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

2006 Cezar Ivănescu, 2
Tags: albegra , fractional part , number theory , algebra
[b]a)[/b] Prove that $ \{ a \} +\{ 1/a \} <3/2, $ for any positive real number $ a. $ [b]b)[/b] Give an example of a number $ b $ satisfying $ \{ b \} +\{ 1/b \} =1. $ [i]{} means fractional part[/i]

1992 China Team Selection Test, 3
Tags: modular arithmetic , number theory , quadratic
For any prime $p$, prove that there exists integer $x_0$ such that $p | (x^2_0 - x_0 + 3)$ $\Leftrightarrow$ there exists integer $y_0$ such that $p | (y^2_0 - y_0 + 25).$

2020-21 IOQM India, 14
Tags: number theory
The product $55\cdot60\cdot65$ is written as a product of 5 distinct numbers. Find the least possible value of the largest number, among these 5 numbers.

2004 Tournament Of Towns, 4
Tags: modular arithmetic , number theory
A positive integer $a > 1$ is given (in decimal notation). We copy it twice and obtain a number $b = \overline{aa}$ which happens to be a multiple of $a^2$. Find all possible values of $b/a^2$.

2024 Malaysia IMONST 2, 3
Tags: number theory
Ivan claims that for all positive integers $n$, $$\left\lfloor\sqrt[2]{\frac{n}{1^3}}\right\rfloor + \left\lfloor\sqrt[2]{\frac{n}{2^3}}\right\rfloor + \left\lfloor\sqrt[2]{\frac{n}{3^3}}\right\rfloor + \cdots = \left\lfloor\sqrt[3]{\frac{n}{1^2}}\right\rfloor + \left\lfloor\sqrt[3]{\frac{n}{2^2}}\right\rfloor + \left\lfloor\sqrt[3]{\frac{n}{3^2}}\right\rfloor + \cdots$$ Why is he correct? (Note: $\lfloor x \rfloor$ denotes the floor function.)

2018 BMT Spring, 8
Tags: number theory
How many $1 < n \le 2018$ such that the set $$\{0, 1, 1+2,...,1+2+3+...+i,..., 1+2+...+n-1\}$$ is a permutation of $\{0, 1, 2, 3, 4,...,; n -1\}$ when reduced modulo $n$?

2015 Thailand TSTST, 1
Tags: number theory , square , perfect square , sum of squares
Prove that there exist infinitely many integers $n$ such that $n, n + 1, n + 2$ are each the sum of two squares of integers.

2013 District Olympiad, 4
Tags: inequalities , algebra , number theory
For a given a positive integer $n$, find all integers $x_1, x_2,... , x_n$ subject to $0 < x_1 < x_2 < ...< x_n < x_{n+1}$ and $$x_nx_{n+1} \le 2(x_1 + x_2 + ... + x_n).$$

2013 Turkey Junior National Olympiad, 2
Tags: number theory
Find all prime numbers $p, q, r$ satisfying the equation \[ p^4+2p+q^4+q^2=r^2+4q^3+1 \]

2006 Paraguay Mathematical Olympiad, 1
Tags: number theory
What are the last two digits of the decimal representation of $21^{2006}$?

2015 Iran MO (3rd round), 1
Tags: number theory , algebra
Prove that there are infinitely natural numbers $n$ such that $n$ can't be written as a sum of two positive integers with prime factors less than $1394$.

2019 Bangladesh Mathematical Olympiad, 10
Tags: number theory
Given $2020*2020$ chessboard, what is the maximum number of warriors you can put on its cells such that no two warriors attack each other. Warrior is a special chess piece which can move either $3$ steps forward and one step sideward and $2$ step forward and $2$ step sideward in any direction.

2024 Brazil Team Selection Test, 1
Tags: number theory , phi function , coprime
Given an integer $n > 1$, let $1 = a_1 < a_2 < \cdots < a_t = n - 1$ be all positive integers less than $n$ that are coprime to $n$. Find all $n$ such that there is no $i \in \{1, 2, \ldots , t - 1\}$ satisfying $3 | a_i + a_{i+1}$.

