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

2016 Junior Regional Olympiad - FBH, 2
Tags: number theory , fraction
Which fraction is bigger: $\frac{5553}{5557}$ or $\frac{6664}{6669}$ ?

2016 Silk Road, 4
Tags: binary representation , number theory , power of 2 , combinatorics
Let $P(n)$ be the number of ways to split a natural number $n$ to the sum of powers of two, when the order does not matter. For example $P(5) = 4$, as $5=4+1=2+2+1=2+1+1+1=1+1+1+1+1$. Prove that for any natural the identity $P(n) + (-1)^{a_1} P(n-1) + (-1)^{a_2} P(n-2) + \ldots + (-1)^{a_{n-1}} P(1) + (-1)^{a_n} = 0,$ is true, where $a_k$ is the number of units in the binary number record $k$ . [url=http://matol.kz/comments/2720/show]source[/url]

2015 India Regional MathematicaI Olympiad, 4
Tags: geometric sequence , algebra , number theory
Find all three digit natural numbers of the form $(abc)_{10}$ such that $(abc)_{10}$, $(bca)_{10}$ and $(cab)_{10}$ are in geometric progression. (Here $(abc)_{10}$ is representation in base $10$.)

2010 District Olympiad, 2
Tags: remainder , number theory
Let $n$ be an integer, $n \ge 2$. Find the remainder of the division of the number $n(n + 1)(n + 2)$ by $n - 1$.

2010 Iran MO (3rd Round), 5
Tags: number theory
prove that if $p$ is a prime number such that $p=12k+\{2,3,5,7,8,11\}$($k \in \mathbb N \cup \{0\}$), there exist a field with $p^2$ elements.($\frac{100}{6}$ points)

2014 IFYM, Sozopol, 2
Tags: number theory , divisor
Find the least natural number $n$, which has at least 6 different divisors $1=d_1<d_2<d_3<d_4<d_5<d_6<...$, for which $d_3+d_4=d_5+6$ and $d_4+d_5=d_6+7$.

2006 Junior Balkan MO, 1
Tags: floor function , number theory
If $n>4$ is a composite number, then $2n$ divides $(n-1)!$.

2015 239 Open Mathematical Olympiad, 2
Tags: binomial coefficient , number theory
Prove that $\binom{n+k}{n}$ can be written as product of $n$ pairwise coprime numbers $a_1,a_2,\dots,a_n$ such that $k+i$ is divisible by $a_i$ for all indices $i$.

2024 Ukraine National Mathematical Olympiad, Problem 1
Tags: number theory , digit
Oleksiy wrote several distinct positive integers on the board and calculated all their pairwise sums. It turned out that all digits from $0$ to $9$ appear among the last digits of these sums. What could be the smallest number of integers that Oleksiy wrote? [i]Proposed by Oleksiy Masalitin[/i]

2015 Canadian Mathematical Olympiad Qualification, 1
Tags: number theory , diophantine
Find all integer solutions to the equation $7x^2y^2 + 4x^2 = 77y^2 + 1260$.

2010 CentroAmerican, 1
Tags: modular arithmetic , number theory
Denote by $S(n)$ the sum of the digits of the positive integer $n$. Find all the solutions of the equation $n(S(n)-1)=2010.$

1995 Bulgaria National Olympiad, 6
Tags: number theory
Suppose that $x$ and $y$ are different real numbers such that $\frac{x^n-y^n}{x-y}$ is an integer for some four consecutive positive integers $n$. Prove that $\frac{x^n-y^n}{x-y}$ is an integer for all positive integers n.

2002 Indonesia MO, 6
Tags: number theory
Find all primes $p$ such that $4p^2+1$ and $6p^2+1$ are both primes.

LMT Speed Rounds, 21
Tags: algebra , number theory , combinatorics
Let $(a_1,a_2,a_3,a_4,a_5)$ be a random permutation of the integers from $1$ to $5$ inclusive. Find the expected value of $$\sum^5_{i=1} |a_i -i | = |a_1 -1|+|a_2 -2|+|a_3 -3|+|a_4 -4|+|a_5 -5|.$$ [i]Proposed by Muztaba Syed[/i]

2021 Peru IMO TST, P1
Tags: number theory
For any positive integer $n$, we define $S(n)$ to be the sum of its digits in the decimal representation. Prove that for any positive integer $m$, there exists a positive integer $n$ such that $S(n)-S(n^2)>m$.

2016 CMIMC, 2
Tags: number theory
Let $a_1$, $a_2$, $\ldots$ be an infinite sequence of (positive) integers such that $k$ divides $\gcd(a_{k-1},a_k)$ for all $k\geq 2$. Compute the smallest possible value of $a_1+a_2+\cdots+a_{10}$.

1997 Israel National Olympiad, 5
Tags: coprime , number theory , prime
The natural numbers $a_1,a_2,...,a_n, n \ge 12$, are smaller than $9n^2$ and pairwise coprime. Show that at least one of these numbers is prime.

2005 AIME Problems, 3
Tags: ratio , geometric series , number theory , relatively prime , algebra , system of equations
An infinite geometric series has sum $2005$. A new series, obtained by squaring each term of the original series, has $10$ times the sum of the original series. The common ratio of the original series is $\frac{m}{n}$ where $m$ and $n$ are relatively prime integers. Find $m+n$.

1979 IMO Longlists, 25
Tags: number theory , relatively prime , series summation , divisibility , prime
If $p$ and $q$ are natural numbers so that \[ \frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+ \ldots -\frac{1}{1318}+\frac{1}{1319}, \] prove that $p$ is divisible with $1979$.

PEN K Problems, 2
Tags: function , induction , strong induction , number theory , prime factorization , functional equation
Find all surjective functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $m,n\in \mathbb{N}$: \[m \vert n \Longleftrightarrow f(m) \vert f(n).\]

1995 Cono Sur Olympiad, 1
Tags: number theory
Find a number with $3$ digits, knowing that the sum of its digits is $9$, their product is $24$ and also the number read from right to left is $\frac{27}{38}$ of the original.

1999 Bundeswettbewerb Mathematik, 2
Tags: number theory , sum of digits , digit , inequalities
For every natural number $n$, let $Q(n)$ denote the sum of the decimal digits of $n$. Prove that there are infinitely many positive integers $k$ with $Q(3^k) \ge Q(3^{k+1})$.

2019 OMMock - Mexico National Olympiad Mock Exam, 2
Tags: number theory , divisibility , algebra
Find all pairs of positive integers $(m, n)$ such that $m^2-mn+n^2+1$ divides both numbers $3^{m+n}+(m+n)!$ and $3^{m^3+n^3}+m+n$. [i]Proposed by Dorlir Ahmeti[/i]

1970 All Soviet Union Mathematical Olympiad, 139
Tags: number theory , digit
Prove that for every natural number $k$ there exists an infinite set of such natural numbers $t$, that the decimal notation of $t$ does not contain zeroes and the sums of the digits of the numbers $t$ and $kt$ are equal.

1999 IMO, 4
Tags: number theory , divisibility , prime
Find all the pairs of positive integers $(x,p)$ such that p is a prime, $x \leq 2p$ and $x^{p-1}$ is a divisor of $ (p-1)^{x}+1$.