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

2012 Princeton University Math Competition, B2
Tags: number theory
Let $M$ be the smallest positive multiple of $2012$ that has $2012$ divisors. Suppose $M$ can be written as $\Pi_{k=1}^{n}p_k^{a_k}$ where the $p_k$’s are distinct primes and the $a_k$’s are positive integers. Find $\Sigma_{k=1}^{n}(p_k + a_k)$

2010 Kazakhstan National Olympiad, 6
Tags: number theory , modular arithmetic
Let numbers $1,2,3,...,2010$ stand in a row at random. Consider row, obtain by next rule: For any number we sum it and it's number in a row (For example for row $( 2,7,4)$ we consider a row $(2+1;7+2;4+3)=(3;9;7)$ ); Proved, that in resulting row we can found two equals numbers, or two numbers, which is differ by $2010$

India EGMO 2025 TST, 6
Tags: number theory
Let $M$ be a positive integer, and let $a,b,c$ be integers in the interval $[M,M+\sqrt{\frac{M}{2}})$ such that $a^3b+b^3c+c^3a$ is divisible by $abc$. Prove that $a=b=c$. Proposed by Shantanu Nene

2008 Federal Competition For Advanced Students, Part 2, 2
Tags: algebra , number theory , vieta , polynomial
(a) Does there exist a polynomial $ P(x)$ with coefficients in integers, such that $ P(d) \equal{} \frac{2008}{d}$ holds for all positive divisors of $ 2008$? (b) For which positive integers $ n$ does a polynomial $ P(x)$ with coefficients in integers exists, such that $ P(d) \equal{} \frac{n}{d}$ holds for all positive divisors of $ n$?

2022 Philippine MO, 5
Tags: number theory
Find all positive integers $n$ for which there exists a set of exactly $n$ distinct positive integers, none of which exceed $n^2$, whose reciprocals add up to $1$.

2021 Spain Mathematical Olympiad, 2
Tags: number theory , number of divisors
Given a positive integer $n$, we define $\lambda (n)$ as the number of positive integer solutions of $x^2-y^2=n$. We say that $n$ is [i]olympic[/i] if $\lambda (n) = 2021$. Which is the smallest olympic positive integer? Which is the smallest olympic positive odd integer?

2021/2022 Tournament of Towns, P1
Tags: number theory
Peter picked a positive integer, multiplied it by 5, multiplied the result by 5,then multiplied the result by 5 again and so on. Altogether $k$ multiplications were made. It so happened that the decimal representations of the original number and of all $k$ resulting numbers in this sequence do not contain digit $7$. Prove that there exists a positive integer such that it can be multiplied $k$ times by $2$ so that no number in this sequence contains digit $7$.

2016 Romanian Master of Mathematics Shortlist, C4
Tags: number theory , combinatorics , sets of integers
Prove that a $46$-element set of integers contains two distinct doubletons $\{u, v\}$ and $\{x,y\}$ such that $u + v \equiv x + y$ (mod $2016$).

2004 Bulgaria Team Selection Test, 1
Tags: number theory , algebra , induction , polynomial
Let $n$ be a positive integer. Find all positive integers $m$ for which there exists a polynomial $f(x) = a_{0} + \cdots + a_{n}x^{n} \in \mathbb{Z}[X]$ ($a_{n} \not= 0$) such that $\gcd(a_{0},a_{1},\cdots,a_{n},m)=1$ and $m|f(k)$ for each $k \in \mathbb{Z}$.

2013 Junior Balkan Team Selection Tests - Romania, 1
Tags: diophantine , diophantine equation , number theory
Let $n$ be a positive integer. Determine all positive integers $p$ for which there exist positive integers $x_1 < x_2 <...< x_n$ such that $\frac{1}{x_1}+\frac{2}{x_2}+ ... +\frac{n}{x_n}= p$ Irish Mathematical Olympiad

2014 Singapore MO Open, 5
Tags: inequalities , induction , number theory
Determine the largest odd positive integer $n$ such that every odd integer $k$ with $1<k<n$ and $\gcd(k, n)=1$ is a prime.

