Let $M$ be an interior point of the tetrahedron $ABCD$. Prove that \[ \begin{array}{c}\ \stackrel{\longrightarrow }{MA} \text{vol}(MBCD) +\stackrel{\longrightarrow }{MB} \text{vol}(MACD) +\stackrel{\longrightarrow }{MC} \text{vol}(MABD) + \stackrel{\longrightarrow }{MD} \text{vol}(MABC) = 0 \end{array}\] ($\text{vol}(PQRS)$ denotes the volume of the tetrahedron $PQRS$).

A painting $18''\ \text{X}\ 24''$ is to be placed into a wooden frame with the longer dimension vertical. The wood at the top and bottom is twice as wide as the wood on the sides. If the frame area equals that of the painting itself, the ratio of the smaller to the larger dimension of the framed painting is: $\textbf{(A)}\ 1:3 \qquad\textbf{(B)}\ 1:2 \qquad\textbf{(C)}\ 2:3 \qquad\textbf{(D)}\ 3:4 \qquad\textbf{(E)}\ 1:1$

In the middle of the number 1996 (i.e. between $19$ and $96$) you need to insert several digits so that the resulting number is divisible by $1997$. In this case, you need to get by with the least number of inserted digits.

We call a [i]ship[/i] a figure made up of unit squares connected by common edges. Prove that if there is an odd number of possible different ships consisting of n unit squares on a $ 10 \times 10$ board, then n is divisible by 4.

There are $4$ piles of stones with the following quantities: $1004$, $1005$, $2009$ and $2010$. A legitimate move is to remove a stone from each from $3$ different piles. Two players $A$ and $B$ play in turns. $A$ begins the game . The player who, on his turn, cannot make a legitimate move, loses. Determine which of the players has a winning strategy and give a strategy for that player.

[u]Round 1[/u] [b]1.1.[/b] A classroom has $29$ students. A teacher needs to split up the students into groups of at most $4$. What is the minimum number of groups needed? [b]1.2.[/b] On his history map quiz, Eric recalls that Sweden, Norway and Finland are adjacent countries, but he has forgotten which is which, so he labels them in random order. The probability that he labels all three countries correctly can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]1.3.[/b] In a class of $40$ sixth graders, the class average for their final test comes out to be $90$ (out of a $100$). However, a student brings up an issue with problem $5$, and $10$ students receive credit for this question, bringing the class average to a $90.75$. How many points was problem $5$ worth? [u]Round 2[/u] [b]2.1.[/b] Compute $1 - 2 + 3 - 4 + ... - 2022 + 2023$. [b]2.2.[/b] In triangle $ABC$, $\angle ABC = 75^o$. Point $D$ lies on side $AC$ such that $BD = CD$ and $\angle BDC$ is a right angle. Compute the measure of $\angle A$. [b]2.3.[/b] Joe is rolling three four-sided dice each labeled with positive integers from $1$ to $4$. The probability the sum of the numbers on the top faces of the dice is $6$ can be written as $\frac{p}{q}$ where $p$ and $q$ are relatively prime integers. Find $p + q$. [u]Round 3[/u] [b]3.1.[/b] For positive integers $a, b, c, d$ that satisfy $a + b + c + d = 23$, what is the maximum value of $abcd$? [b]3.2.[/b] A buckball league has twenty teams. Each of the twenty teams plays exactly five games with each of the other teams. If each game takes 1 hour and thirty minutes, then how many total hours are spent playing games? [b]3.3.[/b] For a triangle $\vartriangle ABC$, let $M, N, O$ be the midpoints of $AB$, $BC$, $AC$, respectively. Let $P, Q, R$ be points on $AB$, $BC$, $AC$ such that $AP =\frac13 AB$, $BQ =\frac13 BC$, and $CR =\frac13 AC$. The ratio of the areas of $\vartriangle MNO$ and $\vartriangle P QR$ can be expressed as $\frac{m}{n}$ , where $ m$ and $n$ are relatively prime positive integers. Find $m + n$. [u]Round 4[/u] [b]4.1.[/b] $2023$ has the special property that leaves a remainder of $1$ when divided by $2$, $21$ when divided by $22$, and $22$ when divided by $23$. Let $n$ equal the lowest integer greater than $2023$ with the above properties. What is $n$? [b]4.2.[/b] Ants $A, B$ are on points $(0, 0)$ and $(3, 3)$ respectively, and ant A is trying to get to $(3, 3)$ while ant $B$ is trying to get to $(0, 0)$. Every second, ant $A$ will either move up or right one with equal probability, and ant $B$ will move down or left one with equal probability. The probability that the ants will meet each other be $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a + b$. [b]4.3.[/b] Find the number of trailing zeros of $100!$ in base $ 49$. PS. You should use hide for answers. Rounds 5-9 have been posted [url=https://artofproblemsolving.com/community/c3h3129723p28347714]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n$ be a positive integer, set $S_n = \{ (a_1,a_2,\cdots,a_{2^n}) \mid a_i=0 \ \text{or} \ 1, 1 \leq i \leq 2^n\}$. For any two elements $a=(a_1,a_2,\cdots,a_{2^n})$ and $b=(b_1,b_2,\cdots,b_{2^n})$ of $S_n$, define \[ d(a,b)= \sum_{i=1}^{2^n} |a_i - b_i| \] We call $A \subseteq S_n$ a $\textsl{Good Subset}$ if $d(a,b) \geq 2^{n-1}$ holds for any two distinct elements $a$ and $b$ of $A$. How many elements can the $\textsl{Good Subset}$ of $S_n$ at most have?

