Katie has a chocolate bar that is a $5$-by-$5$ grid of square pieces, but she only wants to eat the center piece. To get to it, she performs the following operations: i. Take a gridline on the chocolate bar, and split the bar along the line. ii. Remove the piece that doesn’t contain the center. iii. With the remaining bar, repeat steps $1$ and $2$. Determine the number of ways that Katie can perform this sequence of operations so that eventually she ends up with just the center piece.

There are $n\ge 2$ line segments in the plane such that every two segments cross and no three segments meet at a point. Geoff has to choose an endpoint of each segment and place a frog on it facing the other endpoint. Then he will clap his hands $n-1$ times. Every time he claps,each frog will immediately jump forward to the next intersection point on its segment. Frogs never change the direction of their jumps. Geoff wishes to place the frogs in such a way that no two of them will ever occupy the same intersection point at the same time. (a) Prove that Geoff can always fulfill his wish if $n$ is odd. (b) Prove that Geoff can never fulfill his wish if $n$ is even.

There is a regular $2021$-gon. We put a coin with heads up on every vertex of it. Every time, you can choose one vertex, and flip the coin on the vertices adjacent to it. Can you make all the coin tails up?

The expression $1\oplus2\oplus3\oplus4\oplus5\oplus6\oplus7\oplus8\oplus9$ is written on a blackboard. Bill and Peter play the following game. They replace $\oplus$ by $+$ or $\cdot$, making their moves in turn, and one of them can use only $+$, while the other one can use only $\cdot$. At the beginning, Bill selects the sign he will use, and he tries to make the result an even number. Peter tries to make the result an odd number. Prove that Peter can always win. [hide=Original Wording]The expression $1*2*3*4*5*6*7*8*9$ is written on a blackboard. Bill and Peter play the following game. They replace $*$ by $+$ or $\cdot$, making their moves in turn, and one of them can use only $+$, while the other one can use only $\cdot$. At the beginning Bill selects the sign he will use, and he tries to make the result an even number. Peter tries to make the result an odd number. Prove that Peter can always win.[/hide]

All the diagonals of a regular $25$-gon are drawn. Prove that no $9$ of the diagonals pass through one interior point of the $25$-gon. (A Shapovalov)

For each interger $n\geq 4$, we consider the $m$ subsets $A_1, A_2,\dots, A_m$ of $\{1, 2, 3,\dots, n\}$, such that $A_1$ has exactly one element, $A_2$ has exactly two elements,...., $A_m$ has exactly $m$ elements and none of these subsets is contained in any other set. Find the maximum value of $m$.

Juku has the first $100$ volumes of the Harrie Totter book series at his home. For every$ i$ and $j$, where $1 \le i < j \le 100$, call the pair $(i, j)$ reversed if volume No. $j$ is before volume No, $i$ on Juku’s shelf. Juku wants to arrange all volumes of the series to one row on his shelf in such a way that there does not exist numbers $i, j, k$, where $1 \le i < j < k \le 100$, such that pairs $(i, j)$ and $(j, k)$ are both reversed. Find the largest number of reversed pairs that can occur under this condition

$n$ people are in the plane, so that the closest person is unique and each one shoot this closest person with a squirt gun. If $n$ is odd, prove that there exists at least one person that nobody shot. If $n$ is even, will there always be a person who escape? Justify that.

There are $n>1000$ people at a round table. Some of them are knights who always tell the truth, and the rest are liars who always tell lies. Each of those sitting said the phrase: “among the $20$ people sitting clockwise from where I sit there are as many knights as among the $20$ people seated counterclockwise from where I sit”. For what $n$ could this happen?

