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.

AND:
OR:
NO:

Found problems: 15460

2022 Azerbaijan IMO TST, 2
Tags: number theory
Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers. [i]Proposed by Dorlir Ahmeti, Albania[/i]

2025 Chile TST IMO-Cono, 5
Tags: number theory
Let \( u_n \) be the \( n \)-th term of the Fibonacci sequence (where \( u_1 = u_2 = 1 \) and \( u_{n+1} = u_n + u_{n-1} \) for \( n \geq 2 \)). For each prime \( p \), let \( n(p) \) be the smallest integer \( n \) such that \( u_n \) is divisible by \( p \). Find the smallest possible value of \( p - n(p) \).

2022 CIIM, 4
Tags: permutation , counting , divisibility , number theory
Given a positive integer $n$, determine how many permutations $\sigma$ of the set $\{1, 2, \ldots , 2022n\}$ have the following property: for each $i \in \{1, 2, \ldots , 2021n + 1\}$, the number $$\sigma(i) + \sigma(i + 1) + \cdots + \sigma(i + n - 1)$$ is a multiple of $n$.

2000 Manhattan Mathematical Olympiad, 2
Tags: modular arithmetic , least common multiple , number theory
How many zeroes are there at the end the number $9^{999} + 1$?

2018 Middle European Mathematical Olympiad, 8
Tags: number theory
An integer $n $ is called silesian if there exist positive integers $a,b$ and $c$ such that $$n=\frac{a^2+b^2+c^2}{ab+bc+ca}.$$ $(a)$ prove that there are infinitely many silesian integers. $(b)$ prove that not every positive integer is silesian.

2023 Durer Math Competition Finals, 16
Tags: number theory
What is the remainder of $2025\wedge (2024\wedge (2022\wedge (2021\wedge (2020\wedge ...\wedge (2\wedge 1) . . .)))))$ when it is divided by $2023$? Here $\wedge$ is the exponential symbol, for example $2\wedge (3\wedge 2) = 2\wedge 9 = 512$. The power tower contains the integers from $2025$ to $1$ exactly once, except that the number $2023$ is missing.

1997 Moldova Team Selection Test, 3
Tags: pigeonhole principle , decimal representation , divisibility , multiple , digit , number theory
Prove that every integer $ k$ greater than 1 has a multiple that is less than $ k^4$ and can be written in the decimal system with at most four different digits.

2020 Thailand Mathematical Olympiad, 1
Tags: number theory
Show that $\varphi(2n)\mid n!$ for all positive integer $n$.

2025 International Zhautykov Olympiad, 4
Tags: number theory , combinatorics
Vaysha has a board with $999$ consecutive numbers written and $999$ labels of the form [i]"This number is [b]not [/b]divisible by $i$"[/i], for $i \in \{ 2,3, \dots ,1000 \} $. She places each label next to a number on the board, so that each number has exactly one label. For each true statement on the stickers, Vaysha gets a piece of candy. How many pieces of candy can Vaysha guarantee to win, regardless of the numbers written on the board, if she plays optimally?

2024 Malaysian IMO Training Camp, 2
Tags: digit , number theory
A finite sequence of decimal digits from $\{0,1,\cdots, 9\}$ is said to be [i]common[/i] if for each sufficiently large positive integer $n$, there exists a positive integer $m$ such that the expansion of $n$ in base $m$ ends with this sequence of digits. For example, $0$ is common because for any large $n$, the expansion of $n$ in base $n$ is $10$, whereas $00$ is not common because for any squarefree $n$, the expansion of $n$ in any base cannot end with $00$. Determine all common sequences. [i]Proposed by Wong Jer Ren[/i]

2020 Tournament Of Towns, 2
Tags: factorial , combinatorics , game , number theory
Three legendary knights are fighting against a multiheaded dragon. Whenever the first knight attacks, he cuts off half of the current number of heads plus one more. Whenever the second knight attacks, he cuts off one third of the current number of heads plus two more. Whenever the third knight attacks, he cuts off one fourth of the current number of heads plus three more. They repeatedly attack in an arbitrary order so that at each step an integer number of heads is being cut off. If all the knights cannot attack as the number of heads would become non-integer, the dragon eats them. Will the knights be able to cut off all the dragon’s heads if it has $41!$ heads? Alexey Zaslavsky

1975 USAMO, 1
Tags: floor function , inequalities , modular arithmetic , number theory
(a) Prove that \[ [5x]\plus{}[5y] \ge [3x\plus{}y] \plus{} [3y\plus{}x],\] where $ x,y \ge 0$ and $ [u]$ denotes the greatest integer $ \le u$ (e.g., $ [\sqrt{2}]\equal{}1$). (b) Using (a) or otherwise, prove that \[ \frac{(5m)!(5n)!}{m!n!(3m\plus{}n)!(3n\plus{}m)!}\] is integral for all positive integral $ m$ and $ n$.

