Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?

Let $p$ be a prime number and $a$ and $n$ positive nonzero integers. Prove that if $2^p + 3^p = a^n$ then $n=1$

Find all primes $p,q$ such that $p ^3-q^7=p-q$.

A shepherd left, as an inheritance, to his children a flock of $k$ sheep, distributed as follows: the oldest received $\left\lfloor\frac{k}{2}\right\rfloor$ sheep, the middle one $\left\lfloor\frac{k}{3}\right\rfloor$ sheep and the youngest $\left\lfloor\frac{k}{5}\right\rfloor$ sheep. Knowing that there are no sheep left, determine all possible values for $k$.

Let $f(x)$ be a non-constant polynomial with non-negative integer coefficients. Prove that there are infinitely many positive integers $n$, for which $f(n)$ is not divisible by any of $f(2)$, $f(3)$, ..., $f(n-1)$.

Do there exist $10$ positive integers such that each of them is divisible by none of the other numbers but the square of each of these numbers is divisible by each of the other numbers? (Folklore)

Determine whether there exist infinitely many triples $(a, b, c)$ of positive integers such that every prime $p$ divides \[\left\lfloor\left(a+b\sqrt{2024}\right)^p\right\rfloor-c.\]

For each positive integer $n$, let $f(n)$ be the maximal natural number such that: $2^{f(n)}$ divides $\sum^{\left\lfloor \frac{n - 1}{2}\right\rfloor}_{i=0} \binom{n}{2 \cdot i + 1} 3^i$. Find all $n$ such that $f(n) = 1996.$ [hide="old version"]For each positive integer $n$, let $f(n)$ be the maximal natural number such that: $2^{f(n)}$ divides $\sum^{n + 1/2}_{i=1} \binom{2 \cdot i + 1}{n}$. Find all $n$ such that $f(n) = 1996.$[/hide]

Let us consider a simle graph with vertex set $ V$. All ordered pair $ (a,b)$ of integers with $ gcd(a,b) \equal{} 1$, are elements of V. $ (a,b)$ is connected to $ (a,b \plus{} kab)$ by an edge and to $ (a \plus{} kab,b)$ by another edge for all integer k. Prove that for all $ (a,b)\in V$, there exists a path fromm $ (1,1)$ to $ (a,b)$.

Prove that for any positive integer $ m$ there exist an infinite number of pairs of integers $ (x, y)$ such that [i](i)[/i] $ x$ and $ y$ are relatively prime; [i](ii)[/i] $ y$ divides $ x^2 \plus{} m$; [i](iii)[/i] $ x$ divides $ y^2 \plus{} m.$ [i](iv)[/i] $ x \plus{} y \leq m \plus{} 1\minus{}$ (optional condition)

For a positive integer $n$, denote by $g(n)$ the number of strictly ascending triples chosen from the set $\{1, 2, ..., n\}$. Find the least positive integer $n$ such that the following holds:[i] The number $g(n)$ can be written as the product of three different prime numbers which are (not necessarily consecutive) members in an arithmetic progression with common difference $336$.[/i]

Given the equation $$ a^b\cdot b^c=c^a $$ in positive integers $a$, $b$, and $c$. [i](i)[/i] Prove that any prime divisor of $a$ divides $b$ as well. [i](ii)[/i] Solve the equation under the assumption $b\ge a$. [i](iii)[/i] Prove that the equation has infinitely many solutions. [i](I. Voronovich)[/i]

Find all tuples of positive integers $(a,b,c)$ such that $$a^b+b^c+c^a=a^c+b^a+c^b$$

Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\] [i]Proposed by Titu Andreescu, USA[/i]

There are $2025$ positive integers $a_1, a_2, \cdots, a_{2025}$ are placed around a circle. For any $k = 1, 2, \cdots, 2025$, $a_k \mid a_{k-1} + a_{k+1}$ where indices are considered modulo $n$. Prove that there exists a positive integer $N$ such that satisfies the following condition. [list] [*] [b](Condition)[/b] For any positive integer $n > N$, when $a_1 = n^n$, $a_1, a_2, \cdots, a_{2025}$ are all multiples of $n$. [/list]

Find the least positive integer such that the product of its digits is $8! = 8 \cdot 7 \cdot6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$.

