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

2014 Contests, 1
Tags: number theory
On a circle there are $99$ natural numbers. If $a,b$ are any two neighbouring numbers on the circle, then $a-b$ is equal to $1$ or $2$ or $ \frac{a}{b}=2 $. Prove that there exists a natural number on the circle that is divisible by $3$. [i]S. Berlov[/i]

2023 Chile TST Ibero., 3
Tags: number theory
Determine the smallest positive integer \( n \) with the following property: for every triple of positive integers \( x, y, z \), with \( x \) dividing \( y^3 \), \( y \) dividing \( z^3 \), and \( z \) dividing \( x^3 \), it also holds that \( (xyz) \) divides \( (x + y + z)^n \).

1995 AIME Problems, 7
Tags: trigonometry , quadratic , number theory , relatively prime , algebra , quadratic formula
Given that $(1+\sin t)(1+\cos t)=5/4$ and \[ (1-\sin t)(1-\cos t)=\frac mn-\sqrt{k}, \] where $k, m,$ and $n$ are positive integers with $m$ and $n$ relatively prime, find $k+m+n.$

2010 Ukraine Team Selection Test, 6
Tags: number theory , integer , divisibility
Find all pairs of odd integers $a$ and $b$ for which there exists a natural number$ c$ such that the number $\frac{c^n+1}{2^na+b}$ is integer for all natural $n$.

2015 China Team Selection Test, 3
Tags: number theory , function
For all natural numbers $n$, define $f(n) = \tau (n!) - \tau ((n-1)!)$, where $\tau(a)$ denotes the number of positive divisors of $a$. Prove that there exist infinitely many composite $n$, such that for all naturals $m < n$, we have $f(m) < f(n)$.

1999 Baltic Way, 2
Tags: geometry , 3d geometry , number theory
Determine all positive integers $n$ with the property that the third root of $n$ is obtained by removing its last three decimal digits.

Russian TST 2018, P1
Tags: number theory
The natural numbers $a > b$ are such that $a-b=5b^2-4a^2$. Prove that the number $8b + 1$ is composite.

1992 Brazil National Olympiad, 5
Tags: number theory
Let $d(n)=\sum_{0<d|n}{1}$. Show that, for any natural $n>1$, \[ \sum_{2 \leq i \leq n}{\frac{1}{i}} \leq \sum{\frac{d(i)}{n}} \leq \sum_{1 \leq i \leq n}{\frac{1}{i}} \]

2017 Israel National Olympiad, 2
Tags: number theory , product , digit , maximize
Denote by $P(n)$ the product of the digits of a positive integer $n$. For example, $P(1948)=1\cdot9\cdot4\cdot8=288$. [list=a] [*] Evaluate the sum $P(1)+P(2)+\dots+P(2017)$. [*] Determine the maximum value of $\frac{P(n)}{n}$ where $2017\leq n\leq5777$. [/list]

2007 Estonia Team Selection Test, 3
Tags: number theory , power of prime , prime
Let $n$ be a natural number, $n > 2$. Prove that if $\frac{b^n-1}{b-1}$ is a prime power for some positive integer $b$ then $n$ is prime.

1999 China Team Selection Test, 2
Tags: prime number , number theory
Find all prime numbers $p$ which satisfy the following condition: For any prime $q < p$, if $p = kq + r, 0 \leq r < q$, there does not exist an integer $q > 1$ such that $a^{2} \mid r$.

2007 Tuymaada Olympiad, 1
Tags: number theory
Positive integers $ a<b$ are given. Prove that among every $ b$ consecutive positive integers there are two numbers whose product is divisible by $ ab$.

2012 India IMO Training Camp, 2
Tags: number theory
Find the least positive integer that cannot be represented as $\frac{2^a-2^b}{2^c-2^d}$ for some positive integers $a, b, c, d$.

1997 Irish Math Olympiad, 5
Tags: inequalities , number theory
Let $ S$ be the set of odd integers greater than $ 1$. For each $ x \in S$, denote by $ \delta (x)$ the unique integer satisfying the inequality $ 2^{\delta (x)}<x<2^{\delta (x) \plus{}1}$. For $ a,b \in S$, define: $ a \ast b\equal{}2^{\delta (a)\minus{}1} (b\minus{}3)\plus{}a.$ Prove that if $ a,b,c \in S$, then: $ (a)$ $ a \ast b \in S$ and $ (b)$ $ (a \ast b)\ast c\equal{}a \ast (b \ast c)$.

