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

2021 Durer Math Competition Finals, 5
Tags: remainder , divisor , number theory
Let $n$ be a positive integer. Show that every divisors of $2n^2 - 1$ gives a different remainder after division by $2n$.

2021 BMT, 7
Tags: number theory
Alice is counting up by fives, starting with the number $3$. Meanwhile, Bob is counting down by fours, starting with the number $2021$. How many numbers between $3$ and $2021$, inclusive, are counted by both Alice and Bob?

2005 MOP Homework, 1
Tags: number theory
Let $a0$, $a1$, ..., $a_n$ be integers, not all zero, and all at least $-1$. Given that $a_0+2a_1+2^2a_2+...+2^na_n =0$, prove that $a_0+a_1+...+a_n>0$.

2022 Azerbaijan BMO TST, N4*
Tags: number theory , shortlist , floor function
A natural number $n$ is given. Determine all $(n - 1)$-tuples of nonnegative integers $a_1, a_2, ..., a_{n - 1}$ such that $$\lfloor \frac{m}{2^n - 1}\rfloor + \lfloor \frac{2m + a_1}{2^n - 1}\rfloor + \lfloor \frac{2^2m + a_2}{2^n - 1}\rfloor + \lfloor \frac{2^3m + a_3}{2^n - 1}\rfloor + ... + \lfloor \frac{2^{n - 1}m + a_{n - 1}}{2^n - 1}\rfloor = m$$ holds for all $m \in \mathbb{Z}$.

2018 Thailand TST, 3
Tags: vieta jumping , number theory
Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both $$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$ are integers.

2014 ELMO Shortlist, 5
Tags: inequalities , number theory , induction
Define a [i]beautiful number[/i] to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer. Prove that each integer greater than $2$ can be expressed as the sum of pairwise distinct beautiful numbers. [i]Proposed by Matthew Babbitt[/i]

2013 BMT Spring, 7
Tags: number theory
Denote by $S(a,b)$ the set of integers $k$ that can be represented as $k=a\cdot m+b\cdot n$, for some non-negative integers $m$ and $n$. So, for example, $S(2,4)=\{0,2,4,6,\ldots\}$. Then, find the sum of all possible positive integer values of $x$ such that $S(18,32)$ is a subset of $S(3,x)$.

2017 Romanian Master of Mathematics Shortlist, N1
Tags: number theory , sum of digits
For each positive integer $k$, let $S(k)$ the sum of digits of $k$ in decimal system. Show that there is an integer $k$, with no $9$ in it's decimal representation, such that: $$S(2^{24^{2017}}k)=S(k)$$

1980 IMO Shortlist, 9
Tags: modular arithmetic , pascal triangle , representation , number theory
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 $s \geq 0, 0 < q < p$.

2021 CHMMC Winter (2021-22), 4
Tags: number theory
Show that for any three positive integers $a,m,n$ such that $m$ divides $n$, there exists an integer $k$ such that $gcd(a,m) = gcd(a+km,n)$ .

1972 IMO, 3
Tags: binomial coefficient , recurrence relation , factorial , number theory
Prove that $(2m)!(2n)!$ is a multiple of $m!n!(m+n)!$ for any non-negative integers $m$ and $n$.

2004 Purple Comet Problems, 21
Tags: number theory , relatively prime
Define $a_k = (k^2 + 1)k!$ and $b_k = a_1 + a_2 + a_3 + \cdots + a_k$. Let \[\frac{a_{100}}{b_{100}} = \frac{m}{n}\] where $m$ and $n$ are relatively prime natural numbers. Find $n - m$.

2012 Kosovo National Mathematical Olympiad, 3
Tags: number theory
The integers $a_1,a_2,...,a_{2012}$ are given. Exactly $29$ of them are divisible by $3$. Prove that the sum $a_1^2+a_2^2+...+a_{2012}^2$ is divisible by $3$.