1980 IMO, 22
Tags: number theory , pascal triangle , representation , modular arithmetic
Let $p$ be a prime number. Prove that there is no number divisible by $p$ in the $n-th$ row of Pascal's triangle if and only if $n$ can be represented in the form $n = p^sq - 1$, where $s$ and $q$ are integers with $s \geq 0, 0 < q < p$.

2019 District Olympiad, 3
Tags: integer , number theory
Consider the sets $M = \{0,1,2,, 2019\}$ and $$A=\left\{ x\in M\,\, | \frac{x^3-x}{24} \in N\right\} $$ a) How many elements does the set $A$ have? b) Determine the smallest natural number $n$, $n \ge 2$, which has the property that any $n$-element subset of the set $A $contains two distinct elements whose difference is divisible by $40$.

1991 Putnam, B4
Tags: number theory
Let $p>2$ be a prime. Prove that $\sum_{n=0}^p\binom pn\binom{p+n}n\equiv2p+1\pmod{p^2}$.

2015 Princeton University Math Competition, B4
Tags: number theory
A circle with radius $1$ and center $(0, 1)$ lies on the coordinate plane. Ariel stands at the origin and rolls a ball of paint at an angle of $35$ degrees relative to the positive $x$-axis (counting degrees counterclockwise). The ball repeatedly bounces off the circle and leaves behind a trail of paint where it rolled. After the ball of paint returns to the origin, the paint has traced out a star with $n$ points on the circle. What is $n$?

2019 Brazil National Olympiad, 2
Tags: number theory , inequalities
Let $a, b$ and $k$ be positive integers with $k> 1$ such that $lcm (a, b) + gcd (a, b) = k (a + b)$. Prove that $a + b \geq 4k$

2018 BMT Spring, Tie 1
Tags: number theory
Compute the least positive $x$ such that $25x - 6$ is divisible by $1001$.

2009 Bosnia And Herzegovina - Regional Olympiad, 1
Tags: integer , number theory
Find all triplets of integers $(x,y,z)$ such that $$xy(x^2-y^2)+yz(y^2-z^2)+zx(z^2-x^2)=1$$

2009 Junior Balkan Team Selection Tests - Romania, 1
Tags: number theory , diophantine , diophantine equation
Find all non-negative integers $a,b,c,d$ such that $7^a= 4^b + 5^c + 6^d$.

2016 Portugal MO, 5
Tags: number theory , power of 2
Determine all natural numbers $x, y$ and $z$ such that the number $2^x +4^y +8^z +16^2$ is a power of $2$.

1954 Putnam, A7
Tags: number theory , integer
Prove that there are no integers $x$ and $y$ for which $$x^2 +3xy-2y^2 =122.$$

2008 IberoAmerican Olympiad For University Students, 1
Tags: number theory
Let $n$ be a positive integer that is not divisible by either $2$ or $5$. In the decimal expansion of $\frac{1}{n}= 0.a_1a_2a_3\cdots$ a finite number of digits after the decimal point are chosen arbitrarily to be deleted. Clearly the decimal number obtained by this procedure is also rational, so it's equal to $\frac{a}{b}$ for some integers $a,b$. Prove that $b$ is divisible by $n$.

2009 Cuba MO, 9
Tags: divide , number theory
Find all the triples of prime numbers $(p, q, r)$ such that $$p | 2qr + r \,\,\,, \,\,\,q |2pr + p \,\,\, and \,\,\, r | 2pq + q.$$

2001 Saint Petersburg Mathematical Olympiad, 9.6
Tags: exponent , diophanative equations , number theory
Find all positive integer solution: $$k^m+m^n=k^n+1$$ [I]Proposed by V. Frank, F. Petrov[/i]

2019 Iran RMM TST, 4
Tags: least common multiple , number theory
Let $a,b $ be two relatively prime positive integers.Also let $m,n $ be positive integers with $n> m $.\\ Prove that\\ $lcm [am+b,a (m+1)+b,...,an+b]\ge (n+1)\cdot \binom {n}{m}$ [i]Proposed by Navid Safaei[/i]