Some cards each have a pair of numbers written on them. There is just one card for each pair $(a,b)$ with $1 \leq a < b \leq 2003$. Two players play the following game. Each removes a card in turn and writes the product $ab$ of its numbers on the blackboard. The first player who causes the greatest common divisor of the numbers on the blackboard to fall to $1$ loses. Which player has a winning strategy?

[u]Set 2[/u] [b]2.1[/b] What is the last digit of $2022 + 2022^{2022} + 2022^{(2022^{2022})}$? [b]2.2[/b] Let $T$ be the answer to the previous problem. CMIMC executive members are trying to arrange desks for CMWMC. If they arrange the desks into rows of $5$ desks, they end up with $1$ left over. If they instead arrange the desks into rows of $7$ desks, they also end up with $1$ left over. If they instead arrange the desks into rows of $11$ desks, they end up with $T$ left over. What is the smallest possible (non-negative) number of desks they could have? [b]2.3[/b] Let $T$ be the answer to the previous problem. Compute the largest value of $k$ such that $11^k$ divides $$T! = T(T - 1)(T - 2)...(2)(1).$$ PS. You should use hide for answers.

The positive integers $ a$ and $ b$ are such that the numbers $ 15a \plus{} 16b$ and $ 16a \minus{} 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

Let $a, b, c$, and $d$ be prime numbers with $a \le b \le c \le d > 0$. Suppose $a^2 + 2b^2 + c^2 + 2d^2 = 2(ab + bc - cd + da)$. Find $4a + 3b + 2c + d$.

You come across an ancient mathematical manuscript. It reads, "To find out whether a number is divisible by seventeen, take the number formed by the last two digits of the number, subtract the number formed by the third- and fourth-to-last digits of the number, add the number formed by the fifth- and sixth-to-last digits of the number and so on. The resulting number is divisible by seventeen if and only if the original number is divisible by seventeen." What is the sum of the five smallest bases the ancient culture might have been using? (Note that "seventeen" is the number represented by $17$ in base $10$, not $17$ in the base that the ancient culture was using. Express your answer in base $10$.)

Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]

Consider the sequence $a_1, a_2, \dots$ of positive integers such that $a_1=2$ and $a_{n+1}=a_n^4+a_n^3-3a_n^2-a_n+2$, for all $n\geqslant 1$. Prove that there exist infinitely many prime numbers that don't divide any term of the sequence. [i]Proposed by Pavel Ciurea[/i]

If $x,y,z\in\mathbb N$ and $xy=z^2+1$ prove that there exists integers $a,b,c,d$ such that $x=a^2+b^2$, $y=c^2+d^2$, $z=ac+bd$.

Given is a positive integer $n$ divisible by $3$ and such that $2n-1$ is a prime. Does there exist a positive integer $x>n$ such that $$nx^{n+1}+(2n+1)x^n-3(n-1)x^{n-1}-x-3$$ is a product of the first few odd primes?

Each of the four positive integers $N,N +1,N +2,N +3$ has exactly six positive divisors. There are exactly$ 20$ dierent positive numbers which are exact divisors of at least one of the numbers. One of these is $27$. Find all possible values of $N$.(Both $1$ and $m$ are counted as divisors of the number $m$.)

Prove that there are no integers $x$ and $y$ satisfying $x^{2}=y^{5}-4$.

Let $n$ be a positive integer. Let $T_n$ be a set of positive integers such that: \[{T_n={ \{11(k+h)+10(n^k+n^h)| (1 \leq k,h \leq 10)}}\}\] Find all $n$ for which there don't exist two distinct positive integers $a, b \in T_n$ such that $a\equiv b \pmod{110}$

Let[i] $a,b,c$[/i] be positive integers , such that $A=\frac{a^2+1}{bc}+\frac{b^2+1}{ca}+\frac{c^2+1}{ab}$ is, also, an integer. Proof that $\gcd( a, b, c)\leq\lfloor\sqrt[3]{a+ b+ c}\rfloor$.

Suppose $\tan \alpha = \dfrac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Prove that the number $\tan \beta$ for which $\tan {2 \beta} = \tan {3 \alpha}$ is rational only when $p^2 + q^2$ is the square of an integer.

Find all positive integer $n(\ge 2)$ and rational $\beta \in (0,1)$ satisfying the following: There exist positive integers $a_1,a_2,...,a_n$, such that for any set $I \subseteq \{1,2,...,n\}$ which contains at least two elements, $$ S(\sum_{i\in I}a_i)=\beta \sum_{i\in I}S(a_i). $$ where $S(n)$ denotes sum of digits of decimal representation of $n$.

