Find all integer solutions to the equation $$\frac{xyz}{w}+\frac{yzw}{x}+\frac{zwx}{y}+\frac{wxy}{z}=4$$

Given distinct positive integer $ a_1,a_2,…,a_{2020} $. For $ n \ge 2021 $, $a_n$ is the smallest number different from $a_1,a_2,…,a_{n-1}$ which doesn't divide $a_{n-2020}...a_{n-2}a_{n-1}$. Proof that every number large enough appears in the sequence.

(a) Find all pairs $(m, n)$ of integers satisfying $m^3-4mn^2=8n^3-2m^2n$ (b) Among such pairs find those for which $m+n^2=3$

Let $a,b,c,d$ be non-zero integers, such that the only quadruple of integers $(x, y, z, t)$ satisfying the equation \[ax^2+by^2+cz^2+dt^2=0\] is $x=y=z=t=0$. Does it follow that the numbers $a,b,c,d$ have the same sign?

Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define $$x_{k+1} = \begin{cases} x_k + d &\text{if } a \text{ does not divide } x_k \\ x_k/a & \text{if } a \text{ divides } x_k \end{cases}$$ Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.

Find all pairs of natural numbers $(n,k)$ for which $(n+1)^{k}-1 = n!$.

A hexagon inscribed in a circle has three consecutive sides each of length $3$ and three consecutive sides each of length $5$. The chord of the circle that divides the hexagon into two trapezoids, one with three sides each of length $3$ and the other with three sides each of length $5$, has length equal to $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. $\text{(A)}\ 309 \qquad \text{(B)}\ 349 \qquad \text{(C)}\ 369 \qquad \text{(D)}\ 389\qquad \text{(E)}\ 409$

For all primes $p \geq 3,$ define $F(p) = \sum^{\frac{p-1}{2}}_{k=1}k^{120}$ and $f(p) = \frac{1}{2} - \left\{ \frac{F(p)}{p} \right\}$, where $\{x\} = x - [x],$ find the value of $f(p).$

Let $n\ge3$ positive integers $a_1,\ldots,a_n$ be written on a circle so that each of them divides the sum of its two neighbors. Let us denote $$S_n=\frac{a_n+a_2}{a_1}+\frac{a_1+a_3}{a_2}+\ldots+\frac{a_{n-2}+a_n}{a_{n-1}}+\ldots+\frac{a_{n-1}+a_1}{a_n}.$$Determine the minimum and maximum values of $S_n$.

Find all integers $x$ and $y$ satisfying the following equation $x^2 - 2xy + 5y^2 + 2x - 6y - 3 = 0$.

For each positive integer $ n$, let $ f(n)$ denote the greatest common divisor of $ n!\plus{}1$ and $ (n\plus{}1)!$. Find, without proof, a formula for $ f(n)$.

Find non-negative integers $a, b, c, d$ such that $5^a + 6^b + 7^c + 11^d = 1999$.

For a positive integer $K$, define a sequence, $\{a_n\}_n$, as following $a_1=K$, \[ a_{n+1} = \{ \begin{array} {cc} a_n-1 , & \mbox{ if } a_n \mbox{ is even} \\ \frac{a_n-1}2 , & \mbox{ if } a_n \mbox{ is odd} \end{array}, \] for all $n\geq 1$. Find the smallest value of $K$, which makes $a_{2005}$ the first term equal to 0.

Which digit must be substituted instead of the star so that the following large number $$\underbrace{66...66}_{2023} \star \underbrace{55...55}_{2023}$$ is divisible by $7$?

All letters in the word $VUQAR$ are different and chosen from the set $\{1,2,3,4,5\}$. Find all solutions to the equation \[\frac{(V+U+Q+A+R)^2}{V-U-Q+A+R}=V^{{{U^Q}^A}^R}.\]

Let $l, m, n$ be positive integers and $p$ be prime. If $p^{2l-1}m(mn+1)^2 + m^2$ is a perfect square, prove that $m$ is also a perfect square.

$$problem2$$:Determine the number of 100-digit numbers whose all digits are odd, and in which every two consecutive digits differ by 2

Let $a$, $b$, and $c$ be pairwise distinct positive integers, which are side lengths of a triangle. There is a line which cuts both the area and the perimeter of the triangle into two equal parts. This line cuts the longest side of the triangle into two parts with ratio $2:1$. Determine $a$, $b$, and $c$ for which the product abc is minimal.

Find all pairs of natural numbers $ (a, b)$ such that $ 7^a \minus{} 3^b$ divides $ a^4 \plus{} b^2$. [i]Author: Stephan Wagner, Austria[/i]

Let $p$ be a prime number greater than $2$. Patricia wants $7$ not-necessarily different numbers from $\{1, 2, . . . , p\}$ to the black dots in the figure below, on such a way that the product of three numbers on a line or circle always has the same remainder when divided by $p$. [img]https://cdn.artofproblemsolving.com/attachments/3/1/ef0d63b8ff5341ffc340de0cc75b24c7229e23.png[/img] (a) Suppose Patricia uses the number $p$ at least once. How many times does she have the number $p$ then a minimum sum needed? (b) Suppose Patricia does not use the number $p$. In how many ways can she assign numbers? (Two ways are different if there is at least one black one dot different numbers are assigned. The figure is not rotated or mirrored.)

The natural numbers $a_1 <a_2 <.... <a_n\le 2n$ are such that the least common multiple of any two of them is greater than $2n$. Prove that $a_1 >\left[\frac{2n}{3}\right]$.

Suppose $m$ and $n$ are positive integers for which $\bullet$ the sum of the first $m$ multiples of $n$ is $120$, and $\bullet$ the sum of the first $m^3$ multiples of$ n^3$ is $4032000$. Determine the sum of the first $m^2$ multiples of $n^2$

If the set $S = \{1,2,3,…,16\}$ is partitioned into $n$ subsets, there must be a subset in which elements $a, b, c$ (can be the same) exist, satisfying $a+ b=c$. Find the maximum value of $n$.

Let $a, b$ and $c$ be positive integers such that $1 < a < b < c$. Suppose that $(ab-l)(bc-1 )(ca-1)$ is divisible by $abc$. Find the values of $a, b$ and $c$. Justify your answer.

For each positive integer $n$, let $\tau\left(n\right)$ be the number of positive divisors of $n$. It is well-known that if $a$ and $b$ are relatively prime positive integers then $\tau\left(ab\right)=\tau\left(a\right)\tau\left(b\right)$. Does the converse hold? That is, if $a$ and $b$ are positive integers such that $\tau\left(ab\right)=\tau\left(a\right)\tau\left(b\right)$, then is it necessarily true that $a$ and $b$ are relatively prime? Either give a proof, or find a counter-example.