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

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 Purple Comet Problems, 8
Tags: number theory
Find the number of trailing zeros at the end of the base-$10$ representation of the integer $525^{25^2} \cdot 252^{52^5}$ .

Bangladesh Mathematical Olympiad 2020 Final, #12
Tags: number theory
$2^{2921}$ has $581$ digits and starts with a $4$. How many $2^n$'s starts with a $4$, where $0$ is the last digit?

2014 ELMO Shortlist, 1
Tags: modular arithmetic , mod 13 , number theory
Does there exist a strictly increasing infinite sequence of perfect squares $a_1, a_2, a_3, ...$ such that for all $k\in \mathbb{Z}^+$ we have that $13^k | a_k+1$? [i]Proposed by Jesse Zhang[/i]

2019 Durer Math Competition Finals, 7
Tags: number theory , integer
Find the smallest positive integer $n$ with the following property: if we write down all positive integers from $1$ to $10^n$ and add together the reciprocals of every non-zero digit written down, we obtain an integer.

2016 APMC, 5
Tags: factorial , number theory
Let $f(n,k)$ with $n,k\in\mathbb Z_{\geq 2}$ be defined such that $\frac{(kn)!}{(n!)^{f(n,k)}}\in\mathbb Z$ and $\frac{(kn)!}{(n!)^{f(n,k)+1}}\not\in\mathbb Z$ Define $m(k)$ such that for all $k$, $n\geq m(k)\implies f(n,k)=k$. Show that $m(k)$ exists and furthermore that $m(k)\leq \mathcal{O}\left(k^2\right)$

2004 Postal Coaching, 17
Tags: 3d geometry , number theory , geometry
In a system of numeration with base $B$ , there are $n$ one-digit numbers less than $B$ whose cubes have $B-1$ in the units-digits place. Determine the relation between $n$ and $B$

2023 Mexico National Olympiad, 6
Tags: number theory
Find all functions $f: \mathbb{N} \rightarrow \mathbb {N}$ such that for all positive integers $m, n$, $f(m+n)\mid f(m)+f(n)$ and $f(m)f(n) \mid f(mn)$.

PEN E Problems, 23
Tags: induction , number theory , prime number
Let $p_{1}=2, p_{2}={3}, p_{3}=5, \cdots, p_{n}$ be the first $n$ prime numbers, where $n \ge 3$. Prove that \[\frac{1}{{p_{1}}^{2}}+\frac{1}{{p_{2}}^{2}}+\cdots+\frac{1}{{p_{n}}^{2}}+\frac{1}{p_{1}p_{2}\cdots p_{n}}< \frac{1}{2}.\]

2021 BMT, 3
Tags: number theory
How many distinct sums can be made from adding together exactly 8 numbers that are chosen from the set $\{ 1,4,7,10 \}$, where each number in the set is chosen at least once? (For example, one possible sum is $1+1+1+4+7+7+10+10=41$.)

2019 ELMO Shortlist, N1
Tags: number theory
Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=1$, and let $c > 1$ be an integer. Define $x_0=0$ and $x_{i+1} = P(x_i)$ for all integers $i \ge 0$. Show that there are infinitely many positive integers $n$ such that $\gcd (x_n, n+c)=1$. [i]Proposed by Milan Haiman and Carl Schildkraut[/i]

2018 Korea Junior Math Olympiad, 2
Tags: number theory
Find all positive integer $N$ which has not less than $4$ positive divisors, such that the sum of squares of the $4$ smallest positive divisors of $N$ is equal to $N$.

2010 AMC 8, 20
Tags: number theory , least common multiple
In a room, $2/5$ of the people are wearing gloves, and $3/4$ of the people are wearing hats. What is the minimum number of people in the room wearing both a hat and a glove? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 20 $

2017 Costa Rica - Final Round, 2
Tags: gcd , greatest common divisor , number theory
Determine the greatest common divisor of the numbers: $$5^5-5, 7^7-7, 9^9-9 ,..., 2017^{2017}-2017,$$

2019 District Olympiad, 1
Tags: algebra , number theory
Determine the numbers $x,y$, with $x$ integer and $y$ rational, for which equality holds: $$5(x^2+xy+y^2) = 7(x+2y)$$

2005 Postal Coaching, 1
Tags: number theory
Consider the sequence $<{a_n}>$ of natural numbers such that {i} $a_n$ is a square numver for all $n$ ; (ii) $a_{n+1} - a_n$ is either a prime or a square of a prime for each $n$. Show that $<a_n>$ is a finite sequence. Determine the longest such sequence.

2010 Postal Coaching, 4
Tags: number theory
Prove that the following statement is true for two natural nos. $m,n$ if and only $v(m) = v(n)$ where $v(k)$ is the highest power of $2$ dividing $k$. $\exists$ a set $A$ of positive integers such that $(i)$ $x,y \in \mathbb{N}, |x-y| = m \implies x \in A $ or $y \in A$ $(ii)$ $x,y \in \mathbb{N}, |x-y| = n \implies x \not\in A $ or $y \not\in A$

2013 Czech-Polish-Slovak Junior Match, 1
Tags: number theory , infinitely many , prime
Decide whether there are infinitely many primes $p$ having a multiple in the form $n^2 + n + 1$ for some natural number $n$

1992 Baltic Way, 3
Tags: geometry , 3d geometry , arithmetic sequence , algebra , factorization , sum of cubes , number theory
Find an infinite non-constant arithmetic progression of natural numbers such that each term is neither a sum of two squares, nor a sum of two cubes (of natural numbers).

2017 Ecuador NMO (OMEC), 3
Tags: algebra , number theory
Adrian has $2n$ cards numbered from $ 1$ to $2n$. He gets rid of $n$ cards that are consecutively numbered. The sum of the numbers of the remaining papers is $1615$. Find all the possible values of $n$.

II Soros Olympiad 1995 - 96 (Russia), 11.7
Tags: summation , number theory , consecutive integers
Find three consecutive natural numbers, each of which is divisible by the square of the sum of its digits. Prove that there are no five such numbers in a row.

1954 Putnam, B1
Tags: number theory
Show that the equation $x^2 -y^2 =a^3$ has always integral solutions for $x$ and $y$ whenever $a$ is a positive integer.

2021 IMO Shortlist, A2
Tags: algebra , floor function , summation , number theory
Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?

2018 AMC 12/AHSME, 5
Tags: number theory , prime number
How many subsets of $\{2,3,4,5,6,7,8,9\}$ contain at least one prime number? $\textbf{(A)} \text{ 128} \qquad \textbf{(B)} \text{ 192} \qquad \textbf{(C)} \text{ 224} \qquad \textbf{(D)} \text{ 240} \qquad \textbf{(E)} \text{ 256}$

2010 Gheorghe Vranceanu, 4
Tags: number theory , algebra , periodicity , seuqence
Let be two real numbers $ \alpha ,\beta $ and two sequences $ \left(x_n \right)_{n\ge 1} ,\left(y_n \right)_{n\ge 1} $ whose smallest periods are $ p,q, $ respectively. Prove that the sequence $ \left( \alpha x_n+\beta y_n\right)_{n\ge 1} $ is periodic if $ \text{gcd}^2 (p,q) | \text{lcm} (p,q) , $ and in this case find its smallest period.