ checkers, the smaller -- $n$ ($m>n$). Two players are taking checkers in turn from the arbitrary heap. The players are allowed to take from the heap a number of checkers (not zero) divisible by the number of checkers in another heap. The player that takes the last checker in any heap wins. a) Prove that if $m > 2n$, than the first can always win. b) Find all $x$ such that if $m > xn$, than the first can always win.

We will call a rectangular table filled with natural numbers [i]“good”[/i], if for each two rows, there exist a column for which its two cells that are also in these two rows, contain numbers of different parity. Prove that for $\forall$ $n>2$ we can erase a column from a [i]good[/i] $n$ x $n$ table so that the remaining $n$ x $(n-1)$ table is also [i]good[/i].

A positive integer is called [i]good[/i] if it can be written as a sum of two consecutive positive integers and as a sum of three consecutive positive integers. Prove that: a)2001 is [i]good[/i], but 3001 isn't [i]good[/i]. b)the product of two [i]good[/i] numbers is a [i]good[/i] number. c)if the product of two numbers is [i]good[/i], then at least one of the numbers is [i]good[/i]. [i]Bogdan Enescu[/i]