Alice plays the following game of solitaire on a $20 \times 20$ chessboard. She begins by placing $100$ pennies, $100$ nickels, $100$ dimes, and $100$ quarters on the board so that each of the $400$ squares contains exactly one coin. She then chooses $59$ of these coins and removes them from the board. After that, she removes coins, one at a time, subject to the following rules: - A penny may be removed only if there are four squares of the board adjacent to its square (up, down, left, and right) that are vacant (do not contain coins). Squares “off the board” do not count towards this four: for example, a non-corner square bordering the edge of the board has three adjacent squares, so a penny in such a square cannot be removed under this rule, even if all three adjacent squares are vacant. - A nickel may be removed only if there are at least three vacant squares adjacent to its square. (And again, “off the board” squares do not count.) - A dime may be removed only if there are at least two vacant squares adjacent to its square (“off the board” squares do not count). - A quarter may be removed only if there is at least one vacant square adjacent to its square (“off the board” squares do not count). Alice wins if she eventually succeeds in removing all the coins. Prove that it is impossiblefor her to win.

The Nationwide Basketball Society (NBS) has $8001$ teams, numbered $2000$ through $10000$. For each $n$, team $n$ has $n+1$ players, and in a sheer coincidence, this year each player attempted $n$ shots and on team $n$, exactly one player made $0$ shots, one player made $1$ shot, . . ., one player made $n$ shots. A player's [i]field goal percentage[/i] is defined as the percentage of shots the player made, rounded to the nearest tenth of a percent (For instance, $32.45\%$ rounds to $32.5\%$). A player in the NBS is randomly selected among those whose field goal percentage is $66.6\%$. If this player plays for team $k$, the probability that $k\geq 6000$ can be expressed as $\tfrac{p}{q}$ for relatively prime positive integers $p$ and $q$. Find $p+q$.

Find the sum of all $5$-digit positive integers that they have only the digits $1, 2$, and $5$, none repeated more than three consecutive times.

Let $x_1, x_2 \ldots , x_5$ be real numbers. Find the least positive integer $n$ with the following property: if some $n$ distinct sums of the form $x_p+x_q+x_r$ (with $1\le p<q<r\le 5$) are equal to $0$, then $x_1=x_2=\cdots=x_5=0$.

Let $E$ be the set of $1983^3$ points of the space $\mathbb R^3$ all three of whose coordinates are integers between $0$ and $1982$ (including $0$ and $1982$). A coloring of $E$ is a map from $E$ to the set {red, blue}. How many colorings of $E$ are there satisfying the following property: The number of red vertices among the $8$ vertices of any right-angled parallelepiped is a multiple of $4$ ?

Let $n$ be a natural number. The leader of the math team invites $n$ girls for winter training, and each leaves her two gloves in a common box upon entry. The mischievous little brother randomly pairs the gloves into pairs, where each pair consists of one left glove and one right glove. A pairing is called [i]weak[/i] if there is a set of $k < \frac{n}{2}$ pairs containing gloves of exactly $k$ girls. Find the probability that the pairing is not weak.

Let $n$ be a natural number. A regular hexagon with edge length $n$ gets split into equilateral exploded triangles whose edges are $1$ in length and parallel to one side of the hexagon. Find the number of regular hexagons, the angles of which are all angles of these triangles are.

Define two sequences $x_n, y_n$ for $n = 1, 2, \ldots$ by \[x_n = \left(\sum^n_{k=0} \binom{2n}{2k}49^k 48^{n-k} \right) -1, \quad \text{and} \quad y_n = \sum^{n-1}_{k=0} \binom{2n}{2k + 1} 49^k 48^{n-k}\] Prove there is a positive integer $m$ for which for every integer $n > m,$ the greatest common factor of $x_n$ and $y_n$ is more than $10^{2024}.$

Ryan Murphy is playing poker. He is dealt a hand of $5$ cards. Given that the probability that he has a straight hand (the ranks are all consecutive; e.g. $3,4,5,6,7$ or $9,10,J,Q,K$) or $3$ of a kind (at least $3$ cards of the same rank; e.g. $5, 5, 5, 7, 7$ or $5, 5, 5, 7,K$) is $m/n$ , where $m$ and $n$ are relatively prime positive integers, find $m +n$. [i]Proposed by Aditya Rao[/i]

Given a rectangular table with 2 rows and 100 columns. Dima fills the cells of the first row with numbers 1, 2 or 3. Prove that Alex can fill the cells of the second row with numbers 1, 2, 3 in such a way that the numbers in each column are different and the sum of the numbers in the second row equals 200.

