One of the sides of the triangle is divided by the ratio $p: q$, and the other by $m: n: k$. The obtained division points of the sides are connected to the opposite vertices of the triangle by straight lines. Find the ratio of the area of ​​this triangle to the area of ​​the quadrilateral formed by three such lines and one of the sides of the triangle.

Let $ T \in \textsl{SL}(n,\mathbb{Z})$, let $ G$ be a nonsingular $ n \times n$ matrix with integer elements, and put $ S\equal{}G^{\minus{}1}TG$. Prove that there is a natural number $ k$ such that $ S^k \in \textsl{SL}(n,\mathbb{Z})$. [i]Gy. Szekeres[/i]

Given a triangle $ABC$, let $ P$ be the point which minimizes the sum of squares of distances from the sides of the triangle. Let $D, E, F$ the projections of $ P$ on the sides of the triangle $ABC$. Show that $P$ is the barycenter of $DEF$. (Jack D’Aurizio)

Find all $3$-digit numbers which are the sums of the cubes of their digits.

Find the number of ways that a positive integer $n$ can be represented as a sum of one or more consecutive positive integers.

Let $(a_1, a_2, ..., a_8)$ be a permutation of $(1, 2, ... , 8)$. Find, with proof, the maximum possible number of elements of the set $$\{a_1, a_1 + a_2, ... , a_1 + a_2 + ... + a_8\}$$ that can be perfect squares.

We have a complete graph with $n$ vertices. We have to color the vertices and the edges in a way such that: no two edges pointing to the same vertice are of the same color; a vertice and an edge pointing him are coloured in a different way. What is the minimum number of colors we need?

A circle with a circumscribed and an inscribed square centered at the origin $ O$ of a rectangular coordinate system with positive $ x$ and $ y$ axes $ OX$ and $ OY$ is shown in each figure $ I$ to $ IV$ below. [asy] size((400)); draw((0,0)--(22,0), EndArrow); draw((10,-10)--(10,12), EndArrow); draw((25,0)--(47,0), EndArrow); draw((35,-10)--(35,12), EndArrow); draw((-25,0)--(-3,0), EndArrow); draw((-15,-10)--(-15,12), EndArrow); draw((-50,0)--(-28,0), EndArrow); draw((-40,-10)--(-40,12), EndArrow); draw(Circle((-40,0),6)); draw(Circle((-15,0),6)); draw(Circle((10,0),6)); draw(Circle((35,0),6)); draw((-34,0)--(-40,6)--(-46,0)--(-40,-6)--(-34,0)--(-34,6)--(-46,6)--(-46,-6)--(-34,-6)--cycle); draw((-6.5,0)--(-15,8.5)--(-23.5,0)--(-15,-8.5)--cycle); draw((-10.8,4.2)--(-19.2,4.2)--(-19.2,-4.2)--(-10.8,-4.2)--cycle); draw((14.2,4.2)--(5.8,4.2)--(5.8,-4.2)--(14.2,-4.2)--cycle); draw((16,6)--(4,6)--(4,-6)--(16,-6)--cycle); draw((41,0)--(35,6)--(29,0)--(35,-6)--cycle); draw((43.5,0)--(35,8.5)--(26.5,0)--(35,-8.5)--cycle); label("I", (-49,9)); label("II", (-24,9)); label("III", (1,9)); label("IV", (26,9)); label("X", (-28,0), S); label("X", (-3,0), S); label("X", (22,0), S); label("X", (47,0), S); label("Y", (-40,12), E); label("Y", (-15,12), E); label("Y", (10,12), E); label("Y", (35,12), E);[/asy] The inequalities \[ |x| \plus{} |y| \leq \sqrt {2(x^2 \plus{} y^2)} \leq 2\mbox{Max}(|x|, |y|)\] are represented geometrically* by the figure numbered $ \textbf{(A)}\ I \qquad \textbf{(B)}\ II \qquad \textbf{(C)}\ III \qquad \textbf{(D)}\ IV \qquad \textbf{(E)}\ \mbox{none of these}$ *An inequality of the form $ f(x, y) \leq g(x, y)$, for all $ x$ and $ y$ is represented geometrically by a figure showing the containment \[ \{\mbox{The set of points }(x, y)\mbox{ such that }g(x, y) \leq a\} \subset\\ \{\mbox{The set of points }(x, y)\mbox{ such that }f(x, y) \leq a\}\] for a typical real number $ a$.

Triangle $ABC$ has circumcircle $\omega$ where $B'$ is the point diametrically opposite $B$ and $C'$ is the point diametrically opposite $C$. Given $B'C'$ passes through the midpoint of $AB$, if $AC' = 3$ and $BC = 7$, find $AB'^2$..

