[u]Round 1[/u] [b]p1.[/b] Six pirates – Captain Jack and his five crewmen – sit in a circle to split a treasure of $99$ gold coins. Jack must decide how many coins to take for himself and how many to give each crewman (not necessarily the same number to each). The five crewmen will then vote on Jack's decision. Each is greedy and will vote “aye” only if he gets more coins than each of his two neighbors. If a majority vote “aye”, Jack's decision is accepted. Otherwise Jack is thrown overboard and gets nothing. What is the most coins Captain Jack can take for himself and survive? [b]p2[/b]. Rose and Bella take turns painting cells red and blue on an infinite piece of graph paper. On Rose's turn, she picks any blank cell and paints it red. Bella, on her turn, picks any blank cell and paints it blue. Bella wins if the paper has four blue cells arranged as corners of a square of any size with sides parallel to the grid lines. Rose goes first. Show that she cannot prevent Bella from winning. [img]https://cdn.artofproblemsolving.com/attachments/d/6/722eaebed21a01fe43bdd0dedd56ab3faef1b5.png[/img] [b]p3.[/b] A $25\times 25$ checkerboard is cut along the gridlines into some number of smaller square boards. Show that the total length of the cuts is divisible by $4$. For example, the cuts shown on the picture have total length $16$, which is divisible by $4$. [img]https://cdn.artofproblemsolving.com/attachments/c/1/e152130e48b804fe9db807ef4f5cd2cbad4947.png[/img] [b]p4.[/b] Each robot in the Martian Army is equipped with a battery that lasts some number of hours. For any two robots, one's battery lasts at least three times as long as the other's. A robot works until its battery is depleted, then recharges its battery until it is full, then goes back to work, and so on. A battery that lasts $N$ hours takes exactly $N$ hours to recharge. Prove that there will be a moment in time when all the robots are recharging (so you can invade the planet). [b]p5.[/b] A casino machine accepts tokens of $32$ different colors, one at a time. For each color, the player can choose between two fixed rewards. Each reward is up to $\$10$ cash, plus maybe another token. For example, a blue token always gives the player a choice of getting either $\$5$ plus a red token or $\$3$ plus a yellow token; a black token can always be exchanged either for $\$10$ (but no token) or for a brown token (but no cash). A player may keep playing as long as he has a token. Rob and Bob each have one white token. Rob watches Bob play and win $\$500$. Prove that Rob can win at least $\$1000$. [img]https://cdn.artofproblemsolving.com/attachments/6/6/e55614bae92233c9b2e7d66f5f425a18e6475a.png [/img] [u]Round 2[/u] [b]p6.[/b] The sum of $2015$ rational numbers is an integer. The product of every pair of them is also an integer. Prove that they are all integers. (A rational number is one that can be written as $m/n$, where $m$ and $n$ are integers and $n\ne 0$.) [b]p7.[/b] An $N \times N$ table is filled with integers such that numbers in cells that share a side differ by at most $1$. Prove that there is some number that appears in the table at least $N$ times. For example, in the $5 \times 5$ table below the numbers $1$ and $2$ appear at least $5$ times. [img]https://cdn.artofproblemsolving.com/attachments/3/8/fda513bcfbe6834d88fb8ca0bfcdb504d8b859.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[u]Round 1[/u] [b]p1.[/b] Let $ABCDEF$ be a regular hexagon. How many acute triangles have all their vertices among the vertices of $ABCDEF$? [b]p2.[/b] A rectangle has a diagonal of length $20$. If the width of the rectangle is doubled, the length of the diagonal becomes $22$. Given that the width of the original rectangle is $w$, compute $w^2$. [b]p3.[/b] The number $\overline{2022A20B22}$ is divisible by 99. What is $A + B$? [u]Round 2[/u] [b]p4.[/b] How many two-digit positive integers have digits that sum to at least $16$? [b]p5.[/b] For how many integers $k$ less than $10$ do there exist positive integers x and y such that $k =x^2 - xy + y^2$? [b]p6.[/b] Isosceles trapezoid $ABCD$ is inscribed in a circle of radius $2$ with $AB \parallel CD$, $AB = 2$, and one of the interior angles of the trapezoid equal to $110^o$. What is the degree measure of minor arc $CD$? [u]Round 3[/u] [b]p7.[/b] In rectangle $ALEX$, point $U$ lies on side $EX$ so that $\angle AUL = 90^o$. Suppose that $UE = 2$ and $UX = 12$. Compute the square of the area of $ALEX$. [b]p8.[/b] How many digits does $20^{22}$ have? [b]p9.[/b] Compute the units digit of $3 + 3^3 + 3^{3^3} + ... + 3^{3^{...{^3}}}$ , where the last term of the series has $2022$ $3$s. [u]Round 4[/u] [b]p10.[/b] Given that $\sqrt{x - 1} + \sqrt{x} = \sqrt{x + 1}$ for some real number $x$, the number $x^2$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]p11.[/b] Eric the Chicken Farmer arranges his $9$ chickens in a $3$-by-$3$ grid, with each chicken being exactly one meter away from its closest neighbors. At the sound of a whistle, each chicken simultaneously chooses one of its closest neighbors at random and moves $\frac12$ of a unit towards it. Given that the expected number of pairs of chickens that meet can be written as $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers, compute $p + q$. [b]p12.[/b] For a positive integer $n$, let $s(n)$ denote the sum of the digits of $n$ in base $10$. Find the greatest positive integer $n$ less than $2022$ such that $s(n) = s(n^2)$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2949432p26408285]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Define a positive integer $n$ to be [i]almost-cubic [/i] if it becomes a perfect cube upon concatenating the digit $5$. For example, $12$ is almost-cubic because $125 = 5^3$. Find the remainder when the sum of all almost-cubic $n < 10^8$ is divided by $1000$.

