This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 15460

1997 IMO, 5
Tags: equation , prime factorization , number theory
Find all pairs $ (a,b)$ of positive integers that satisfy the equation: $ a^{b^2} \equal{} b^a$.

2018 May Olympiad, 1
Tags: number theory , digit , perfect square
You have a $4$-digit whole number that is a perfect square. Another number is built adding $ 1$ to the unit's digit, subtracting $ 1$ from the ten's digit, adding $ 1$ to the hundred's digit and subtracting $ 1$ from the ones digit of one thousand. If the number you get is also a perfect square, find the original number. It's unique?

2003 Paraguay Mathematical Olympiad, 2
Tags: number theory , digit
With three different digits, all greater than $0$, six different three-digit numbers are formed. If we add these six numbers together the result is $4.218$. The sum of the three largest numbers minus the sum of the three smallest numbers equals $792$. Find the three digits.

2018 Hong Kong TST, 3
Tags: number theory , diophantine equation
Find all primes $p$ and all positive integers $a$ and $m$ such that $a\leq 5p^2$ and $(p-1)!+a=p^m$

2015 Latvia Baltic Way TST, 13
Tags: floor function , number theory , prime
Are there positive real numbers $a$ and $b$ such that $[an+b]$ is prime for all natural values of $n$ ? $[x]$ denotes the integer part of the number $x$, the largest integer that does not exceed $x$.

2001 All-Russian Olympiad, 1
Tags: ceiling function , floor function , number theory
The integers from $1$ to $999999$ are partitioned into two groups: the first group consists of those integers for which the closest perfect square is odd, whereas the second group consists of those for which the closest perfect square is even. In which group is the sum of the elements greater?

1991 National High School Mathematics League, 3
Tags: number theory
Let $a_n$ be the number of such numbers $N$: sum of all digits of $N$ is $n$, and each digit can only be $1,3,4$. Prove that $a_{2n}$ is a perfect square for all $n\in\mathbb{Z}_+$.

1992 Austrian-Polish Competition, 6
Tags: functional , functional equation , algebra , number theory , divisor
A function $f: Z \to Z$ has the following properties: $f (92 + x) = f (92 - x)$ $f (19 \cdot 92 + x) = f (19 \cdot 92 - x)$ ($19 \cdot 92 = 1748$) $f (1992 + x) = f (1992 - x)$ for all integers $x$. Can all positive divisors of $92$ occur as values of f?

2018 Peru Iberoamerican Team Selection Test, P10
Tags: number theory
Does there exist a sequence of positive integers $a_1,a_2,...$ such that every positive integer occurs exactly once and that the number $\tau (na_{n+1}^n+(n+1)a_n^{n+1})$ is divisible by $n$ for all positive integer. Here $\tau (n)$ denotes the number of positive divisor of $n$.

2015 India IMO Training Camp, 3
Tags: floor function , sequence , parity , number theory
Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

Oliforum Contest V 2017, 1
Tags: number theory , digit , multiple
We know that there exists a positive integer with $7$ distinct digits which is multiple of each of them. What are its digits? (Paolo Leonetti)

2025 China Team Selection Test, 11
Tags: number theory
Let \( n \geq 4 \). Proof that \[ (2^x - 1)(5^x - 1) = y^n \] have no positive integer solution \((x, y)\).

2023 Dutch IMO TST, 1
Tags: number theory
Find all prime numbers $p$ such that the number $$3^p+4^p+5^p+9^p-98$$ has at most $6$ positive divisors.

2004 Iran MO (3rd Round), 28
Tags: number theory
Find all prime numbers $p$ such that $ p = m^2 + n^2$ and $p\mid m^3+n^3-4$.

2000 Chile National Olympiad, 3
Tags: number theory , divide , divisible , periodic
A number $N_k$ is defined as [i]periodic[/i] if it is composed in number base $N$ of a repeated $k$ times . Prove that $7$ divides to infinite periodic numbers of the set $N_1, N_2, N_3,...$

2012 IMAC Arhimede, 4
Tags: diophantine equation , number theory
Solve the following equations in the set of natural numbers: a) $(5+11\sqrt2)^p=(11+5\sqrt2)^q$ b) $1005^x+2011^y=1006^z$

2006 Estonia National Olympiad, 3
Tags: modular arithmetic , number theory
Prove or disprove the following statements. a) For every integer $ n \ge 3$, there exist $ n$ pairwise distinct positive integers such that the product of any two of them is divisible by the sum of the remaining $ n \minus{} 2$ numbers. b) For some integer $ n \ge 3$, there exist $ n$ pairwise distinct positive integers, such that the sum of any $ n \minus{} 2$ of them is divisible by the product of the remaining two numbers.

2018 Hanoi Open Mathematics Competitions, 11
Tags: number theory , divisible
Find all positive integers $k$ such that there exists a positive integer $n$, for which $2^n + 11$ is divisible by $2^k - 1$.

2015 Regional Competition For Advanced Students, 2
Tags: number theory
Let $x$, $y$, and $z$ be positive real numbers with $x+y+z = 3$. Prove that at least one of the three numbers $$x(x+y-z)$$ $$y(y+z-x)$$ $$z(z+x-y)$$ is less or equal $1$. (Karl Czakler)

2004 Poland - Second Round, 1
Tags: number theory
Find all positive integers $n$ which have exactly $\sqrt{n}$ positive divisors.

2022 Moldova Team Selection Test, 12
Tags: number theory
Let $(x_n)_{n\geq1}$ be a sequence that verifies: $$x_1=1, \quad x_2=7, \quad x_{n+1}=x_n+3x_{n-1}, \forall n \geq 2.$$ Prove that for every prime number $p$ the number $x_p-1$ is divisible by $3p.$

2004 Finnish National High School Mathematics Competition, 4
Tags: prime number , number theory
The numbers $2005! + 2, 2005! + 3, ... , 2005! + 2005$ form a sequence of $2004$ consequtive integers, none of which is a prime number. Does there exist a sequence of $2004$ consequtive integers containing exactly $12$ prime numbers?

1972 IMO Longlists, 31
Tags: number theory
Find values of $n\in \mathbb{N}$ for which the fraction $\frac{3^n-2}{2^n-3}$ is reducible.

2019 Taiwan TST Round 3, 2
Tags: number theory
Given a prime $ p = 8k+1 $ for some integer $ k $. Let $ r $ be the remainder when $ \binom{4k}{k} $ is divided by $ p $. Prove that $ \sqrt{r} $ is not an integer. [i]Proposed by Evan Chen[/i]

1930 Eotvos Mathematical Competition, 1
Tags: number theory , digit
How many five-digit multiples of 3 end with the digit 6 ?