A number $n$ is satisfying the conditions below i) $n$ is a positive odd integer; ii) there are some odd integers such that their squares' sum is equal to $n^{4}$. Find all such numbers.

Let $a>1$ be a positive integer. Show that the set of integers \[\{a^2+a-1,a^3+a^2-1,\ldots ,a^{n+1}+a^n-1,\ldots\}\] contains an infinite subset of pairwise coprime integers. [i]Mircea Becheanu[/i]

Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Carlos and Yue play the following game: First Carlos writes a $+$ sign or a $-$ sign in front of each of the $50$ numbers $1,2,\cdots,50$. Then, in turns, each one chooses a number from the sequence obtained; Start by choosing Yue. If the absolute value of the sum of the $25$ numbers that Carlos chose is greater than or equal to the absolute value of the sum of the $25$ numbers that Yue chose, Carlos wins. In the other case, Yue wins. Determine which of the two players can develop a strategy that will ensure victory, no matter how well their opponent plays, and describe said strategy.

For every positive integer $N$, let $\sigma(N)$ denote the sum of the positive integer divisors of $N$. Find all integers $m\geq n\geq 2$ satisfying \[\frac{\sigma(m)-1}{m-1}=\frac{\sigma(n)-1}{n-1}=\frac{\sigma(mn)-1}{mn-1}.\] [i]Ankan Bhattacharya[/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 pairs of positive integers $x,y$ such that $\frac{4}{x}+\frac{2}{y}=1$

Strings $a_1, a_2, ... , a_{2016}$ and $b_1, b_2, ... , b_{2016}$ each contain all natural numbers from $1$ to $2016$ exactly once each (in other words, they are both permutations of the numbers $1, 2, ..., 2016$). Prove that different indices $i$ and $j$ can be found such that $a_ib_i- a_jb_j$ is divisible by $2017$.

The number $201212200619$ has a factor $m$ such that $6 \cdot 10^9 <m <6.5 \cdot 10^9$. Find $m$.

[u]Round 1[/u] [b]p1.[/b] Goldilocks enters the home of the three bears – Papa Bear, Mama Bear, and Baby Bear. Each bear is wearing a different-colored shirt – red, green, or blue. All the bears look the same to Goldilocks, so she cannot otherwise tell them apart. The bears in the red and blue shirts each make one true statement and one false statement. The bear in the red shirt says: “I'm Blue's dad. I'm Green's daughter.” The bear in the blue shirt says: “Red and Green are of opposite gender. Red and Green are my parents.” Help Goldilocks find out which bear is wearing which shirt. [b]p2.[/b] The University of Washington is holding a talent competition. The competition has five contests: math, physics, chemistry, biology, and ballroom dancing. Any student can enter into any number of the contests but only once for each one. For example, a student may participate in math, biology, and ballroom. It turned out that each student participated in an odd number of contests. Also, each contest had an odd number of participants. Was the total number of contestants odd or even? [b]p3.[/b] The $99$ greatest scientists of Mars and Venus are seated evenly around a circular table. If any scientist sees two colleagues from her own planet sitting an equal number of seats to her left and right, she waves to them. For example, if you are from Mars and the scientists sitting two seats to your left and right are also from Mars, you will wave to them. Prove that at least one of the $99$ scientists will be waving, no matter how they are seated around the table. [b]p4.[/b] One hundred boys participated in a tennis tournament in which every player played each other player exactly once and there were no ties. Prove that after the tournament, it is possible for the boys to line up for pizza so that each boy defeated the boy standing right behind him in line. [b]p5.[/b] To celebrate space exploration, the Science Fiction Museum is going to read Star Wars and Star Trek stories for $24$ hours straight. A different story will be read each hour for a total of $12$ Star Wars stories and $12$ Star Trek stories. George and Gene want to listen to exactly $6$ Star Wars and $6$ Star Trek stories. Show that no matter how the readings are scheduled, the friends can find a block of $12$ consecutive hours to listen to the stories together. [u]Round 2[/u] [b]p6.[/b] $2013$ people attended Cinderella's ball. Some of the guests were friends with each other. At midnight, the guests started turning into mice. After the first minute, everyone who had no friends at the ball turned into a mouse. After the second minute, everyone who had exactly one friend among the remaining people turned into a mouse. After the third minute, everyone who had two human friends left in the room turned into a mouse, and so on. What is the maximal number of people that could have been left at the ball after $2013$ minutes? [b]p7.[/b] Bill and Charlie are playing a game on an infinite strip of graph paper. On Bill’s turn, he marks two empty squares of his choice (not necessarily adjacent) with crosses. Charlie, on his turn, can erase any number of crosses, as long as they are all adjacent to each other. Bill wants to create a line of $2013$ crosses in a row. Can Charlie stop him? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].