Let $x, y, z$ be positive integers and $p$ a prime such that $x <y <z <p$. Also $x^3, y^3, z^3$ leave the same remainder when divided by $p$. Prove that $x + y + z$ divides $x^2 + y^2 + z^2$.

Show that the equation $$a^2b=2017(a+b)$$ has no solutions for positive integers $a$ and $b$. [i]Proposed by Oriol Solé[/i]

Find all pairs of integers $(x,y)$ for which $x^z+z^x=(x+z)!$.

A sequence $a_1, a_2, a_3, \ldots$ of positive integers satisfies $a_1 > 5$ and $a_{n+1} = 5 + 6 + \cdots + a_n$ for all positive integers $n$. Determine all prime numbers $p$ such that, regardless of the value of $a_1$, this sequence must contain a multiple of $p$.

Find all positive integer solutions $(x,y,z,n)$ of equation $x^{2n+1}-y^{2n+1}=xyz+2^{2n+1}$, where $n\ge 2$ and $z \le 5\times 2^{2n}$.

15. Let $a_{n}$ denote the number of ternary strings of length $n$ so that there does not exist a $k<n$ such that the first $k$ digits of the string equals the last $k$ digits. What is the largest integer $m$ such that $3^{m} \mid a_{2023}$ ?

A positive proper divisor is a positive divisor of a number, excluding itself. For positive integers $n \ge 2$, let $f(n)$ denote the number that is one more than the largest proper divisor of $n$. Determine all positive integers $n$ such that $f(f(n)) = 2$.

$(FRA 5)$ Let $\alpha(n)$ be the number of pairs $(x, y)$ of integers such that $x+y = n, 0 \le y \le x$, and let $\beta(n)$ be the number of triples $(x, y, z)$ such that$ x + y + z = n$ and $0 \le z \le y \le x.$ Find a simple relation between $\alpha(n)$ and the integer part of the number $\frac{n+2}{2}$ and the relation among $\beta(n), \beta(n -3)$ and $\alpha(n).$ Then evaluate $\beta(n)$ as a function of the residue of $n$ modulo $6$. What can be said about $\beta(n)$ and $1+\frac{n(n+6)}{12}$? And what about $\frac{(n+3)^2}{6}$? Find the number of triples $(x, y, z)$ with the property $x+ y+ z \le n, 0 \le z \le y \le x$ as a function of the residue of $n$ modulo $6.$What can be said about the relation between this number and the number $\frac{(n+6)(2n^2+9n+12)}{72}$?

A [i]Benelux n-square[/i] (with $n\geq 2$) is an $n\times n$ grid consisting of $n^2$ cells, each of them containing a positive integer, satisfying the following conditions: $\bullet$ the $n^2$ positive integers are pairwise distinct. $\bullet$ if for each row and each column we compute the greatest common divisor of the $n$ numbers in that row/column, then we obtain $2n$ different outcomes. (a) Prove that, in each Benelux n-square (with $n \geq 2$), there exists a cell containing a number which is at least $2n^2.$ (b) Call a Benelux n-square [i]minimal[/i] if all $n^2$ numbers in the cells are at most $2n^2.$ Determine all $n\geq 2$ for which there exists a minimal Benelux n-square.

Prove that if the function $ ax^2 + bx + c $ takes an integer value for every integer value of the variable $ x $, then $ 2a $, $ a + b $, $ c $ are integers and vice versa.

Let $X= a1b9$ and $Y ab = 51ab$ be two positive integers where $a$ and $b$ are digits. $X$ is known to be multiple of a positive two-digit number $n$ and $Y$ is the next multiple of that number $n$. Find the number $n$ and the digits $a$ and $b$. Justify why there are no other possibilities.

For all positive integer $ n \ge 2 $, prove that $ 2^n -1 $ can't be a divisor of $ 3^n -1 $.

Prove that the product of four consecutive positive integers cannot be a perfect square or cube.

Are there any positive integers $m$ and $n$ satisfying the equation $m^3 = 9n^4 + 170n^2 + 289$ ?

Find all functions $f : \mathbb{N}\rightarrow{\mathbb{N}}$ such that for all positive integers $m$ and $n$ the number $f(m)+n-m$ is divisible by $f(n)$.

Show that if $m$ and $n$ are integers such that $m^2 + mn + n^2$ is divisible by $9$, then they must both be divisible by $3$.

$S$ is any sequence of at least $3$ positive integers. A move is to take any $a, b$ in the sequence such that neither divides the other and replace them by gcd $(a,b)$ and lcm $(a,b)$. Show that only finitely many moves are possible and that the final result is independent of the moves made, except possibly for order.

[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].

Let $v_0 = 0, v_1 = 1$ and $v_{n+1} = 8 \cdot v_n - v_{n-1},$ $n = 1,2, ...$. Prove that in the sequence $\{v_n\}$ there aren't terms of the form $3^{\alpha} \cdot 5^{\beta}$ with $\alpha, \beta \in \mathbb{N}.$

Prove that there exists infinitely many pairs of rational numbers $(\frac{p_1}{q},\frac{p_2}{q})$ with $p_1,p_2,q\in \mathbb N$ with the following condition: \[|\sqrt{3}-\frac{p_1}{q}|<q^{-\frac{3}{2}}, |\sqrt{2}-\frac{p_2}{q}|< q^{-\frac{3}{2}}.\] [i]Proposed by Mohammad Gharakhani[/i]

Does there exist an even positive integer $n$ for which $n+1$ is divisible by $5$ and the two numbers $2^n + n$ and $2^n -1$ are co-prime?

Prove that the equation \[ x^z + y^z = z^z \] has no solutions in postive integers.

A person flips $2010$ coins at a time. He gains one penny every time he flips a prime number of heads, but must stop once he flips a non-prime number. If his expected amount of money gained in dollars is $\frac{a}{b}$, where $a$ and $b$ are relatively prime, compute $\lceil\log_{2}(100a+b)\rceil$. [i]Proposed by Lewis Chen[/i]

Solve in equation: $ x^2+y^2+z^2+w^2=3(x+y+z+w) $ where $ x,y,z,w $ are positive integers.

Let $a_1, a_2, a_3, \dots, a_{20}$ be a permutation of the numbers $1, 2, \dots, 20$. How many different values can the expression $a_1-a_2+a_3-\dots - a_{20}$ have?