Three rational number $x,p,q$ satisfy $p^2-xq^2$=1. Prove that there are integers $a,b$ such that $p=\frac{a^2+xb^2}{a^2-xb^2}$ and $q=\frac{2ab}{a^2-xb^2}$.

Find the remainder when $8^{2014}$ + $6^{2014}$ is divided by 100.

Show that for any positive integer $n\ge 4$, there exists a multiple of $n^3$ between $n!$ and $(n + 1)!$

[b]p1.[/b] Consider this statement: An equilateral polygon circumscribed about a circle is also equiangular. Decide whether this statement is a true or false proposition in euclidean geometry. If it is true, prove it; if false, produce a counterexample. [b]p2.[/b] Show that the fraction $\frac{x^2-3x+1}{x-3}$ has no value between $1$ and $5$, for any real value of $x$. [b]p3.[/b] A man walked a total of $5$ hours, first along a level road and then up a hill, after which he turned around and walked back to his starting point along the same route. He walks $4$ miles per hour on the level, three miles per hour uphill, and $r$ miles per hour downhill. For what values of $r$ will this information uniquely determine his total walking distance? [b]p4.[/b] A point $P$ is so located in the interior of a rectangle that the distance of $P$ from one comer is $5$ yards, from the opposite comer is $14$ yards, and from a third comer is $10$ yards. What is the distance from $P$ to the fourth comer? [b]p5.[/b] Each small square in the $5$ by $5$ checkerboard shown has in it an integer according to the following rules: $\begin{tabular}{|l|l|l|l|l|} \hline & & & & \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & \\ \hline \end{tabular}$ i. Each row consists of the integers $1, 2, 3, 4, 5$ in some order. ii. Тhе order of the integers down the first column has the same as the order of the integers, from left to right, across the first row and similarly fог any other column and the corresponding row. Prove that the diagonal squares running from the upper left to the lower right contain the numbers $1, 2, 3, 4, 5$ in some order. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all integer solutions to the equation $7x^2y^2 + 4x^2 = 77y^2 + 1260$.

When $12{}^1{}^8$ is divided by $18{}^1{}^2$, the result is $(\tfrac{m}{n})^3$, where $m$ and $n$ are relatively prime integers. Find $m-n$.

