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

2006 Croatia Team Selection Test, 1
Tags: number theory
Find all natural numbers that can be expressed in a unique way as a sum of five or less perfect squares.

2001 Mediterranean Mathematics Olympiad, 3
Tags: number theory
Show that there exists a positive integer $N$ such that the decimal representation of $2000^N$ starts with the digits $200120012001.$

2018 Caucasus Mathematical Olympiad, 5
Tags: number theory , c.r.t
Baron Munсhausen discovered the following theorem: "For any positive integers $a$ and $b$ there exists a positive integer $n$ such that $an$ is a perfect cube, while $bn$ is a perfect fifth power". Determine if the statement of Baron’s theorem is correct.

2008 Regional Olympiad of Mexico Northeast, 3
Tags: number theory , digit
Consider the sequence $1,9,8,3,4,3,…$ in which $a_{n+4}$ is the units digit of $a_n+a_{n+3}$, for $n$ positive integer. Prove that $a^2_{1985}+a^2_{1986}+…+a^2_{2000}$ is a multiple of $2$.

1996 Balkan MO, 2
Tags: floor function , number theory
Let $ p$ be a prime number with $ p>5$. Consider the set $ X \equal{} \left\{p \minus{} n^2 \mid n\in \mathbb{N} ,\ n^2 < p\right\}$. Prove that the set $ X$ has two distinct elements $ x$ and $ y$ such that $ x\neq 1$ and $ x\mid y$. [i]Albania[/i]

1994 IMO Shortlist, 2
Tags: number theory
Find all ordered pairs $ (m,n)$ where $ m$ and $ n$ are positive integers such that $ \frac {n^3 \plus{} 1}{mn \minus{} 1}$ is an integer.

2011 Indonesia TST, 4
Tags: modular arithmetic , number theory
Prove that there exists infinitely many positive integers $n$ such that $n^2+1$ has a prime divisor greater than $2n+\sqrt{5n+2011}$.

2007 Croatia Team Selection Test, 2
Tags: floor function , irrational number , number theory
Prove that the sequence $a_{n}=\lfloor n\sqrt 2 \rfloor+\lfloor n\sqrt 3 \rfloor$ contains infintely many even and infinitely many odd numbers.

1964 All Russian Mathematical Olympiad, 051
Tags: number theory
Given natural $a,b,n$. It is known, that for every natural $k$ ($k\ne b$) the number $a-k^n$ is divisible by $b-k$. Prove that $$a=b^n$$

2023 Serbia Team Selection Test, P5
Tags: factorial , nt , weird , modular arithmetic , number theory
For positive integers $a$ and $b$, define \[a!_b=\prod_{1\le i\le a\atop i \equiv a \mod b} i\] Let $p$ be a prime and $n>3$ a positive integer. Show that there exist at least 2 different positive integers $t$ such that $1<t<p^n$ and $t!_p\equiv 1\pmod {p^n}$.

1993 Taiwan National Olympiad, 3
Tags: number theory
Find all $ x,y,z\in\mathbb{N}_{0}$ such that $ 7^{x} \plus{} 1 \equal{} 3^{y} \plus{} 5^{z}$. [i]Alternative formulation:[/i] Solve the equation $ 1\plus{}7^{x}\equal{}3^{y}\plus{}5^{z}$ in nonnegative integers $ x$, $ y$, $ z$.

PEN P Problems, 43
Tags: number theory , prime factorization , additive number theory
A positive integer $n$ is abundant if the sum of its proper divisors exceeds $n$. Show that every integer greater than $89 \times 315$ is the sum of two abundant numbers.