Find three consecutive odd numbers $a,b,c$ such that $a^2+b^2+c^2$ is a four digit number with four equal digits.

The sequence of positive integers $a_1,a_2,a_3,\dots$ is [i]brazilian[/i] if $a_1=1$ and $a_n$ is the least integer greater than $a_{n-1}$ and $a_n$ is [b]coprime[/b] with at least half elements of the set $\{a_1,a_2,\dots, a_{n-1}\}$. Is there any odd number which does [b]not[/b] belong to the brazilian sequence?

Let $u$ be a positive rational number and $m$ be a positive integer. Define a sequence $q_1,q_2,q_3,\dotsc$ such that $q_1=u$ and for $n\geqslant 2$: $$\text{if }q_{n-1}=\frac{a}{b}\text{ for some relatively prime positive integers }a\text{ and }b, \text{ then }q_n=\frac{a+mb}{b+1}.$$ Determine all positive integers $m$ such that the sequence $q_1,q_2,q_3,\dotsc$ is eventually periodic for any positive rational number $u$. [i]Remark:[/i] A sequence $x_1,x_2,x_3,\dotsc $ is [i]eventually periodic[/i] if there are positive integers $c$ and $t$ such that $x_n=x_{n+t}$ for all $n\geqslant c$. [i]Proposed by Petar Nizié-Nikolac[/i]

Say a number is [i]tico[/i] if the sum of it's digits is a multiple of $2003$. $\text{(i)}$ Show that there exists a positive integer $N$ such that the first $2003$ multiples, $N,2N,3N,\ldots 2003N$ are all tico. $\text{(ii)}$ Does there exist a positive integer $N$ such that all it's multiples are tico?

(a). Given any natural number \(N\), prove that there exists a strictly increasing sequence of \(N\) positive integers in harmonic progression. (b). Prove that there cannot exist a strictly increasing infinite sequence of positive integers which is in harmonic progression.

In the coordinate plane the points with both coordinates being rational numbers are called rational points. For any positive integer $n$, is there a way to use $n$ colours to colour all rational points, every point is coloured one colour, such that any line segment with both endpoints being rational points contains the rational points of every colour?

[b]p1.[/b] There are $5$ weights of masses $1,2,3,5$, and $10$ grams. One of the weights is counterfeit (its weight is different from what is written, it is unknown if the weight is heavier or lighter). How to find the counterfeit weight using simple balance scales only twice? [b]p2.[/b] There are $998$ candies and chocolate bars and $499$ bags. Each bag may contain two items (either two candies, or two chocolate bars, or one candy and one chocolate bar). Ann distributed candies and chocolate bars in such a way that half of the candies share a bag with a chocolate bar. Helen wants to redistribute items in the same bags in such a way that half of the chocolate bars would share a bag with a candy. Is it possible to achieve that? [b]p3.[/b] Insert in sequence $2222222222$ arithmetic operations and brackets to get the number $999$ (For instance, from the sequence $22222$ one can get the number $45$: $22*2+2/2 = 45$). [b]p4.[/b] Put numbers from $15$ to $23$ in a $ 3\times 3$ table in such a way to make all sums of numbers in two neighboring cells distinct (neighboring cells share one common side). [b]p5.[/b] All integers from $1$ to $200$ are colored in white and black colors. Integers $1$ and $200$ are black, $11$ and $20$ are white. Prove that there are two black and two white numbers whose sums are equal. [b]p6.[/b] Show that $38$ is the sum of few positive integers (not necessarily, distinct), the sum of whose reciprocals is equal to $1$. (For instance, $11=6+3+2$, $1/16+1/13+1/12=1$.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$m, n$ are relatively prime. We have three jugs which contain $m$, $n$ and $m+n$ liters. Initially the largest jug is full of water. Show that for any $k$ in $\{1, 2, ... , m+n\}$ we can get exactly $k$ liters into one of the jugs.

Let $T=\text{TNFTPP}$. Fermi and Feynman play the game $\textit{Probabicloneme}$ in which Fermi wins with probability $a/b$, where $a$ and $b$ are relatively prime positive integers such that $a/b<1/2$. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play $\textit{Probabicloneme}$ so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is $(T-332)/(2T-601)$. Find the value of $a$.