A regular $n$-gon $P_1P_2...P_n$ satisfies $\angle P_1P_7P_8 = 178^o$. Compute $n$.

In a sport competition, $m$ teams have participated. We know that each two teams have competed exactly one time and the result is winning a team and losing the other team (i.e. there is no equal result). Prove that there exists a team $x$ such that for each team $y,$ either $x$ wins $y$ or there exists a team $z$ for which $x$ wins $z$ and $z$ wins $y.$ [i][i.e. prove that in every tournament there exists a king.][/i]

Let $n$ be a positive integer. A regular hexagon $ABCDEF$ with side length $n$ is partitioned into $6n^2$ equilateral triangles with side length $1$. The hexagon is covered by $3n^2$ rhombuses with internal angles $60^{\circ}$ and $120^{\circ}$ such that each rhombus covers exactly two triangles and every triangle is covered by exactly one rhombus. Show that the diagonal $AD$ divides in half exactly $n$ rhombuses.

On the side $AB$ of a triangle $ABC$, two points $X, Y$ are chosen so that $AX = BY$. Lines $CX$ and $CY$ meet the circumcircle of the triangle, for the second time, at points $U$ and $V$. Prove that all lines $UV$ (for all $X, Y$, given $A, B, C$) have a common point.

Let $M$ and $I$ be the centroid and the incenter of a scalene triangle $ABC$, and let $r$ be its inradius. Prove that $MI = r/3$ if and only if $MI$ is perpendicular to one of the sides of the triangle. (A.Karlyuchenko)

[b]p1.[/b] Eric is playing Brawl Stars. If he starts playing at $11:10$ AM, and plays for $2$ hours total, then how many minutes past noon does he stop playing? [b]p2.[/b] James is making a mosaic. He takes an equilateral triangle and connects the midpoints of its sides. He then takes the center triangle formed by the midsegments and connects the midpoints of its sides. In total, how many equilateral triangles are in James’ mosaic? [b]p3.[/b] What is the greatest amount of intersections that $3$ circles and $3$ lines can have, given that they all lie on the same plane? [b]p4.[/b] In the faraway land of Arkesia, there are two types of currencies: Silvers and Gold. Each Silver is worth $7$ dollars while each Gold is worth $17$ dollars. In Daniel’s wallet, the total dollar value of the Silvers is $1$ more than that of the Golds. What is the smallest total dollar value of all of the Silvers and Golds in his wallet? [b]p5.[/b] A bishop is placed on a random square of a $8$-by-$8$ chessboard. On average, the bishop is able to move to $s$ other squares on the chessboard. Find $4s$. Note: A bishop is a chess piece that can move diagonally in any direction, as far as it wants. [b]p6.[/b] Andrew has a certain amount of coins. If he distributes them equally across his $9$ friends, he will have $7$ coins left. If he apportions his coins for each of his $15$ classmates, he will have $13$ coins to spare. If he splits the coins into $4$ boxes for safekeeping, he will have $2$ coins left over. What is the minimum number of coins Andrew could have? [b]p7.[/b] A regular polygon $P$ has three times as many sides as another regular polygon $Q$. The interior angle of $P$ is $16^o$ greater than the interior angle of $Q$. Compute how many more diagonals $P$ has compared to $Q$. [b]p8.[/b] In an certain airport, there are three ways to switch between the ground floor and second floor that are 30 meters apart: either stand on an escalator, run on an escalator, or climb the stairs. A family on vacation takes 65 seconds to climb up the stairs. A solo traveller late for their flight takes $25$ seconds to run upwards on the escalator. The amount of time (in seconds) it takes for someone to switch floors by standing on the escalator can be expressed as $\frac{u}{v}$ , where $u$ and $v$ are relatively prime. Find $u + v$. (Assume everyone has the same running speed, and the speed of running on an escalator is the sum of the speeds of riding the escalator and running on the stairs.) [b]p9.[/b] Avanish, being the studious child he is, is taking practice tests to improve his score. Avanish has a $60\%$ chance of passing a practice test. However, whenever Avanish passes a test, he becomes more confident and instead has a $70\%$ chance of passing his next immediate test. If Avanish takes $3$ practice tests in a row, the expected number of practice tests Avanish will pass can be expressed as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime. Find $a + b$. [b]p10.[/b] Triangle $\vartriangle ABC$ has sides $AB = 51$, $BC = 119$, and $AC = 136$. Point $C$ is reflected over line $\overline{AB}$ to create point $C'$. Next, point $B$ is reflected over line $\overline{AC'}$ to create point $B'$. If $[B'C'C]$ can be expressed in the form of $a\sqrt{b}$, where $b$ is not divisible by any perfect square besides $1$, find $a + b$. [b]p11[/b]. Define the following infinite sequence $s$: $$s = \left\{\frac{1}{1},\frac{1}{1 + 3},\frac{1}{1 + 3 + 6}, ... ,\frac{1}{1 + 3 + 6 + ...+ t_k},...\right\},$$ where $t_k$ denotes the $k$th triangular number. The sum of the first $2024$ terms of $s$, denoted $S$, can be expressed as $$S = 3 \left(\frac{1}{2}+\frac{1}{a}-\frac{1}{b}\right),$$ where $a$ and $b$ are positive integers. Find the minimal possible value of $a + b$. [b]p12.[/b] Omar writes the numbers from $1$ to $1296$ on a whiteboard and then converts each of them into base $6$. Find the sum of all of the digits written on the whiteboard (in base $10$), including both the base $10$ and base $6$ numbers. [b]p13.[/b] A mountain number is a number in a list that is greater than the number to its left and right. Let $N$ be the amount of lists created from the integers $1$ - $100$ such that each list only has one mountain number. $N$ can be expressed as $$N = 2^a(2^b - c^2),$$ where $a$, $b$ and $c$ are positive integers and $c$ is not divisible by $2$. Find $a + b+c$. (The numbers at the beginning or end of a list are not considered mountain numbers.)[hide]Original problem was voided because the original format of the answer didn't match the result's format. So I changed it in the wording, in order the problem to be correct[/hide] [b]p14.[/b] A circle $\omega$ with center $O$ has a radius of $25$. Chords $\overline{AB}$ and $\overline{CD}$ are drawn in $\omega$ , intersecting at $X$ such that $\angle BXC = 60^o$ and $AX > BX$. Given that the shortest distance of $O$ with $\overline{AB}$ and $\overline{CD}$ is $7$ and $15$ respectively, the length of $BX$ can be expressed as $x - \frac{y}{\sqrt{z}}$ , where $x$, $y$, and $z$ are positive integers such that $z$ is not divisible by any perfect square. Find $x + y + z.$ [hide]two answers were considered correct according to configuration[/hide] [b]p15.[/b] How many ways are there to split the first $10$ natural numbers into $n$ sets (with $n \ge 1$) such that all the numbers are used and each set has the same average? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Viorel is $10$ years old, and his mother is $31$ years old. In how many years will Viorel be half his mother's age?