LMT Speed Rounds, 23
Tags: number theory
Let $S$ be the set of all positive integers $n$ such that the sum of all factors of $n$, including $1$ and $n$, is $120$. Compute the sum of all numbers in $S$. [i]Proposed by Evin Liang[/i]

2011 Tokio University Entry Examination, 2
Tags: algorithm , number theory , euclidean algorithm
Define real number $y$ as the fractional part of real number $x$ such that $0\leq y<1$ and $x-y$ is integer. Denote this by $<x>$. For real number $a$, define an infinite sequence $\{a_n\}\ (n=1,\ 2,\ 3,\ \cdots)$ inductively as follows. (i) $a_1=<a>$ (ii) If $a\n\neq 0$, then $a_{n+1}=\left<\frac{1}{a_n}\right>$, if $a_n=0$, then $a_{n+1}=0$. (1) For $a=\sqrt{2}$, find $a_n$. (2) For any natural number $n$, find real number $a\geq \frac 13$ such that $a_n=a$. (3) Let $a$ be a rational number. When we express $a=\frac{p}{q}$ with integer $p$, natural number $q$, prove that $a_n=0$ for any natural number $n\geq q$. [i]2011 Tokyo University entrance exam/Science, Problem 2[/i]

2007 Baltic Way, 19
Tags: number theory
Let $r$ and $k$ be positive integers such that all prime divisors of $r$ are greater than $50$. A positive integer, whose decimal representation (without leading zeroes) has at least $k$ digits, will be called [i]nice[/i] if every sequence of $k$ consecutive digits of this decimal representation forms a number (possibly with leading zeroes) which is a multiple of $r$. Prove that if there exist infinitely many nice numbers, then the number $10^k-1$ is nice as well.

1997 Denmark MO - Mohr Contest, 1
Tags: number theory , algebra , digit
Let $n =123456789101112 ... 998999$ be the natural number where is obtained by writing the natural numbers from $1$ to $999$ one after the other. What is the $1997$-th digit number in $n$?

1951 Kurschak Competition, 2
Tags: number theory , divide , divisible
For which $m > 1$ is $(m -1)!$ divisible by $m$?

2015 Greece National Olympiad, 1
Tags: diophantine equation , number theory , prime number
Find all triplets $(x,y,p)$ of positive integers such that $p$ be a prime number and $\frac{xy^3}{x+y}=p$

1978 IMO Shortlist, 8
Tags: pigeonhole principle , number theory , divisibility
Let $S$ be the set of all the odd positive integers that are not multiples of $5$ and that are less than $30m$, $m$ being an arbitrary positive integer. What is the smallest integer $k$ such that in any subset of $k$ integers from $S$ there must be two different integers, one of which divides the other?

2011 Postal Coaching, 1
Tags: modular arithmetic , number theory
Does the sequence \[11, 111, 1111, 11111, \ldots\] contain any fifth power of a positive integer? Justify your answer.

2011 Bulgaria National Olympiad, 1
Tags: number theory
Prove whether or not there exist natural numbers $n,k$ where $1\le k\le n-2$ such that \[\binom{n}{k}^2+\binom{n}{k+1}^2=\binom{n}{k+2}^4 \]

2011 Singapore Senior Math Olympiad, 2
Tags: greatest common divisor , least common multiple , number theory
Determine if there is a set $S$ of 2011 positive integers so that for every pair $m,n$ of distinct elements of $S$, $|m-n|=(m,n)$. Here $(m,n)$ denotes the greatest common divisor of $m$ and $n$.

2021 BMT, 21
Tags: number theory , algebra
There exist integers $a$ and $b$ such that $(1 +\sqrt2)^{12}= a + b\sqrt2$. Compute the remainder when $ab$ is divided by $13$.

1998 China Team Selection Test, 3
Tags: induction , number theory
For any $h = 2^{r}$ ($r$ is a non-negative integer), find all $k \in \mathbb{N}$ which satisfy the following condition: There exists an odd natural number $m > 1$ and $n \in \mathbb{N}$, such that $k \mid m^{h} - 1, m \mid n^{\frac{m^{h}-1}{k}} + 1$.

2002 Estonia National Olympiad, 4
Tags: integer , number theory , sum , max
Let $a_1, ... ,a_5$ be real numbers such that at least $N$ of the sums $a_i+a_j$ ($i < j$) are integers. Find the greatest value of $N$ for which it is possible that not all of the sums $a_i+a_j$ are integers.