Find all positive integer solutions of the equation $k!l! = k!+l!+m!.$

Let $n \ge 50 $ be a natural number. Prove that $n$ is expressible as sum of two natural numbers $n=x+y$, so that for every prime number $p$ such that $ p\mid x$ or $p\mid y $ we have $ \sqrt{n} \ge p $. For example for $n=94$ we have $x=80, y=14$.

Let $P$ be the set of positive integers that are prime numbers. Find the number of subsets of $P$ that have the property that the sum of their elements is $34$ such as $\{3, 31\}$.

Let $n$ be an integer that is greater than $1$. Prove that the following two statements are equivalent: (A) There are positive integers $a, b$ and $c$ that are not greater than $n$ and for which that polynomial $ax^2 + bx + c$ has two different real roots $x_1$ and $x_2$ with $| x_2- x_1 | \le \frac{1}{n}$ (B) The number $n$ has at least two different prime divisors.

The sum of the first $m$ positive odd integers is $212$ more than the sum of the first $n$ positive even integers. What is the sum of all possible values of $n$? $ \textbf{(A)}\ 255 \qquad\textbf{(B)}\ 256 \qquad\textbf{(C)}\ 257 \qquad\textbf{(D)}\ 258 \qquad\textbf{(E)}\ 259 $

Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.

Given a non-negative integer \( n \), determine the values of \( c \) for which the sequence of numbers \[ a_n = 4^n c + \frac{4^n - (-1)^n}{5} \] contains at least one perfect square.

A positive natural number $n$ is said to be [i]comico[/i] if its prime factorization is $n = p_1p_2...p_k$, with $k\ge 3$, and also the primes $p_1,..., p_k$ they fulfill that $p_1 + p_2 = c^2_1$ $p_1 + p_2 + p_3 = c^2_2$ $...$ $p_1 + p_2 + ...+ p_n = c^2_{n-1}$ where $c_1, c_2, ..., c_{n-1}$ are positive integers where $c_1$ is not divisible by $7$. Find all comico numbers less than $10,000$.

Determine the number of three-digit numbers with the following property: The number formed by the first two digits is prime and the number formed by the last two digits is prime.

Let $A$ be a set of $20$ distinct positive integers which are all no greater than $397$. Prove that for any positive integer $n$ it is possible to pick four (not necessarily distinct) elements $x_1, x_2, x_3, x_4$ of $A$ satisfying $x_1 \neq x_2$ and $$(x_1-x_2)n\equiv x_3-x_4 \pmod{397}.$$

Let $m,k,n$ be positive integers. Determine all triples $(m,k,n)$ satisfying the following equation: $3^m5^k=n^3+125$

Prove that for every non-negative integer $n$ there exist integers $x, y, z$ with $gcd(x, y, z) = 1$, such that $x^2 + y^2 + z^2 = 3^{2^n}$.(Poland)

Let $a,b,c,d,$ be real and $ad-bc = 1$. Show that $Q = a^2 + b^2 + c^2 + d^2 + ac +bd$ $\not= 0, 1, -1$

Find the integer $\sqrt[5]{55^5 + 3183^5 + 28969^5 + 85282^5}$.

Let $S(M)$ denote the sum of the digits of a positive integer $M$ written in base $10$. Let $N$ be the smallest positive integer such that $S(N) = 2013$. What is the value of $S(5N + 2013)$?

Prove that for some positive integer $n$ there exist positive integers $a$,$b$ and $c$ such that $a^2-n=xy$, $b^2-n=yz$ and $c^2-n=xz$ where $x,y$ and $z$ - some pairwise different positive integers.

Find all ordered pairs of positive integers $(r, s)$ for which there are exactly $35$ ordered pairs of positive integers $(a, b)$ such that the least common multiple of $a$ and $b$ is $2^r \cdot 3^s$.