2002 Tournament Of Towns, 1
Tags: number theory , combinatorics , greatest common divisor
All the species of plants existing in Russia are catalogued (numbered by integers from $2$ to $2000$ ; one after another, without omissions or repetitions). For any pair of species the gcd of their catalogue numbers was calculated and recorded but the catalogue numbers themselves were lost. Is it possible to restore the catalogue numbers from the data in hand?

1968 AMC 12/AHSME, 33
Tags: number theory , greatest common divisor
A number $N$ has three digits when expressed in base $7$. When $N$ is expressed in base $9$ the digits are reversed. Then the middle digit is: $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5$

2017 Estonia Team Selection Test, 11
Tags: polynomial , number theory , functional equation , sum of digits
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$ [i]Proposed by Warut Suksompong, Thailand[/i]

2016 BAMO, 3
Tags: prime factorization , prime factor , factor , number theory
The ${\textit{distinct prime factors}}$ of an integer are its prime factors listed without repetition. For example, the distinct prime factors of $40$ are $2$ and $5$. Let $A=2^k - 2$ and $B= 2^k \cdot A$, where $k$ is an integer ($k \ge 2$). Show that, for every integer $k$ greater than or equal to $2$, [list=i] [*] $A$ and $B$ have the same set of distinct prime factors. [*] $A+1$ and $B+1$ have the same set of distinct prime factors. [/list]

1996 French Mathematical Olympiad, Problem 5
Tags: france , combinatorics , number theory
Let $n$ be a positive integer. We say that a natural number $k$ has the property $C_n$ if there exist $2k$ distinct positive integers $a_1,b_1,\ldots,a_k,b_k$ such that the sums $a_1+b_1,\ldots,a_k+b_k$ are distinct and strictly smaller than $n$. (a) Prove that if $k$ has the property $C_n$ then $k\le \frac{2n-3}{5}$. (b) Prove that $5$ has the property $C_{14}$. (c) If $(2n-3)/5$ is an integer, prove that it has the property $C_n$.

2017 Balkan MO Shortlist, C2
Tags: congruent triangles , coloring , combinatorics , number theory , lattice points
Let $n,a,b,c$ be natural numbers. Every point on the coordinate plane with integer coordinates is colored in one of $n$ colors. Prove there exists $c$ triangles whose vertices are colored in the same color, which are pairwise congruent, and which have a side whose lenght is divisible by $a$ and a side whose lenght is divisible by $b$.

2017 India PRMO, 1
Tags: sum of digits , divide , number theory , divisible
How many positive integers less than $1000$ have the property that the sum of the digits of each such number is divisible by $7$ and the number itself is divisible by $3$?

2016 Moldova Team Selection Test, 4
Tags: number theory
Show that for every prime number $p$ and every positive integer $n\geq2$ there exists a positive integer $k$ such that the decimal representation of $p^k$ contains $n$ consecutive equal digits.

2000 AIME Problems, 5
Tags: probability , relatively prime , number theory
Each of two boxes contains both black and white marbles, and the total number of marbles in the two boxes is $25.$ One marble is taken out of each box randomly. The probability that both marbles are black is $27/50,$ and the probability that both marbles are white is $m/n,$ where $m$ and $n$ are relatively prime positive integers. What is $m+n?$

2017 Korea National Olympiad, problem 2
Tags: number theory
Find all primes $p$ such that there exist an integer $n$ and positive integers $k, m$ which satisfies the following. $$ \frac{(mk^2+2)p-(m^2+2k^2)}{mp+2} = n^2$$

2017 India PRMO, 23
Tags: prime number , algebra , number theory
Suppose an integer $x$, a natural number $n$ and a prime number $p$ satisfy the equation $7x^2-44x+12=p^n$. Find the largest value of $p$.

2011 Princeton University Math Competition, A1 / B3
Tags: number theory
The only prime factors of an integer $n$ are 2 and 3. If the sum of the divisors of $n$ (including itself) is $1815$, find $n$.