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 Middle European Mathematical Olympiad, 7
Tags: number theory , quadratic , ceiling function
The numbers from 1 to $ 2013^2 $ are written row by row into a table consisting of $ 2013 \times 2013 $ cells. Afterwards, all columns and all rows containing at least one of the perfect squares $ 1, 4, 9, \cdots, 2013^2 $ are simultaneously deleted. How many cells remain?

2019 IberoAmerican, 1
Tags: number theory , digit
For each positive integer $n$, let $s(n)$ be the sum of the squares of the digits of $n$. For example, $s(15)=1^2+5^2=26$. Determine all integers $n\geq 1$ such that $s(n)=n$.

2020 LIMIT Category 1, 8
Tags: number theory , limit , algebra
Find the greatest integer which doesn't exceed $\frac{3^{100}+2^{100}}{3^{96}+2^{96}}$ (A)$81$ (B)$80$ (C)$79$ (D)$82$

2016 Romania Team Selection Test, 3
Tags: number theory
Given a prime $p$, prove that the sum $\sum_{k=1}^{\lfloor \frac{q}{p} \rfloor}{k^{p-1}}$ is not divisible by $q$ for all but finitely many primes $q$.

2016 Taiwan TST Round 1, 1
Tags: floor function , number theory , sequence
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.

2015 Danube Mathematical Competition, 2
Tags: combinatorics , number theory , absolute value
Consider the set $A=\{1,2,...,120\}$ and $M$ a subset of $A$ such that $|M|=30$.Prove that there are $5$ different subsets of $M$,each of them having two elements,such that the absolute value of the difference of the elements of each subset is the same.

2013 Canada National Olympiad, 2
Tags: number theory , modular arithmetic
The sequence $a_1, a_2, \dots, a_n$ consists of the numbers $1, 2, \dots, n$ in some order. For which positive integers $n$ is it possible that the $n+1$ numbers $0, a_1, a_1+a_2, a_1+a_2+a_3,\dots, a_1 + a_2 +\cdots + a_n$ all have di fferent remainders when divided by $n + 1$?

2022 Romania Team Selection Test, 4
Tags: shortlist , number theory
Can every positive rational number $q$ be written as $$\frac{a^{2021} + b^{2023}}{c^{2022} + d^{2024}},$$ where $a, b, c, d$ are all positive integers? [i]Proposed by Dominic Yeo, UK[/i]

2013 JBMO Shortlist, 5
Tags: number theory , positive integer , diophantine equation
Solve in positive integers: $\frac{1}{x^2}+\frac{y}{xz}+\frac{1}{z^2}=\frac{1}{2013}$ .

2013 NIMO Problems, 2
Tags: number theory , relatively prime
If $\frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6} = \frac{m}{n}$ for relatively prime integers $m$ and $n$, compute $100m+n$. [i]Proposed by Evan Chen[/i]

1997 Baltic Way, 7
Tags: algebra , number theory , polynomial
Let $P$ and $Q$ be polynomials with integer coefficients. Suppose that the integers $a$ and $a+1997$ are roots of $P$, and that $Q(1998)=2000$. Prove that the equation $Q(P(x))=1$ has no integer solutions.

2024 Malaysia IMONST 2, 3
Tags: number theory
Janson wants to find a sequence of positive integers $a_{1}, a_{2}, . . . , a_{2024}$ such that each term is at least $10$, and $a_{i}$ has exactly $a_{i+1}$ divisors for all $1 \leq i \leq 2023$. Can you help him find one such sequence, or is this task impossible?

2016 PUMaC Team, 3
Tags: number theory
Compute the sum of all positive integers $n < 200$ such that $gcd(n, k) \ne 1$ for every $k \in\{2 \cdot 11 \cdot 19, 3 \cdot 13 \cdot 17, 5 \cdot 11 \cdot 13, 7 \cdot 17 \cdot 19\}$.

1978 Chisinau City MO, 155
Tags: number theory , perfect square
Find the base of the number system less than $100$, in which $2101$ is a perfect square.

