Near the end of a game of Fish, Celia is playing against a team consisting of Alice and Betsy. Each of the three players holds two cards in their hand, and together they have the Nine, Ten, Jack, Queen, King, and Ace of Spades (this set of cards is known by all three players). Besides the two cards she already has, each of them has no information regarding the other two's hands (In particular, teammates Alice and Betsy do not know each other's cards). It is currently Celia's turn. On a player's turn, the player must ask a player on the other team whether she has a certain card that is in the set of six cards but [i]not[/i] in the asker's hand. If the player being asked does indeed have the card, then she must reveal the card and put it in the asker’s hand, and the asker shall ask again (but may ask a different player on the other team); otherwise, she refuses and it is now her turn. Moreover, a card may not be asked if it is known (to the asker) to be not in the asked person's hand. The game ends when all six cards belong to one team, and the team with all the cards wins. Under optimal play, the probability that Celia wins the game is $\frac{p}{q}$ for relatively prime positive integers $p$ and $q$. Find $100p+q$. [i]Proposed by Yannick Yao[/i]

In the convex pentagon $ABCDE$, $AE = AD$, $AB = AC$, and angle $CAD$ equals the sum of angles $AEB$ and $ABE$. Prove that segment $CD$ is double the length of median $AM$ of triangle $ABE$.

[b]p1.[/b] Find the unknown angle $a$ of the triangle inscribed in the square. [img]https://cdn.artofproblemsolving.com/attachments/b/1/4aab5079dea41637f2fa22851984f886f034df.png[/img] [b]p2.[/b] Draw a polygon in the plane and a point outside of it with the following property: no edge of the polygon is completely visible from that point (in other words, the view is obstructed by some other edge). [b]p3.[/b] This problem has two parts. In each part, $2022$ real numbers are given, with some additional property. (a) Suppose that the sum of any three of the given numbers is an integer. Show that the total sum of the $2022$ numbers is also an integer. (b) Suppose that the sum of any five of the given numbers is an integer. Show that 5 times the total sum of the $2022$ numbers is also an integer, but the sum itself is not necessarily an integer. [b]p4.[/b] Replace stars with digits so that the long multiplication in the example below is correct. [img]https://cdn.artofproblemsolving.com/attachments/9/7/229315886b5f122dc0675f6d578624e83fc4e0.png[/img] [b]p5.[/b] Five nodes of a square grid paper are marked (called marked points). Show that there are at least two marked points such that the middle point of the interval connecting them is also a node of the square grid paper [b]p6.[/b] Solve the system $$\begin{cases} \dfrac{xy}{x+y}=\dfrac{8}{3} \\ \dfrac{yz}{y+z}=\dfrac{12}{5} \\\dfrac{xz}{x+z}=\dfrac{24}{7} \end{cases}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Solve $p^n+144=m^2$ where $m,n\in \mathbb{N}$ and $p$ is a prime number.

Let $n,k$ be positive integers such that $n>k$. There is a square-shaped plot of land, which is divided into $n\times n$ grid so that each cell has the same size. The land needs to be plowed by $k$ tractors; each tractor will begin on the lower-left corner cell and keep moving to the cell sharing a common side until it reaches the upper-right corner cell. In addition, each tractor can only move in two directions: up and right. Determine the minimum possible number of unplowed cells.

In a group of $n$ people each one has an Easter Egg. They exchange their eggs in the following way: On each exchange two people exchange the eggs they currently have. Each two exchange eggs between themselves at least once. After a certain amount of such exchanges it turned out that each one of the $n$ people had the same egg he had from the beginning. Determine the least amount of exchanges needed, if: a) $n=5$; b) $n=6$.

The intramural squash league has 5 players, namely Albert, Bassim, Clara, Daniel, and Eugene. Albert has played one game, Bassim has played two games, Clara has played 3 games, and Daniel has played 4 games. Assuming no two players in the league play each other more than one time, how many games has Eugene played?

Suppose that $ a_1$, $ a_2$, $ \ldots$, $ a_n$ are integers such that $ n\mid a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n$. Prove that there exist two permutations $ \left(b_1,b_2,\ldots,b_n\right)$ and $ \left(c_1,c_2,\ldots,c_n\right)$ of $ \left(1,2,\ldots,n\right)$ such that for each integer $ i$ with $ 1\leq i\leq n$, we have \[ n\mid a_i \minus{} b_i \minus{} c_i \] [i]Proposed by Ricky Liu & Zuming Feng, USA[/i]

For all positive real numbers $a,b,c$, prove that \[\sqrt{\frac{a^4 + 2b^2c^2}{a^2+2bc}} + \sqrt{\frac{b^4+2c^2a^2}{b^2+2ca}} + \sqrt{\frac{c^4 + 2a^2b^2}{c^2 + 2ab}} \geq a + b + c.\] [i]In-Sung Na.[/i]

Let $S$ be a square of side length $1$, one of whose vertices is $A$. Let $S^+$ be the square obtained by rotating S clockwise about $A$ by $30^o$ . Let $S^-$ be the square obtained by rotating S counterclockwise about $A$ by $30^o$. Compute the total area that is covered by exactly two of the squares $S$, $S^+$, $S^-$. Express your answer in the form $a + b\sqrt3$ where $a, b$ are rational numbers.