2013 Tuymaada Olympiad, 7
Tags: modular arithmetic , number theory
Solve the equation $p^2-pq-q^3=1$ in prime numbers. [i]A. Golovanov[/i]

2005 AIME Problems, 8
Tags: quadratic , geometry , trapezoid , number theory , relatively prime , pythagorean theorem , algebra
Circles $C_1$ and $C_2$ are externally tangent, and they are both internally tangent to circle $C_3$. The radii of $C_1$ and $C_2$ are $4$ and $10$, respectively, and the centers of the three circles are all collinear. A chord of $C_3$ is also a common external tangent of $C_1$ and $C_2$. Given that the length of the chord is $\frac{m\sqrt{n}}{p}$ where $m,n,$ and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime, find $m+n+p$.

2015 Kyiv Math Festival, P3
Tags: positive integer , number theory , sum
Is it true that every positive integer greater than $100$ is a sum of $4$ positive integers such that each two of them have a common divisor greater than $1$?

2013 Harvard-MIT Mathematics Tournament, 11
Tags: hmmt , number theory , prime factorization , prime number , binomial coefficient , pascal triangle
Compute the prime factorization of $1007021035035021007001$. (You should write your answer in the form $p_1^{e_1}p_2^{e_2}\ldots p_k^{e_k}$ where $p_1,\ldots,p_k$ are distinct prime numbers and $e_1,\ldots,e_k$ are positive integers.)

2013 ELMO Shortlist, 2
Tags: algebra , polynomial , 3d geometry , modular arithmetic , number theory , geometry
For what polynomials $P(n)$ with integer coefficients can a positive integer be assigned to every lattice point in $\mathbb{R}^3$ so that for every integer $n \ge 1$, the sum of the $n^3$ integers assigned to any $n \times n \times n$ grid of lattice points is divisible by $P(n)$? [i]Proposed by Andre Arslan[/i]

2013 Hanoi Open Mathematics Competitions, 5
Tags: number theory , diophantine equation , diophantine
The number of integer solutions $x$ of the equation below $(12x -1)(6x - 1)(4x -1)(3x - 1) = 330$ is (A): $0$, (B): $1$, (C): $2$, (D): $3$, (E): None of the above.

2018 239 Open Mathematical Olympiad, 8-9.1
Tags: number theory
Given a prime number $p$. A positive integer $x$ is divided by $p$ with a remainder, and the number $p^2$ is divided by $x$ with a remainder. The remainders turned out to be equal. Find them [i]Proposed by Sergey Berlov[/i]

2025 Serbia Team Selection Test for the IMO 2025, 6
Tags: number theory
For an $n \times n$ table filled with natural numbers, we say it is a [i]divisor table[/i] if: - the numbers in the $i$-th row are exactly all the divisors of some natural number $r_i$, - the numbers in the $j$-th column are exactly all the divisors of some natural number $c_j$, - $r_i \ne r_j$ for every $i \ne j$. A prime number $p$ is given. Determine the smallest natural number $n$, divisible by $p$, such that there exists an $n \times n$ divisor table, or prove that such $n$ does not exist. [i]Proposed by Pavle Martinović[/i]

2007 Iran MO (3rd Round), 1
Tags: modular arithmetic , number theory
Let $ n$ be a natural number, such that $ (n,2(2^{1386}\minus{}1))\equal{}1$. Let $ \{a_{1},a_{2},\dots,a_{\varphi(n)}\}$ be a reduced residue system for $ n$. Prove that:\[ n|a_{1}^{1386}\plus{}a_{2}^{1386}\plus{}\dots\plus{}a_{\varphi(n)}^{1386}\]

2016 Irish Math Olympiad, 1
Tags: number theory , divisibility
If the three-digit number $ABC$ is divisible by $27$, prove that the three-digit numbers $BCA$ and $CAB$ are also divisible by $27$.

2024 Czech-Polish-Slovak Junior Match, 2
Tags: product , perfect cube , geometry , 3d geometry , number theory
How many non-empty subsets of $\{1,2,\dots,11\}$ are there with the property that the product of its elements is the cube of an integer?