1989 Turkey Team Selection Test, 2
Tags: number theory
A positive integer is called a "double number" if its decimal representation consists of a block of digits, not commencing with $0$, followed immediately by an identical block. So, for instance, $360360$ is a double number, but $36036$ is not. Show that there are infinitely many double numbers which are perfect squares.

2016 Hanoi Open Mathematics Competitions, 5
Tags: diophantine , diophantine equation , number theory
There are positive integers $x, y$ such that $3x^2 + x = 4y^2 + y$, and $(x - y)$ is equal to (A): $2013$ (B): $2014$ (C): $2015$ (D): $2016$ (E): None of the above.

2021 AMC 12/AHSME Fall, 8
Tags: number theory , least common multiple
Let $M$ be the least common multiple of all the integers $10$ through $30,$ inclusive. Let $N$ be the least common multiple of $M,$ $32,$ $33,$ $34,$ $35,$ $36,$ $37,$ $38,$ $39,$ and $40.$ What is the value of $\frac{N}{M}?$ $(\textbf{A})\: 1\qquad(\textbf{B}) \: 2\qquad(\textbf{C}) \: 37\qquad(\textbf{D}) \: 74\qquad(\textbf{E}) \: 2886$

2024 Romanian Master of Mathematics, 4
Tags: remainder , size , number theory
Fix integers $a$ and $b$ greater than $1$. For any positive integer $n$, let $r_n$ be the (non-negative) remainder that $b^n$ leaves upon division by $a^n$. Assume there exists a positive integer $N$ such that $r_n < \frac{2^n}{n}$ for all integers $n\geq N$. Prove that $a$ divides $b$. [i]Pouria Mahmoudkhan Shirazi, Iran[/i]

1995 IMO Shortlist, 1
Tags: number theory , algebra , modular arithmetic
Let $ k$ be a positive integer. Show that there are infinitely many perfect squares of the form $ n \cdot 2^k \minus{} 7$ where $ n$ is a positive integer.

2013 Hanoi Open Mathematics Competitions, 5
Tags: diophantine , diophantine equation , number theory
The number of integer solutions $x$ of the equation below $(12x -1)(6x - 1)(4x -1)(3x - 1) = 330$ is (A): $0$, (B): $1$, (C): $2$, (D): $3$, (E): None of the above.

2019 Korea Junior Math Olympiad., 3
Tags: prime number , number theory
Find all pairs of prime numbers $p,\,q(p\le q)$ satisfying the following condition: There exists a natural number $n$ such that $2^{n}+3^{n}+\cdots+(2pq-1)^{n}$ is a multiple of $2pq$.

2025 Harvard-MIT Mathematics Tournament, 10
Tags: algebra , number theory
Let $a,b,$ and $c$ be pairwise distinct complex numbers such that $$a^2=b+6, \quad b^2=c+6, \quad \text{and} \quad c^2=a+6.$$ Compute the two possible values of $a+b+c.$

2008 Kyiv Mathematical Festival, 3
Tags: number theory
Prove that among any 7 integers there exist three numbers $ a,b,c$ such that $ a^2\plus{}b^2\plus{}c^2\minus{}ab\minus{}bc\minus{}ac$ is divisible by 7.

2016 Croatia Team Selection Test, Problem 4
Tags: decimal representation , prime , prime number , primitive root , number theory
Let $p > 10^9$ be a prime number such that $4p + 1$ is also prime. Prove that the decimal expansion of $\frac{1}{4p+1}$ contains all the digits $0,1, \ldots, 9$.

2013 Romania Team Selection Test, 1
Tags: algebra , symmetry , geometry , 3d geometry , number theory , polynomial , modular arithmetic
Given an integer $n\geq 2,$ let $a_{n},b_{n},c_{n}$ be integer numbers such that \[ \left( \sqrt[3]{2}-1\right) ^{n}=a_{n}+b_{n}\sqrt[3]{2}+c_{n}\sqrt[3]{4}. \] Prove that $c_{n}\equiv 1\pmod{3} $ if and only if $n\equiv 2\pmod{3}.$