[u]Round 1[/u] [b]p1.[/b] Ash is running around town catching Pokémon. Each day, he may add $3, 4$, or $5$ Pokémon to his collection, but he can never add the same number of Pokémon on two consecutive days. What is the smallest number of days it could take for him to collect exactly $100$ Pokémon? [b]p2.[/b] Jack and Jill have ten buckets. One bucket can hold up to $1$ gallon of water, another can hold up to $2$ gallons, and so on, with the largest able to hold up to $10$ gallons. The ten buckets are arranged in a line as shown below. Jack and Jill can pour some amount of water into each bucket, but no bucket can have less water than the one to its left. Is it possible that together, the ten buckets can hold 36 gallons of water? [img]https://cdn.artofproblemsolving.com/attachments/f/8/0b6524bebe8fe859fe7b1bc887ac786106fc17.png[/img] [b]p3.[/b] There are $2023$ knights and liars standing in a row. Knights always tell the truth and liars always lie. Each of them says, “the number of liars to the left of me is greater than the number of knights to the right.” How many liars are there? [b]p4.[/b] Camila has a deck of $101$ cards numbered $1, 2, ..., 101$. She starts with $50$ random cards in her hand and the rest on a table with the numbers visible. In an exchange, she replaces all $50$ cards in her hand with her choice of $50$ of the $51$ cards from the table. Show that Camila can make at most 50 exchanges and end up with cards $1, 2, ..., 50$. [img]https://cdn.artofproblemsolving.com/attachments/0/6/c89e65118764f3b593da45264bfd0d89e95067.png[/img] [b]p5.[/b] There are $101$ pirates on a pirate ship: the captain and $100$ crew. Each pirate, including the captain, starts with $1$ gold coin. The captain makes proposals for redistributing the coins, and the crew vote on these proposals. The captain does not vote. For every proposal, each crew member greedily votes “yes” if he gains coins as a result of the proposal, “no” if he loses coins, and passes otherwise. If strictly more crew members vote “yes” than “no,” the proposal takes effect. The captain can make any number of proposals, one after the other. What is the largest number of coins the captain can accumulate? [u]Round 2[/u] [b]p6.[/b] The town of Lumenville has $100$ houses and is preparing for the math festival. The Tesla wiring company will lay lengths of power wire in straight lines between the houses so that power flows between any two houses, possibly by passing through other houses. The Edison lighting company will hang strings of lights in straight lines between pairs of houses so that each house is connected by a string to exactly one other. Show that however the houses are arranged, the Edison company can always hang their strings of lights so that the total length of the strings is no more than the total length of the power wires the Tesla company used. [img]https://cdn.artofproblemsolving.com/attachments/9/2/763de9f4138b4dc552247e9316175036c649b6.png[/img] [b]p7.[/b] You are given a sequence of $16$ digits. Is it always possible to select one or more digits in a row, so that multiplying them results in a square number? [img]https://cdn.artofproblemsolving.com/attachments/d/1/f4fcda2e1e6d4a1f3a56cd1a04029dffcd3529.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a_1,a_2,...$ be an arithmetic sequence with the common difference between terms is positive. Assume there are $k$ terms of this sequence creates an geometric sequence with common ratio $d$. Prove that $n\ge 2^{k-1}$.

Prove that for every odd integer $n > 1$, there exist integers $a, b > 0$ such that, if we let $Q(x) = (x + a)^ 2 + b$, then the following conditions hold: $\bullet$ we have $\gcd(a, n) = gcd(b, n) = 1$; $\bullet$ the number $Q(0)$ is divisible by $n$; and $\bullet$ the numbers $Q(1), Q(2), Q(3), \dots$ each have a prime factor not dividing $n$.

Solve in integers the equation $xy + 3x - 5y = - 3$.

Find all natural numbers $ n,x,y $ such that $ \big| 2^x-n^{y+1}\big| =1 . $

For a positive integer $n$, let $d(n)$ be the number of positive divisors of $n$, and let $\varphi(n)$ be the number of positive integers not exceeding $n$ which are coprime to $n$. Does there exist a constant $C$ such that $$ \frac {\varphi ( d(n))}{d(\varphi(n))}\le C$$ for all $n\ge 1$ [i]Cyprus[/i]

$300$ people took part in the drawing for the main prize of the television lottery. They lined up in a circle, then, starting with someone who received number $1$, they began to count them. Moreover, every third person dropped out every time. (So, in the first round, everyone with numbers divisible by $3$ dropped out). The counting continued until there was only one person left. (It is clear that more than one circle was made). This person received the main prize. (It “accidentally” turned out to be the TV director’s mother-in-law). What number did this person have in the initial lineup?

Find all triples of primes $(p,q,r)$ such that $p^q=2021+r^3$