Let $c\neq 1$ be a positive rational number. Show that it is possible to partition $\mathbb{N}$, the set of positive integers, into two disjoint nonempty subsets $A,B$ so that $x/y\neq c$ holds whenever $x$ and $y$ lie both in $A$ or both in $B$.

A $20 \times 20$ checkered board is cut into several squares with integer side length. The size of a square is it's side length. What is the maximum amount of different sizes this squares can have?

Rijul and Rohinee are playing a game on an $n\times n$ board alternating turns, with Rijul going first. In each turn, they fill an unfilled cell with a number from $1,2,\cdots, n^2$ such that no number is used twice. Rijul wins if there is any column such that the sum of all its elements is divisible by $n$. Rohinee wins otherwise. For what positive integers $n$ does he have a winning strategy? Proposed by Rohan Goyal

In how many ways can the letters in BEEKEEPER be rearranged so that two or more Es do not appear together? $\textbf{(A)} ~1\qquad\textbf{(B)} ~4\qquad\textbf{(C)} ~12\qquad\textbf{(D)} ~24\qquad\textbf{(E)} ~120\qquad$

Distinct pebbles are placed on a $1001 \times 1001$ board consisting of $1001^2$ unit tiles, such that every unit tile consists of at most one pebble. The [i]pebble set[/i] of a unit tile is the set of all pebbles situated in the same row or column with said unit tile. Determine the minimum amount of pebbles that must be placed on the board so that no two distinct tiles have the same [i]pebble set[/i]. [hide=Where's the Algebra Problem?]It's already posted [url=https://artofproblemsolving.com/community/c6h2742895_simple_inequality]here[/url].[/hide]

[b]p1.[/b] Evaluate $1^2 - 2^2 + 3^2 - 4^2 + ...+ 19^2 - 20^2 + 21^2$. [b]p2.[/b] Kevin is playing in a table-tennis championship against Vincent. Kevin wins the championship if he wins two matches against Vincent, while Vincent must win three matches to win the championship. Given that both players have a $50\%$ chance of winning each match and there are no ties, the probability that Vincent loses the championship can be written in the form $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$. [b]p3.[/b] For how many positive integers $n$ less than $2000$ is $n^{3n}$ a perfect fourth power? [b]p4.[/b] Given that a coin of radius $\sqrt{3}$ cm is tossed randomly onto a plane tiled by regular hexagons of side length $14$ cm, the chance that it lands strictly inside of a hexagon can be written in the form $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p + q$. [b]p5.[/b] Given that $A,C,E,I, P,$ and $M$ are distinct nonzero digits such that $$EPIC + EMCC + AMC = PEACE,$$ what is the least possible value of $PEACE$? [b]p6.[/b] A palindrome is a number that reads the same forwards and backwards. Call a number palindrome-ish if it is not a palindrome but we can make it a palindrome by changing one digit (we cannot change the first digit to zero). For instance, $4009$ is palindrome-ish because we can change the $4$ to a $9$. How many palindrome-ish four-digit numbers are there? [b]p7.[/b] Given that the heights of triangle $ABC$ have lengths $\frac{15}{7}$ , $5$, and $3$, what is the square of the area of $ABC$? [b]p8.[/b] Suppose that cubic polynomial $P(x)$ has leading coecient $1$ and three distinct real roots in the interval $[-20, 2]$. Given that the equation $P\left(x + \frac{1}{x} \right) = 0$ has exactly two distinct real solutions, the range of values that $P(3)$ can take is the open interval $(a, b)$. Compute $b - a$. [b]p9.[/b] Vincent the Bug has $17$ students in his class lined up in a row. Every day, starting on January $1$, $2021$, he performs the same series of swaps between adjacent students. One example of a series of swaps is: swap the $4$th and the $5$th students, then swap the $2$nd and the $3$rd, then the $3$rd and the $4$th. He repeats this series of swaps every day until the students are in the same arrangement as on January $1$. What is the greatest number of days this process could take? [b]p10.[/b] The summation $$\sum^{18}_{i=1}\frac{1}{i}$$ can be written in the form $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Compute the number of divisors of $b$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].