2009 Turkey MO (2nd round), 3
Tags: modular arithmetic , floor function , number theory
If $1<k_1<k_2<...<k_n$ and $a_1,a_2,...,a_n$ are integers such that for every integer $N,$ $k_i \mid N-a_i$ for some $1 \leq i \leq n,$ find the smallest possible value of $n.$

2010 Swedish Mathematical Competition, 4
Tags: arithmetic sequence , number theory
We create a sequence by setting $a_1 = 2010$ and requiring that $a_n-a_{n-1}\leq n$ and $a_n$ is also divisible by $n$. Show that $a_{100},a_{101},a_{102},\dots$ form an arithmetic sequence.

2018 European Mathematical Cup, 1
Tags: algebra , number theory
A partition of a positive integer is even if all its elements are even numbers. Similarly, a partition is odd if all its elements are odd. Determine all positive integers $n$ such that the number of even partitions of $n$ is equal to the number of odd partitions of $n$. Remark: A partition of a positive integer $n$ is a non-decreasing sequence of positive integers whose sum of elements equals $n$. For example, $(2; 3; 4), (1; 2; 2; 2; 2)$ and $(9) $ are partitions of $9.$

2023 Durer Math Competition Finals, 3
Tags: digit , number theory
Which is the largest four-digit number that has all four of its digits among its divisors and its digits are all different?

2022 Dutch BxMO TST, 3
Tags: diophantine equation , diophantine , number theory
Find all pairs $(p, q)$ of prime numbers such that $$p(p^2 -p - 1) = q(2q + 3).$$

2011 CentroAmerican, 3
Tags: inequalities , number theory
A [i]slip[/i] on an integer $n\geq 2$ is an operation that consists in choosing a prime divisor $p$ of $n$ and replacing $n$ by $\frac{n+p^2}{p}.$ Starting with an arbitrary integer $n\geq 5$, we successively apply the slip operation on it. Show that one eventually reaches $5$, no matter the slips applied.

1999 Korea - Final Round, 3
Tags: modular arithmetic , number theory
Find all intengers n such that $2^n - 1$ is a multiple of 3 and $(2^n - 1)/3$ is a divisor of $4m^2 + 1$ for some intenger m.

2000 Tuymaada Olympiad, 4
Tags: factorial , number theory
Prove that no number of the form $10^{-n}$, $n\geq 1,$ can be represented as the sum of reciprocals of factorials of different positive integers.

2007 Olympic Revenge, 1
Tags: function , search , number theory
Let $a$, $b$, $n$ be positive integers with $a,b > 1$ and $\gcd(a,b) = 1$. Prove that $n$ divides $\phi\left(a^{n}+b^{n}\right)$.

2008 Bosnia And Herzegovina - Regional Olympiad, 2
Tags: number theory
Find all positive integers $ a$ and $ b$ such that $ \frac{a^{4}\plus{}a^{3}\plus{}1}{a^{2}b^{2}\plus{}ab^{2}\plus{}1}$ is an integer.

1983 IMO Longlists, 27
Tags: modular arithmetic , frobenius , additive number theory , additive combinatorics , number theory
Let $a,b$ and $c$ be positive integers, no two of which have a common divisor greater than $1$. Show that $2abc-ab-bc-ca$ is the largest integer which cannot be expressed in the form $xbc+yca+zab$, where $x,y,z$ are non-negative integers.

2007 AIME Problems, 5
Tags: number theory
The formula for converting a Fahrenheit temperature $F$ to the corresponding Celsius temperature $C$ is $C=\frac{5}{9}(F-32)$. An integer Fahrenheit temperature is converted to Celsius and rounded to the nearest integer; the resulting integer Celsius temperature is converted back to Fahrenheit and rounded to the nearest integer. For how many integer Fahrenheit temperatures $T$ with $32 \leq T \leq 1000$ does the original temperature equal the final temperature?

2006 Vietnam Team Selection Test, 3
Tags: induction , number theory
The real sequence $\{a_n|n=0,1,2,3,...\}$ defined $a_0=1$ and \[ a_{n+1}=\frac{1}{2}\left (a_{n}+\frac{1}{3 \cdot a_{n}} \right ). \] Denote \[ A_n=\frac{3}{3 \cdot a_n^2-1}. \] Prove that $A_n$ is a perfect square and it has at least $n$ distinct prime divisors.