Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Does there exist a sequence of $2017$ consecutive integers which contains exactly $17$ primes?

Let $n$ be a positive integer. Prove that the number of ways to express $n$ as a sum of distinct positive integers (up to order) and the number of ways to express $n$ as a sum of odd positive integers (up to order) are the same.

For any positive integer $n$, define the subset $S_n$ of natural numbers as follow $$ S_n = \left\{x^2+ny^2 : x,y \in \mathbb{Z} \right\}.$$ Find all positive integers $n$ such that there exists an element of $S_n$ which [u]doesn't belong[/u] to any of the sets $S_1, S_2,\dots,S_{n-1}$. [i]Proposed by Yahya Motevassel[/i]

A positive integer $N$ has base-eleven representation $\underline{a}\,\underline{b}\,\underline{c}$ and base-eight representation $\underline{1}\,\underline{b}\,\underline{c}\,\underline{a}$, where $a$, $b$, and $c$ represent (not necessarily distinct) digits. Find the least such $N$ expressed in base ten.

There were $n{}$ positive integers. For each pair of those integers Boris wrote their arithmetic mean onto a blackboard and their geometric mean onto a whiteboard. It so happened that for each pair at least one of those means was integer. Prove that on at least one of the boards all the numbers are integer. [i]Boris Frenkin[/i]

Fix an odd prime number $p$. Find the largest positive integer $n$ such that there exist points $A_1,A_2,\cdots,A_n$ in the plane with integral coordinates, no three points are collinear. Moreover, for any $1\le i<j<k\le n$, $p \nmid 2S_{\Delta A_iA_j A_k}.$

Consider the set $E$ of all natural numbers $n$ such that whenn divided by $11, 12, 13$, respectively, the remainders, int that order, are distinct prime numbers in an arithmetic progression. If $N$ is the largest number in $E$, find the sum of digits of $N$.

[b]p1.[/b] Prove that no matter what digits are placed in the four empty boxes, the eight-digit number $9999\Box\Box\Box\Box$ is not a perfect square. [b]p2.[/b] Prove that the number $m/3+m^2/2+m^3/6$ is integral for all integral values of $m$. [b]p3.[/b] An elevator in a $100$ store building has only two buttons: UP and DOWN. The UP button makes the elevator go $13$ floors up, and the DOWN button makes it go $8$ floors down. Is it possible to go from the $13$th floor to the $8$th floor? [b]p4.[/b] Cut the triangle shown in the picture into three pieces and rearrange them into a rectangle. (Pieces can not overlap.) [img]https://cdn.artofproblemsolving.com/attachments/4/b/ca707bf274ed54c1b22c4f65d3d0b0a5cfdc56.png[/img] [b]p5.[/b] Two players Tom and Sid play the following game. There are two piles of rocks, $7$ rocks in the first pile and $9$ rocks in the second pile. Each of the players in his turn can take either any amount of rocks from one pile or the same amount of rocks from both piles. The winner is the player who takes the last rock. Who does win in this game if Tom starts the game? [b]p6.[/b] In the next long multiplication example each letter encodes its own digit. Find these digits. $\begin{tabular}{ccccc} & & & a & b \\ * & & & c & d \\ \hline & & c & e & f \\ + & & a & b & \\ \hline & c & f & d & f \\ \end{tabular}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The sequence $\left(a_{n}\right)_{n\in\mathbb{N}}$ is defined recursively as $a_{0}=a_{1}=1$, $a_{n+2}=5a_{n+1}-a_{n}-1$, $\forall n\in\mathbb{N}$ Prove that $$a_{n}\mid a_{n+1}^{2}+a_{n+1}+1$$ for any $n\in\mathbb{N}$

Determine all the pairs $ (p , n )$ of a prime number $ p$ and a positive integer $ n$ for which $ \frac{ n^p + 1 }{p^n + 1} $ is an integer.

How many pairs of positive integers $(a,b)$ are there such that $a < b$ and $a,b$ can be the legs of a right triangle with hypotenuse $340$?

Let $M$ be a subset of $\mathbb{R}$ such that the following conditions are satisfied: a) For any $x \in M, n \in \mathbb{Z}$, one has that $x+n \in \mathbb{M}$. b) For any $x \in M$, one has that $-x \in M$. c) Both $M$ and $\mathbb{R}$ \ $M$ contain an interval of length larger than $0$. For any real $x$, let $M(x) = \{ n \in \mathbb{Z}^{+} | nx \in M \}$. Show that if $\alpha,\beta$ are reals such that $M(\alpha) = M(\beta)$, then we must have one of $\alpha + \beta$ and $\alpha - \beta$ to be rational.

The sum $\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}=\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Find all the two-digit numbers $\overline{ab}$ that squared give a result where the last two digits are $\overline{ab}$.

Let $n > 1$ be an odd integer. Prove that a necessary and sufficient condition for the existence of positive integers $x$ and $y$ satisfying $$\frac{4}{n}=\frac{1}{x}+\frac{1}{y}$$ is that $n$ has a prime divisor of the form $4k - 1$.

Let $b$ be a positive integer. Find all $2002$−tuples $(a_1, a_2,\ldots , a_{2002})$, of natural numbers such that \[\sum_{j=1}^{2002} a_j^{a_j}=2002b^b.\]

Suppose that we have two natural numbers $x , y \le 100!$ with undetermined values. Prove that there exist natural numbers $m , n$ such that values of $x , y$ get uniquely determined according to value of $\varphi(d(my))+d(\varphi(nx))$. ( for each natural number $n$ , $d(n)$ is number of its positive divisors and $\varphi(n)$ is the number of the numbers less that $n$ which are relatively prime to $n$. ) [i]Proposed by Mehran Talaei[/i]

There are $n$ natural numbers written on a blackboard, where $n\in\mathbb{N},\ n\geq 2$. During each step two chosen numbers $a,b$, having the property that none of them divides the other, are replaced by their greatest common divisor and least common multiple. Prove that after a number of steps, all the numbers on the blackboard cease modifying. Prove that the respective number of steps is at most $(n-1)!$.

Let $p,q$ denote distinct prime numbers and $k,l$ positive integers. Find all positive divisors of the numbers: (a) $A = p^k$ (b) $B=p^kq^l$ (c) $1944$

Let $n \geq 2$ be a positive integer. Call a sequence $a_1, a_2, \cdots , a_k$ of integers an $n$[i]-chain[/i] if $1 = a_2 < a_ 2 < \cdots < a_k =n$, $a_i$ divides $a_{i+1}$ for all $i$, $1 \leq i \leq k-1$. Let $f(n)$ be the number of $n$[i]-chains[/i] where $n \geq 2$. For example, $f(4) = 2$ corresponds to the $4$-chains $\{1,4\}$ and $\{1,2,4\}$. Prove that $f(2^m \cdot 3) = 2^{m-1} (m+2)$ for every positive integer $m$.

Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.

Prove that $\prod\limits_{k=0}^{n-1}{({{2}^{n}}-{{2}^{k}})}$ is divisible by $n!$ for all positive integers $n$.

To a manufacturer of three products whose unit prices are $50$, $70$, and $65$ pta, a retailer asks him for $100$ units, remitting him $6850$ pta as payment, on the condition that you send as many of the higher-priced product as possible and the rest of the other two. How many of each product should he send to serve the request?

Let $N$ be the set of positive integers. Determine if there is a function $f: N\to N$ such that $f(f(n))=2n$, for all $n$ belongs to $N$.