Let $S$ be the set of all the odd positive integers that are not multiples of $5$ and that are less than $30m$, $m$ being an arbitrary positive integer. What is the smallest integer $k$ such that in any subset of $k$ integers from $S$ there must be two different integers, one of which divides the other?

The real sequence $\{a_n|n=0,1,2,3,...\}$ defined $a_0=1$ and \[ a_{n+1}=\frac{1}{2}\left (a_{n}+\frac{1}{3 \cdot a_{n}} \right ). \] Denote \[ A_n=\frac{3}{3 \cdot a_n^2-1}. \] Prove that $A_n$ is a perfect square and it has at least $n$ distinct prime divisors.

Let $n>1$ be a positive integer. Show that the number of residues modulo $n^2$ of the elements of the set $\{ x^n + y^n : x,y \in \mathbb{N} \}$ is at most $\frac{n(n+1)}{2}$. [I]Proposed by N. Safaei (Iran)[/i]

Find the sum of all integers $n$ such that $1 < n < 30$ and $n$ divides $$1+\sum^{n-1}_{k=1}k^{2k}.$$

How many positive integers less than $1998$ are relatively prime to $1547$? (Two integers are relatively prime if they have no common factors besides 1.)

Show that there exist no integer solutions $(x, y, z)$ to the equation $$x^3+2y^3+4z^3=9$$

Find all solutions of the equation $(a^a)^5 = b^b$ in positive integers.

Does there exist a natural number $n$ in whose decimal representation each digit occurs at least $2006$ times and which has the property that you can find two different digits in its decimal representation such that the number obtained from $n$ by interchanging these two digits is different from $n$ and has the same set of prime divisors as $n$ ?

Prove that for any integers $ n> m> 0 $ the number $ 2 ^n-1 $ has a prime divisor not dividing $ 2 ^m-1 $.

Show that for each integer $a$, there is a unique decomposition \[ a = \sum_{j=0}^{n} d_j 2^j , d_j \in (-1,0,1) \] such that no two consecutive $d_j$'s are nonzero. Show further that if $f$ is nondecreasing function from the set of all non-negative integers in to the set of all non-negative real numbers, and if $a = \sum_{j=0}^{n} c_j 2^j$ is any other decomposition of $a$ with $c_j \in (-1,0,1)$ , then \[ \sum_{j=0}^{n} |d_j| f(j) \leq \sum_{j=0}^{n} |c_j| f(j) \]

How many permutations $a_1, a_2, a_3, a_4$ of $1, 2, 3, 4$ satisfy the condition that for $k = 1, 2, 3,$ the list $a_1,. . . , a_k$ contains a number greater than $k$?

Anna, Ben and Chris sit at the round table passing and eating nuts. At first only Anna has the nuts that she divides equally between Ben and Chris, eating a leftover (if there is any). Then Ben does the same with his pile. Then Chris does the same with his pile. The process repeats itself: each of the children divides his/her pile of nuts equally between his/her neighbours eating the leftovers if there are any. Initially, the number of nuts is large enough (more than 3). Prove that a) at least one nut is eaten; [i](3 points)[/i] b) all nuts cannot be eaten. [i](3 points)[/i]

Rothschild the benefactor has a certain number of coins. A man comes, and Rothschild wants to share his coins with him. If he has an even number of coins, he gives half of them to the man and goes away. If he has an odd number of coins, he donates one coin to charity so he can have an even number of coins, but meanwhile another man comes. So now he has to share his coins with two other people. If it is possible to do so evenly, he does so and goes away. Otherwise, he again donates a few coins to charity (no more than 3). Meanwhile, yet another man comes. This goes on until Rothschild is able to divide his coins evenly or until he runs out of money. Does there exist a natural number $N$ such that if Rothschild has at least $N$ coins in the beginning, he will end with at least one coin?

Solve in N $1/x+2/y-4/z=1$

Determine whether there exists a positive integer $a$ such that $$2015a,2016a,\dots,2558a$$ are all perfect power.

Positive integers $a$ and $b$ are given such that each of the numbers $ab$ and $(a+1)(b+1)$ is a perfect square. Prove that there exists an integer $n>1$ such that the number $(a+n)(b+n)$ is a perfect square.