Found problems: 15460
2021 Polish Junior MO Second Round, 1
The numbers $a, b$ satisfy the condition $2a + a^2= 2b + b^2$. Prove that if $a$ is an integer, $b$ is also an integer.
2023 Indonesia MO, 5
Let $a$ and $b$ be positive integers such that $\text{gcd}(a, b) + \text{lcm}(a, b)$ is a multiple of $a+1$. If $b \le a$, show that $b$ is a perfect square.
2007 Nordic, 1
Find a solution to the equation $x^2-2x-2007y^2=0$ in positive integers.
2021 Princeton University Math Competition, 2
Let $k \in Z_{>0}$ be the smallest positive integer with the property that $k\frac{gcd(x,y)gcd(y,z)}{lcm (x,y^2,z)}$ is a positive integer for all values $1 \le x \le y \le z \le 121$. If k' is the number of divisors of $k$, find the number of divisors of $k'$.
1993 All-Russian Olympiad Regional Round, 9.2
Find the largest natural number which cannot be turned into a multiple of $11$ by reordering its (decimal) digits.
2011 Saudi Arabia Pre-TST, 3.1
Let $n$ be a positive integer such that $2011^{2011}$ divides $n!$. Prove that $2011^{2012} $divides $n!$ .
2020 Dutch IMO TST, 4
Let $a, b \ge 2$ be positive integers with $gcd (a, b) = 1$. Let $r$ be the smallest positive value that $\frac{a}{b}- \frac{c}{d}$ can take, where $c$ and $d$ are positive integers satisfying $c \le a$ and $d \le b$. Prove that $\frac{1}{r}$ is an integer.
2022 Dutch BxMO TST, 3
Find all pairs $(p, q)$ of prime numbers such that $$p(p^2 -p - 1) = q(2q + 3).$$
2022 CMIMC, 1.5
Grant is standing at the beginning of a hallway with infinitely many lockers, numbered, $1, 2, 3, \ldots$ All of the lockers are initially closed. Initially, he has some set $S = \{1, 2, 3, \ldots\}$.
Every step, for each element $s$ of $S$, Grant goes through the hallway and opens each locker divisible by $s$ that is closed, and closes each locker divisible by $s$ that is open. Once he does this for all $s$, he then replaces $S$ with the set of labels of the currently open lockers, and then closes every door again.
After $2022$ steps, $S$ has $n$ integers that divide ${10}^{2022}$. Find $n$.
[i]Proposed by Oliver Hayman[/i]
2010 Indonesia TST, 1
find all pairs of relatively prime natural numbers $ (m,n) $ in such a way that there exists non constant polynomial f satisfying \[ gcd(a+b+1, mf(a)+nf(b) > 1 \]
for every natural numbers $ a $ and $ b $
1963 Swedish Mathematical Competition., 3
What is the remainder on dividing $1234^{567} + 89^{1011}$ by $12$?
1987 IberoAmerican, 1
The sequence $(p_n)$ is defined as follows: $p_1=2$ and for all $n$ greater than or equal to $2$, $p_n$ is the largest prime divisor of the expression $p_1p_2p_3\ldots p_{n-1}+1$.
Prove that every $p_n$ is different from $5$.
2024 PErA, P6
For each positive integer $k$, define $a_k$ as the number obtained from adding $k$ zeroes and a $1$ to the right of $2024$, all written in base $10$. Determine wether there's a $k$ such that $a_k$ has at least $2024^{2024}$ distinct prime divisors.
2014 Contests, 2
Given the rational numbers $r$, $q$, and $n$, such that $\displaystyle\frac1{r+qn}+\frac1{q+rn}=\frac1{r+q}$, prove that $\displaystyle\sqrt{\frac{n-3}{n+1}}$ is a rational number.
Kvant 2022, M2694
Call a natural number $n{}$ [i]interesting[/i] if any natural number not exceeding $n{}$ can be represented as the sum of several (possibly one) pairwise distinct positive divisors of $n{}$.
[list=a]
[*]Find the largest three-digit interesting number.
[*]Prove that there are arbitrarily large interesting numbers other than the powers of two.
[/list]
[i]Proposed by N. Agakhanov[/i]
2014 ELMO Shortlist, 11
Let $p$ be a prime satisfying $p^2\mid 2^{p-1}-1$, and let $n$ be a positive integer. Define
\[ f(x) = \frac{(x-1)^{p^n}-(x^{p^n}-1)}{p(x-1)}. \]
Find the largest positive integer $N$ such that there exist polynomials $g(x)$, $h(x)$ with integer coefficients and an integer $r$ satisfying $f(x) = (x-r)^N g(x) + p \cdot h(x)$.
[i]Proposed by Victor Wang[/i]
2011 Cuba MO, 7
Find a set of positive integers with the greatest possible number of elements such that the least common multiple of all of them is less than $2011$.
2016 Bangladesh Mathematical Olympiad, 4
Consider the set of integers $ \left \{ 1, 2, \dots , 100 \right \} $. Let $ \left \{ x_1, x_2, \dots , x_{100} \right \}$ be some arbitrary arrangement of the integers $ \left \{ 1, 2, \dots , 100 \right \}$, where all of the $x_i$ are different. Find the smallest possible value of the sum
$$S = \left | x_2 - x_1 \right | + \left | x_3 - x_2 \right | + \cdots+ \left |x_{100} - x_{99} \right | + \left |x_1 - x_{100} \right | .$$
2022 Swedish Mathematical Competition, 3
Let $n$ be a positive integer divisible by $39$. What is the smallest possible sum of digits that $n$ can have (in base $10$)?
2021 Science ON all problems, 3
A nonnegative integer $n$ is said to be $\textit{squarish}$ is it satisfies the following conditions:
$\textbf{(i)}$ it contains no digits of $9$;
$\textbf{(ii)}$ no matter how we increase exactly one of its digits with $1$, the outcome is a square.
Find all squarish nonnegative integers.
$\textit{(Vlad Robu)}$
2000 Switzerland Team Selection Test, 7
Show that the equation $14x^2 +15y^2 = 7^{2000}$ has no integer solutions.
2007 Estonia National Olympiad, 1
The seven-digit integer numbers are different in pairs and this number is divided by each of its own numbers.
a) Find all possibilities for the three numbers that are not included in this number.
b) Give an example of such a number.
2008 IMO Shortlist, 6
Prove that there are infinitely many positive integers $ n$ such that $ n^{2} \plus{} 1$ has a prime divisor greater than $ 2n \plus{} \sqrt {2n}$.
[i]Author: Kestutis Cesnavicius, Lithuania[/i]
1972 IMO Longlists, 22
Show that for any $n \not \equiv 0 \pmod{10}$ there exists a multiple of $n$ not containing the digit $0$ in its decimal expansion.
Russian TST 2017, P3
Let $a_1,\ldots , a_{p-2}{}$ be nonzero residues modulo an odd prime $p{}$. For every $d\mid p - 1$ there are at least $\lfloor(p - 2)/d\rfloor$ indices $i{}$ for which $p{}$ does not divide $a_i^d-1$. Prove that the product of some of $a_1,\ldots , a_{p-2}$ gives the remainder two modulo $p{}$.