Find all positive integers $ a$ and $ b$ for which \[ \left \lfloor \frac{a^2}{b} \right \rfloor \plus{} \left \lfloor \frac{b^2}{a} \right \rfloor \equal{} \left \lfloor \frac{a^2 \plus{} b^2}{ab} \right \rfloor \plus{} ab.\]

When counting from $3$ to $201$, $53$ is the $51^{\text{st}}$ number counted. When counting backwards from $201$ to $3$, $53$ is the $n^{\text{th}}$ number counted. What is $n$? $\textbf{(A) }146\qquad \textbf{(B) } 147\qquad\textbf{(C) } 148\qquad\textbf{(D) }149\qquad\textbf{(E) }150$

Given $5n$ real numbers $r_i, s_i, t_i, u_i, v_i \geq 1 (1 \leq i \leq n)$, let $R = \frac {1}{n} \sum_{i=1}^{n} r_i$, $S = \frac {1}{n} \sum_{i=1}^{n} s_i$, $T = \frac {1}{n} \sum_{i=1}^{n} t_i$, $U = \frac {1}{n} \sum_{i=1}^{n} u_i$, $V = \frac {1}{n} \sum_{i=1}^{n} v_i$. Prove that $\prod_{i=1}^{n}\frac {r_i s_i t_i u_i v_i + 1}{r_i s_i t_i u_i v_i - 1} \geq \left(\frac {RSTUV +1}{RSTUV - 1}\right)^n$.

Compute the largest integer not exceeding $$\frac{2549^3}{2547\cdot 2548} -\frac{2547^3}{2548\cdot 2549}$$

Let $F_0,F_1,\dots$ be the sequence of Fibonacci numbers, with $F_0=0,F_1=1$, and $F_n=F_{n-1}+F_{n-2}$ for $n \ge 2$. For $m>2$, let $R_m$ be the remainder when the product $\prod_{k=1}^{F_m-1} k^k$ is divided by $F_m$. Prove that $R_m$ is also a Fibonacci number.

Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that $$(z + 1)f(x + y) = f(xf(z) + y) + f(yf(z) + x),$$ for all positive real numbers $x, y, z$. [i]Fajar Yuliawan, Indonesia[/i]

Is there a gambling game with an honest coin for two players, in which the probability of one of them winning is $\frac{1}{{\pi}^e}$.

Let $ f: \mathbb{R}\rightarrow \mathbb{R}$ a strictly increasing function such that $ f\circ f$ is continuos. Prove that $ f$ is continuos.

Assume $ 0 < r < 3$. Below are five equations for $ x$. Which equation has the largest solution $ x$? $ \textbf{(A)}\ 3(1 \plus{} r)^x \equal{} 7\qquad \textbf{(B)}\ 3(1 \plus{} r/10)^x \equal{} 7\qquad \textbf{(C)}\ 3(1 \plus{} 2r)^x \equal{} 7$ $ \textbf{(D)}\ 3(1 \plus{} \sqrt {r})^x \equal{} 7\qquad \textbf{(E)}\ 3(1 \plus{} 1/r)^x \equal{} 7$

Let $0<p<1$ be given. Initially, we have $n$ coins, all of which have probability $p$ of landing on heads, and probability $1-p$ of landing on tails (the results of the tosses are independent of each other). In each round, we toss our coins and remove those that result in heads. We keep repeating this until all our coins are removed. Let $k_n$ denote the expected number of rounds that are needed to get rid of all the coins. Prove that there exists $c>0$ for which the following inequality holds for all $n>0$ \[c\bigg(1+\frac{1}{2}+\cdots+\frac{1}{n}\bigg)<k_n<1+c\bigg(1+\frac{1}{2}+\cdots+\frac{1}{n}\bigg).\]

Let $(a_n), n = 0, 1, . . .,$ be a sequence of real numbers such that $a_0 = 0$ and \[a^3_{n+1} = \frac{1}{2} a^2_n -1, n= 0, 1,\cdots\] Prove that there exists a positive number $q, q < 1$, such that for all $n = 1, 2, \ldots ,$ \[|a_{n+1} - a_n| \leq q|a_n - a_{n-1}|,\] and give one such $q$ explicitly.

$ A,B$ are fixed points. Variable line $ l$ passes through the fixed point $ C$. There are two circles passing through $ A,B$ and tangent to $ l$ at $ M,N$. Prove that circumcircle of $ AMN$ passes through a fixed point.

$30$ volleyball teams have participated in a league. Any two teams have played a match with each other exactly once. At the end of the league, a match is called [b]unusual[/b] if at the end of the league, the winner of the match have a smaller amount of wins than the loser of the match. A team is called [b]astonishing[/b] if all its matches are [b]unusual[/b] matches. Find the maximum number of [b]astonishing[/b] teams.

An arrangement of chips in the squares of $ n\times n$ table is called [i]sparse[/i] if every $ 2\times 2$ square contains at most 3 chips. Serge put chips in some squares of the table (one in a square) and obtained a sparse arrangement. He noted however that if any chip is moved to any free square then the arrangement is no more sparce. For what $ n$ is this possible? [i]Proposed by S. Berlov[/i]

$(GBR 1)$ The polynomial $P(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k$, where $a_0,\cdots, a_k$ are integers, is said to be divisible by an integer $m$ if $P(x)$ is a multiple of $m$ for every integral value of $x$. Show that if $P(x)$ is divisible by $m$, then $a_0 \cdot k!$ is a multiple of $m$. Also prove that if $a, k,m$ are positive integers such that $ak!$ is a multiple of $m$, then a polynomial $P(x)$ with leading term $ax^k$can be found that is divisible by $m.$