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

1998 Romania National Olympiad, 1
Tags: inequalities , algebra , perfect square , number theory
Let $n$ be a positive integer and $x_1,x_2,...,x_n$ be integer numbers such that $$x_1^2+x_2^2+...+x_n^2+ n^3 \le (2n - 1)(x_1+x_2+...+x_n ) + n^2$$ . Show that : a) $x_1,x_2,...,x_n$ are non-negative integers b) the number $x_1+x_2+...+x_n+n+1$ is not a perfect square.

2013 NIMO Problems, 6
Tags: number theory , relatively prime
Let $f(n)=\varphi(n^3)^{-1}$, where $\varphi(n)$ denotes the number of positive integers not greater than $n$ that are relatively prime to $n$. Suppose \[ \frac{f(1)+f(3)+f(5)+\dots}{f(2)+f(4)+f(6)+\dots} = \frac{m}{n} \] where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$. [i]Proposed by Lewis Chen[/i]

2016 CentroAmerican, 1
Tags: number theory , digit
Find all positive integers $n$ that have 4 digits, all of them perfect squares, and such that $n$ is divisible by 2, 3, 5 and 7.

2015 Princeton University Math Competition, B1
Tags: number theory
What is the remainder when \[\sum_{k=0}^{100}10^k\] is divided by $9$?

2013 China Second Round Olympiad, 4
Tags: number theory
Let $n,k$ be integers greater than $1$, $n<2^k$. Prove that there exist $2k$ integers none of which are divisible by $n$, such that no matter how they are separated into two groups there exist some numbers all from the same group whose sum is divisible by $n$.

2017 Bosnia And Herzegovina - Regional Olympiad, 4
Tags: number theory , number of divisors , divisor
It is given positive integer $N$. Let $d_1$, $d_2$,...,$d_n$ be its divisors and let $a_i$ be number of divisors of $d_i$, $i=1,2,...n$. Prove that $$(a_1+a_2+...+a_n)^2={a_1}^3+{a_2}^3+...+{a_n}^3$$

MathLinks Contest 4th, 6.2
Tags: number theory , function , geometry , square
Let $P$ be the set of points in the plane, and let $f : P \to P$ be a function such that the image through $f$ of any triangle is a square (any polygon is considered to be formed by the reunion of the points on its sides). Prove that $f(P)$ is a square.

2024 Ukraine National Mathematical Olympiad, Problem 2
Tags: number theory , combinatorics , game , geometry
You are given positive integers $m, n>1$. Vasyl and Petryk play the following game: they take turns marking on the coordinate plane yet unmarked points of the form $(x, y)$, where $x, y$ are positive integers with $1 \leq x \leq m, 1 \leq y \leq n$. The player loses if after his move there are two marked points, the distance between which is not a positive integer. Who will win this game if Vasyl moves first and each player wants to win? [i]Proposed by Mykyta Kharin[/i]

2024 Israel TST, P1
Tags: factorial , number theory , floor function
For each positive integer $n$ let $a_n$ be the largest positive integer satisfying \[(a_n)!\left| \prod_{k=1}^n \left\lfloor \frac{n}{k}\right\rfloor\right.\] Show that there are infinitely many positive integers $m$ for which $a_{m+1}<a_m$.

2021 Francophone Mathematical Olympiad, 4
Tags: number theory , functional equation , functional , algebra
Let $\mathbb{N}_{\ge 1}$ be the set of positive integers. Find all functions $f \colon \mathbb{N}_{\ge 1} \to \mathbb{N}_{\ge 1}$ such that, for all positive integers $m$ and $n$: (a) $n = \left(f(2n)-f(n)\right)\left(2 f(n) - f(2n)\right)$, (b)$f(m)f(n) - f(mn) = \left(f(2m)-f(m)\right)\left(2 f(n) - f(2n)\right) + \left(f(2n)-f(n)\right)\left(2 f(m) - f(2m)\right)$, (c) $m-n$ divides $f(2m)-f(2n)$ if $m$ and $n$ are distinct odd prime numbers.

