Prove that there are infinitely many positive integers $m$ such that the number of odd distinct prime factor of $m(m+3)$ is a multiple of $3$.

For some positive integers $m,n$, the system $x+y^2 = m$ and $x^2+y = n$ has exactly one integral solution $(x,y)$. Determine all possible values of $m-n$.

Let $A$ be a set of $16$ positive integers with the property that the product of any two distinct members of $A$ will not exceed 1994. Show that there are numbers $a$ and $b$ in the set $A$ such that the gcd of $a$ and $b$ is greater than 1.

Find all triples of positive integers $(a, b, c)$ such that $$(2^a-1)(3^b-1)=c!.$$

Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.

[u]Round 1[/u] [b]p1.[/b] Ten children arrive at a birthday party and leave their shoes by the door. All the children have different shoe sizes. Later, as they leave one at a time, each child randomly grabs a pair of shoes their size or larger. After some kids have left, all of the remaining shoes are too small for any of the remaining children. What is the greatest number of shoes that might remain by the door? [b]p2.[/b] Turans, the king of Saturn, invented a new language for his people. The alphabet has only $6$ letters: A, N, R, S, T, U; however, the alphabetic order is different than in English. A word is any sequence of $6$ different letters. In the dictionary for this language, the first word is SATURN. Which word follows immediately after TURANS? [b]p3.[/b] Benji chooses five integers. For each pair of these numbers, he writes down the pair's sum. Can all ten sums end with different digits? [b]p4.[/b] Nine dwarves live in a house with nine rooms arranged in a $3\times3$ square. On Monday morning, each dwarf rubs noses with the dwarves in the adjacent rooms that share a wall. On Monday night, all the dwarves switch rooms. On Tuesday morning, they again rub noses with their adjacent neighbors. On Tuesday night, they move again. On Wednesday morning, they rub noses for the last time. Show that there are still two dwarves who haven't rubbed noses with one another. [b]p5.[/b] Anna and Bobby take turns placing rooks in any empty square of a pyramid-shaped board with $100$ rows and $200$ columns. If a player places a rook in a square that can be attacked by a previously placed rook, he or she loses. Anna goes first. Can Bobby win no matter how well Anna plays? [img]https://cdn.artofproblemsolving.com/attachments/7/5/b253b655b6740b1e1310037da07a0df4dc9914.png[/img] [u]Round 2[/u] [b]p6.[/b] Some boys and girls, all of different ages, had a snowball fight. Each girl threw one snowball at every kid who was older than her. Each boy threw one snowball at every kid who was younger than him. Three friends were hit by the same number of snowballs, and everyone else took fewer hits than they did. Prove that at least one of the three is a girl. [b]p7.[/b] Last year, jugglers from around the world travelled to Jakarta to participate in the Jubilant Juggling Jamboree. The festival lasted $32$ days, with six solo performances scheduled each day. The organizers noticed that for any two days, there was exactly one juggler scheduled to perform on both days. No juggler performed more than once on a single day. Prove there was a juggler who performed every day. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Compute $g(2005)$ where $g$ is a function defined on the natural numbers that has the following properties: i) $g(1) = 0$ ii) $g(nm) = g(n) + g(m) + g(n)g(m)$ for any pair of integers $n, m$. iii) $g(n^2 + 1) = (g(n) + 1)^2$ for every integer $n$.

Let $n$ be a positive integer of the form $4k+1$, $k\in \mathbb N$ and $A = \{ a^2 + nb^2 \mid a,b \in \mathbb Z\}$. Prove that there exist integers $x,y$ such that $x^n+y^n \in A$ and $x+y \notin A$.

Let $x_0=a,x_1=b$ and $x_{n+1}=2x_n-9x_{n-1}$ for each $n\in\mathbb N$, where $a,b$ are integers. Find the necessary and sufficient condition on $a$ and $b$ for the existence of an $x_n$ which is a multiple of $7$.

Prove that in any set of $17$ distinct natural numbers one can either find five numbers so that four of them are divisible into the other or five numbers none of which is divisible into any other. (An established theorem)

The numbers $1,2,3,\dots,2017$ are on the blackboard. Amelie and Boris take turns removing one of those until only two numbers remain on the board. Amelie starts. If the sum of the last two numbers is divisible by $8$, then Amelie wins. Else Boris wins. Who can force a victory?

For a positive integer $n$ let $a_n$ denote the last digit of $n^{(n^n)}$. Prove that the sequence $(a_n)$ is periodic and determine the length of the minimal period.

A palindrome is a number that remains the same when its digits are reversed. Let $n$ be a product of distinct primes not divisible by $10$. Prove that infinitely many multiples of $n$ are palindromes.

Let $a_1,a_2,\ldots,a_n,\ldots$ be any permutation of all positive integers. Prove that there exist infinitely many positive integers $i$ such that $\gcd(a_i,a_{i+1})\leq \frac{3}{4} i$.

Determine all integers $n>1$ such that every power of $n$ has an odd number of digits.

For even positive integers $a>1$. Prove that there are infinite positive integers $n$ that makes $n | a^n+1$. [i](tomoyo-jung)[/i]

Suppose that $m, n, k$ are positive integers satisfying $$3mk=(m+3)^n+1.$$ Prove that $k$ is odd.

Given six consecutive positive integers, prove that there exists a prime such that one and only one of these six integers is divisible by this prime.

Determine the number of positive divisors of $2008^8$ that are less than $2008^4$.

Let $K$ be an odd number st $S_2{(K)} = 2$ and let $ab=K$ where $a,b$ are positive integers. Show that if $a,b>1$ and $l,m >2$ are positive integers st:$S_2{(a)} < l$ and $S_2{(b)} < m$ then : $$K \leq 2^{lm-6} +1$$ ($S_2{(n)}$ is the sum of digits of $n$ written in base 2)

Let $a,b$ and $c$ be integers and $r$ a real number such that $ar^2+br+c=0$ with $ac\not =0$.Prove that $\sqrt{r^2+c^2}$ is an irrational number

Prove that there exists a number divisible by $5^{1000}$ not containing a single zero in its decimal notation.

A sequence of integers is constructed as follows: $a_1$ is an arbitrary three-digit number, $a_2$ is the sum of squares of the digits of $a_1, a_3$ is the sum of squares of the digits of $a_2$, etc. Prove that either $1$ or $4$ must occur in the sequence $a_1, a_2, a_3, ....$

Define the function $f(n)$ for positive integers $n$ as follows: if $n$ is prime, then $f(n) = 1$; and $f(ab) = a \cdot f(b)+f(a)\cdot b$ for all positive integers $a$ and $b$. How many positive integers $n$ less than $5^{50}$ have the property that $f(n) = n$?

Let $\{a_i\}_{i=1}^{\infty}$ be a sequence of positive integers such that $a_1<a_2<a_3\cdots$ and all of primes are members of this sequence. Prove that for every $n<m$ \[\dfrac{1}{a_n} + \dfrac{1}{a_{n+1}} + \cdots + \dfrac{1}{a_m} \not \in \mathbb N\]