Calculate $1+2+3+4-5-6-7-8+9+\cdots-96+97+98+99+100$

Determine all positive integers that are equal to $300$ times the sum of their digits.

Let $n>1$ be an integer. Find all non-constant real polynomials $P(x)$ satisfying , for any real $x$ , the identy \[P(x)P(x^2)P(x^3)\cdots P(x^n)=P(x^{\frac{n(n+1)}{2}})\]

Farmer James wishes to cover a circle with circumference $10\pi$ with six different types of colored arcs. Each type of arc has radius $5$, has length either $\pi$ or $2\pi$, and is colored either red, green, or blue. He has an unlimited number of each of the six arc types. He wishes to completely cover his circle without overlap, subject to the following conditions: [list][*] Any two adjacent arcs are of different colors. [*] Any three adjacent arcs where the middle arc has length $\pi$ are of three different colors. [/list] Find the number of distinct ways Farmer James can cover his circle. Here, two coverings are equivalent if and only if they are rotations of one another. In particular, two colorings are considered distinct if they are reflections of one another, but not rotations of one another. [i]Proposed by James Lin.[/i]

Let $ABC$ be a triangle with $AB=13,BC=15,AC=14$, circumcenter $O$, and orthocenter $H$, and let $M,N$ be the midpoints of minor and major arcs $BC$ on the circumcircle of $ABC$. Suppose $P\in AB, Q\in AC$ satisfy that $P,O,Q$ are collinear and $PQ||AN$, and point $I$ satisfies $IP\perp AB,IQ\perp AC$. Let $H'$ be the reflection of $H$ over line $PQ$, and suppose $H'I$ meets $PQ$ at a point $T$. If $\frac{MT}{NT}$ can be written in the form $\frac{\sqrt{m}}{n}$ for positive integers $m,n$ where $m$ is not divisible by the square of any prime, then find $100m+n$. [i]Proposed by Vincent Huang[/i]

Let $ABCDE$ be a right quadrangular pyramid with vertex $E$ and height $EO$. Point $S$ divides this height in the ratio $ES: SO=m$. In which ratio does the plane $(ABC)$ divide the lateral area of the pyramid.

The following sum of three four digits numbers is divisible by $75$, $7a71 + 73b7 + c232$, where $a, b, c$ are decimal digits. Find the necessary conditions in $a, b, c$.

Let $n$ be an integer greater than or equal to 3. Prove that there is a set of $n$ points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with a rational area.

In the isosceles triangle $ABC$, $ (AB = BC)$ the bisector $AD$ was drawn, and in the triangle $ABD$ the bisector $DE$ was drawn. Find the values of the angles of the triangle $ABC$, if it is known that the bisectors of the angles $ABD$ and $AED$ intersect on the line $AD$. (Fedak Ivan)

Given an acute triangle $ABC$. The incircle with center $I$ touches $BC,CA,AB$ at $D,E,F$ respectively. Let $M,N$ be the midpoint of the minor arc of $AB$ and $AC$ respectively. Prove that $M,F,E,N$ are collinear if and only if $\angle BAC =90$$^{\circ}$

Prove that there exists an infinite sequence of perfect squares with the following properties: (i) The arithmetic mean of any two consecutive terms is a perfect square, (ii) Every two consecutive terms are coprime, (iii) The sequence is strictly increasing.

Determine the angular acceleration of the disk when $t=\text{2.0 s}$. (A) $\text{-12 rad/s}^2$. (B) $\text{-8 rad/s}^2$. (C) $\text{-4 rad/s}^2$. (D) $\text{-2 rad/s}^2$. (E) $\text{0 rad/s}^2$.

For all positive integers $r$ and $s,$ let $\text{Top}(r,s)$ denote the top number (i.e., numerator) when $r$ $s$ is written in simplified form. For instance, $\text{Top}(20,24) = 5.$ Compute the number of ordered pairs of positive integers $(a,z)$ such that $200 \le a \le 300$ and $\text{Top}(a,z) = \text{Top}(z,a-1).$

In a triangle $ABC$, let $D$ be the point on the segment $BC$ such that $AB+BD=AC+CD$. Suppose that the points $B$, $C$ and the centroids of triangles $ABD$ and $ACD$ lie on a circle. Prove that $AB=AC$.

Let $p>2$ be a prime number and $n$ a positive integer. Prove that $pn^2$ has at most one positive divisor $d$ for which $n^2+d$ is a square number.

Let $A$ be the set of the $16$ first positive integers. Find the least positive integer $k$ satisfying the condition: In every $k$-subset of $A$, there exist two distinct $a, b \in A$ such that $a^2 + b^2$ is prime.

Function $f$ satisfies the equation $f(\cos x) = \cos (17x)$. Prove that it also satisfies the equation $f(\sin x) = \sin (17x)$.

Let $\mathbb{Q}^+$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}^+\to \mathbb{Q}^+$ satisfying$$f(x^2f(y)^2)=f(x^2)f(y)$$for all $x,y\in\mathbb{Q}^+$