1994 Dutch Mathematical Olympiad, 3
Tags: modular arithmetic , number theory , geometry , 3d geometry
$ (a)$ Prove that every multiple of $ 6$ can be written as a sum of four cubes. $ (b)$ Prove that every integer can be written as a sum of five cubes.

2019 India PRMO, 30
Tags: floor function , number theory
For any real number $x$, let $\lfloor x \rfloor$ denote the integer part of $x$; $\{ x \}$ be the fractional part of $x$ ($\{x\}$ $=$ $x-$ $\lfloor x \rfloor$). Let $A$ denote the set of all real numbers $x$ satisfying $$\{x\} =\frac{x+\lfloor x \rfloor +\lfloor x + (1/2) \rfloor }{20}$$ If $S$ is the sume of all numbers in $A$, find $\lfloor S \rfloor$

2009 Moldova Team Selection Test, 4
Tags: number theory
[color=darkred]Let $ p$ be a prime divisor of $ n\ge 2$. Prove that there exists a set of natural numbers $ A \equal{} \{a_1,a_2,...,a_n\}$ such that product of any two numbers from $ A$ is divisible by the sum of any $ p$ numbers from $ A$.[/color]

2005 Baltic Way, 17
Tags: algebraic identity , number theory
A sequence $(x_n)_{n\ge 0}$ is defined as follows: $x_0=a,x_1=2$ and $x_n=2x_{n-1}x_{n-2}-x_{n-1}-x_{n-2}+1$ for all $n>1$. Find all integers $a$ such that $2x_{3n}-1$ is a perfect square for all $n\ge 1$.

2010 Kazakhstan National Olympiad, 1
Tags: modular arithmetic , number theory , geometry
It is given that for some $n \in \mathbb{N}$ there exists a natural number $a$, such that $a^{n-1} \equiv 1 \pmod{n}$ and that for any prime divisor $p$ of $n-1$ we have $a^{\frac{n-1}{p}} \not \equiv 1 \pmod{n}$. Prove that $n$ is a prime.

2023 Auckland Mathematical Olympiad, 6
Tags: combinatorics , number theory
Suppose there is an infi nite sequence of lights numbered $1, 2, 3,...,$ and you know the following two rules about how the lights work: $\bullet$ If the light numbered $k$ is on, the lights numbered $2k$ and $2k + 1$ are also guaranteed to be on. $\bullet$ If the light numbered $k$ is off, then the lights numbered $4k + 1$ and $4k + 3$ are also guaranteed to be off. Suppose you notice that light number $2023$ is on. Identify all the lights that are guaranteed to be on?

2018 Saudi Arabia IMO TST, 2
Tags: number theory
A non-empty subset of $\{1,2, ..., n\}$ is called [i]arabic [/i] if arithmetic mean of its elements is an integer. Show that the number of arabic subsets of $\{1,2, ..., n\}$ has the same parity as $n$.

2012 Princeton University Math Competition, A6
Tags: number theory
Let $p_1 = 2012$ and $p_n = 2012^{p_{n-1}}$ for $n > 1$. Find the largest integer $k$ such that $p_{2012}- p_{2011}$ is divisible by $2011^k$.

1979 Chisinau City MO, 174
Tags: number theory , odd , divide , divisible
Prove that for any odd number $a$ there exists an integer $b$ such that $2^b-1$ is divisible by $a$.

2000 Italy TST, 1
Tags: number theory
Determine all triples $(x,y,z)$ of positive integers such that \[\frac{13}{x^2}+\frac{1996}{y^2}=\frac{z}{1997} \]

2010 Contests, 2
Tags: induction , number theory
Every non-negative integer is coloured white or red, so that: • there are at least a white number and a red number; • the sum of a white number and a red number is white; • the product of a white number and a red number is red. Prove that the product of two red numbers is always a red number, and the sum of two red numbers is always a red number.

2014 India IMO Training Camp, 2
Tags: number theory
Find all positive integers $x$ and $y$ such that $x^{x+y}=y^{3x}$.