Find all pairwise relatively prime positive integers $l, m, n$ such that \[(l+m+n)\left( \frac{1}{l}+\frac{1}{m}+\frac{1}{n}\right)\] is an integer.

Determine all the positive integers with more than one digit, all distinct, such that the sum of its digits is equal to the product of its digits.

Find the smallest prime number $p$ that cannot be represented in the form $|3^{a} - 2^{b}|$, where $a$ and $b$ are non-negative integers.

$a,b$ are natural numbers such that $\frac{a+1}{b}$ and $\frac{b+1}{a}$ are integers.Let $d=GCD(a;b)$.Prove that $d^2\le a+b$

For every positive integer $ n$ let \[ a_n\equal{}\sum_{k\equal{}n}^{2n}\frac{(2k\plus{}1)^n}{k}\] Show that there exists no $ n$, for which $ a_n$ is a non-negative integer.

A triangle has sides of length $48$, $55$, and $73$. Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the length of the shortest altitude of the triangle. Find the value of $a+b$.

All sequences of $k$ elements $(a_1,a_2,...,a_k)$ are considered, where each $a_i$ belongs to the set $\{1,2,... ,2021\}$. What is the sum of the smallest elements of all these sequences?

Determine all pairs $(x, y)$ of integers for which satisfy the equality $\sqrt{x-\sqrt{y}}+ \sqrt{x+\sqrt{y}}= \sqrt{xy}$

Determine, with proof, all the integer solutions of the equation $x^3 + 2y^3 + 4z^3 - 6xyz = 1$.

Let $p$ be an odd prime and let $a$ be an integer not divisible by $p$. Show that there are $p^2+1$ triples of integers $(x,y,z)$ with $0 \le x,y,z < p$ and such that $(x+y+z)^2 \equiv axyz \pmod p$

Let $p$ be a prime, and let $a_1, a_2, a_3, . . .$ be a sequence of positive integers so that $a_na_{n+2} = a^2_{n+1} + p$ for all positive integers $n$. Show that $a_{n+1}$ divides $a_n + a_{n+2}$ for all positive integers $n$.

Let $n$ be a natural number. The finite sequence $\alpha$ of positive integer terms, there are $n$ different numbers ($\alpha$ can have repeated terms). Moreover, if from one from its terms any we subtract 1, we obtain a sequence which has, between its terms, at least $n$ different positive numbers. What's the minimum value of the sum of all the terms of $\alpha$?

Prove that there exists a positive integer $k$ such that $k\cdot2^n+1$ is composite for every integer $n$.

[b]p1.[/b] A right triangle has hypotenuse of length $12$ cm. The height corresponding to the right angle has length $7$ cm. Is this possible? [img]https://cdn.artofproblemsolving.com/attachments/0/e/3a0c82dc59097b814a68e1063a8570358222a6.png[/img] [b]p2.[/b] Prove that from any $5$ integers one can choose $3$ such that their sum is divisible by $3$. [b]p3.[/b] Two players play the following game on an $8\times 8$ chessboard. The first player can put a knight on an arbitrary square. Then the second player can put another knight on a free square that is not controlled by the first knight. Then the first player can put a new knight on a free square that is not controlled by the knights on the board. Then the second player can do the same, etc. A player who cannot put a new knight on the board loses the game. Who has a winning strategy? [b]p4.[/b] Consider a regular octagon $ABCDEGH$ (i.e., all sides of the octagon are equal and all angles of the octagon are equal). Show that the area of the rectangle $ABEF$ is one half of the area of the octagon. [img]https://cdn.artofproblemsolving.com/attachments/d/1/674034f0b045c0bcde3d03172b01aae337fba7.png[/img] [b]p5.[/b] Can you find a positive whole number such that after deleting the first digit and the zeros following it (if they are) the number becomes $24$ times smaller? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that if $(a,b,c,d)$ are positive integers such that $(a+2^{\frac13}b+2^{\frac23}c)^2=d$ then $d$ is a perfect square (i.e is the square of a positive integer).

In base ten, the number $100! = 100 \cdot 99 \cdot 98 \cdot 97... 2 \cdot 1$ has $158$ digits, and the last $24$ digits are all zeros. Find the number of zeros there are at the end of this same number when it is written in base $24$.

A positive integer $n$ is called [i]egyptian[/i] if there exists a strictly increasing sequence $0<a_1<a_2<\dots<a_k=n$ of integers with last term $n$ such that \[\frac{1}{a_1}+\frac{1}{a_2}+\dots+\frac{1}{a_k}=1.\] (a) Determine if $n=72$ is egyptian. (b) Determine if $n=71$ is egyptian. (c) Determine if $n=72^{71}$ is egyptian.

An integer $n > 1$ is called $\emph{good}$ if there exists a permutation $a_1, a_2, a_3, \dots, a_n$ of the numbers $1, 2, 3, \dots, n$, such that: $(i)$ $a_i$ and $a_{i+1}$ have different parities for every $1 \leq i \leq n-1$; $(ii)$ the sum $a_1 + a_2 + \cdots + a_k$ is a quadratic residue modulo $n$ for every $1 \leq k \leq n$. Prove that there exist infinitely many good numbers, as well as infinitely many positive integers which are not good.

Find all primes $p,q, r$ such that $\frac{p^{2q}+q^{2p}}{p^3-pq+q^3} = r$. Titu Andreescu, Mathematics Department, College of Texas, USA

find all positive integer n such that there exists positive integers (a,b) such that (a^n + b^n)/n! is a positive integer smaller than 101

A pen costs $11$ € and a notebook costs $13$ €. Find the number of ways in which a person can spend exactly $1000$ € to buy pens and notebooks.

For each positive integer $n$ denote: \[n!=1\cdot 2\cdot 3\dots n\] Find all positive integers $n$ for which $1!+2!+3!+\cdots+n!$ is a perfect square.

Find the greatest positive integer $A$ with the following property: For every permutation of $\{1001,1002,...,2000\}$ , the sum of some ten consecutive terms is great than or equal to $A$.

Let $k$ be a fixed positive integer. Prove that the sequence $\binom{2}{1},\binom{4}{2},\binom{8}{4},\ldots, \binom{2^{n+1}}{2^n},\ldots$ is eventually constant modulo $2^k$. [i]Proposed by V. Rastorguyev[/i]

For each positive integer $n$ let \[x_n=p_1+\cdots+p_n\] where $p_1,\ldots,p_n$ are the first $n$ primes. Prove that for each positive integer $n$, there is an integer $k_n$ such that $x_n<k_n^2<x_{n+1}$