There are $2n$ cards. On each card some real number $x$, $(1\le x\le 2n)$, is written (there can be different numbers on different cards). Prove that the cards can be divided into two heaps with sums $s_1$ and $s_2$ so that $\frac{n}{n+1}\le\frac{s_1}{s_2}\le 1$.

Show that the equation, $2x^4+2x^2y^2+y^4=z^2$, does not have integer solution when $x \neq 0$.

Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_1<\cdots<a_n$ and a real number $r>0$ such that the $\tfrac{1}{2}n(n-1)$ differences $a_j-a_i$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^1,r^2,\ldots,r^{\frac{1}{2}n(n-1)}$.

Let $ABC$ be an acute triangle, and let $D$ be the foot of the altitude through $A$. On $AD$, there are distinct points $E$ and $F$ such that $|AE| = |BE|$ and $|AF| =|CF|$. A point$ T \ne D$ satises $\angle BTE = \angle CTF = 90^o$. Show that $|TA|^2 =|TB| \cdot |TC|$.

For how many positive integers $100 < n \le 10000$ does $\lfloor \sqrt{n-100} \rfloor$ divide $n$? [i]Proposed by Michael Tang[/i]

Let $n$ and $b$ be positive integers. We say $n$ is $b$-discerning if there exists a set consisting of $n$ different positive integers less than $b$ that has no two different subsets $U$ and $V$ such that the sum of all elements in $U$ equals the sum of all elements in $V$. (a) Prove that $8$ is $100$-discerning. (b) Prove that $9$ is not $100$-discerning. [i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]

Find all positive integers \(n\) such that among the \(n\) numbers \[ 2n + 1, \, 2^2 n + 1, \, \ldots, \, 2^n n + 1 \] there are \(n\), \(n - 1\), or \(n - 2\) primes.

Determine, if they exist, the real values of $x$ and $y$ that satisfy that $$\frac{x^2}{y^2} +\frac{y^2}{x^2} +\frac{x}{y}+\frac{y}{x} = 0$$ such that $x + y <0.$

Frankie the Frog starts his morning at the origin in $\mathbb{R}^2$. He decides to go on a leisurely stroll, consisting of $3^1+3^{10}+3^{11}+3^{100}+3^{111}+3^{1000}$ moves, starting with the first move. On the $n$th move, he hops a distance of $$\max\{k\in\mathbb{Z}:3^k|n\}+1,$$ then turns $90^{\circ}$ counterclockwise. What is the square of the distance from his final position to the origin?