2010 Singapore MO Open, 3
Tags: ratio , prime number , arithmetic sequence , number theory
Suppose that $a_1,...,a_{15}$ are prime numbers forming an arithmetic progression with common difference $d > 0$ if $a_1 > 15$ show that $d > 30000$

2004 Switzerland - Final Round, 6
Tags: number theory , last digit
Determine all $k$ for which there exists a natural number n such that $1^n + 2^n + 3^n + 4^n$ with exactly $k$ zeros at the end.

2014 Kazakhstan National Olympiad, 1
Tags: number theory
$a_1,a_2,...,a_{2014}$ is a permutation of $1,2,3,...,2014$. What is the greatest number of perfect squares can have a set ${ a_1^2+a_2,a_2^2+a_3,a_3^2+a_4,...,a_{2013}^2+a_{2014},a_{2014}^2+a_1 }?$

2013 European Mathematical Cup, 2
Tags: induction , number theory
Palindrome is a sequence of digits which doesn't change if we reverse the order of its digits. Prove that a sequence $(x_n)^{\infty}_{n=0}$ defined as $x_n=2013+317n$ contains infinitely many numbers with their decimal expansions being palindromes.

1990 Iran MO (2nd round), 3
Tags: number theory
[b](a)[/b] For every positive integer $n$ prove that \[1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2} <2\] [b](b)[/b] Let $X=\{1, 2, 3 ,\ldots, n\} \ ( n \geq 1)$ and let $A_k$ be non-empty subsets of $X \ (k=1,2,3, \ldots , 2^n -1).$ If $a_k$ be the product of all elements of the set $A_k,$ prove that \[\sum_{i=1}^{m} \sum_{j=1}^m \frac{1}{a_i \cdot j^2} <2n+1\]

2023 Auckland Mathematical Olympiad, 8
Tags: combinatorics , number theory
How few numbers is it possible to cross out from the sequence $$1, 2,3,..., 2023$$ so that among those left no number is the product of any two (distinct) other numbers?

2017 Argentina National Olympiad, 2
Tags: sum , algebra , number theory , consecutive
In a row there are $51$ written positive integers. Their sum is $100$ . An integer is [i]representable [/i] if it can be expressed as the sum of several consecutive numbers in a row of $51$ integers. Show that for every $k$ , with $1\le k \le 100$ , one of the numbers $k$ and $100-k$ is representable.

1998 Czech And Slovak Olympiad IIIA, 4
Tags: number theory , max , power of 3
For each date of year $1998$, we calculate day$^{month}$ −year and determine the greatest power of $3$ that divides it. For example, for April $21$ we get $21^4 - 1998 =192483 = 3^3 \cdot 7129$, which is divisible by $3^3$ and not by $3^4$ . Find all dates for which this power of $3$ is the greatest.

2024 Miklos Schweitzer, 9
Tags: number theory , finite field
Let $q > 1$ be a power of $2$. Let $f: \mathbb{F}_{q^2} \to \mathbb{F}_{q^2}$ be an affine map over $\mathbb{F}_2$. Prove that the equation \[ f(x) = x^{q+1} \] has at most $2q - 1$ solutions.

2007 Estonia Math Open Junior Contests, 3
Tags: number theory
Find all positive integers N with at most 4 digits such that the number obtained by reversing the order of digits of N is divisible by N and differs from N.

2010 Germany Team Selection Test, 1
Tags: function , algebra , modular arithmetic , divisibility , number theory
Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$. [i]Proposed by Juhan Aru, Estonia[/i]

2007 Junior Balkan MO, 4
Tags: quadratic , modular arithmetic , number theory
Prove that if $ p$ is a prime number, then $ 7p+3^{p}-4$ is not a perfect square.

2023 Ukraine National Mathematical Olympiad, 11.1
Tags: number theory , divisibility
Set $M$ contains $n \ge 2$ positive integers. It's known that for any two different $a, b \in M$, $a^2+1$ is divisible by $b$. What is the largest possible value of $n$? [i]Proposed by Oleksiy Masalitin[/i]

2018 Baltic Way, 17
Tags: number theory
Prove that for any positive integers $p,q$ such that $\sqrt{11}>\frac{p}{q}$, the following inequality holds: \[\sqrt{11}-\frac{p}{q}>\frac{1}{2pq}.\]

2013 IFYM, Sozopol, 5
Tags: number theory
Find all positive integers $n$ satisfying $2n+7 \mid n! -1$.