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

1985 IMO Longlists, 54
Tags: number theory , greatest common divisor , induction , algorithm , modular arithmetic , algebra , binomial theorem
Set $S_n = \sum_{p=1}^n (p^5+p^7)$. Determine the greatest common divisor of $S_n$ and $S_{3n}.$

2015 BMT Spring, 3
Tags: algebra , system of equations , number theory
Find all integer solutions to \begin{align*} x^2+2y^2+3z^2&=36,\\ 3x^2+2y^2+z^2&=84,\\ xy+xz+yz&=-7. \end{align*}

2014 Contests, 1
Tags: number theory
Tarik and Sultan are playing the following game. Tarik thinks of a number that is greater than $100$. Then Sultan is telling a number greater than $1$. If Tarik’s number is divisible by Sultan’s number, Sultan wins, otherwise Tarik subtracts Sultan’s number from his number and Sultan tells his next number. Sultan is forbidden to repeat his numbers. If Tarik’s number becomes negative, Sultan loses. Does Sultan have a winning strategy?

2016 Taiwan TST Round 1, 3
Tags: binomial coefficient , function , algebra , number theory , surjective function
Let $\mathbb{Z}^+$ denote the set of all positive integers. Find all surjective functions $f:\mathbb{Z}^+ \times \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ that satisfy all of the following conditions: for all $a,b,c \in \mathbb{Z}^+$, (i)$f(a,b) \leq a+b$; (ii)$f(a,f(b,c))=f(f(a,b),c)$ (iii)Both $\binom{f(a,b)}{a}$ and $\binom{f(a,b)}{b}$ are odd numbers.(where $\binom{n}{k}$ denotes the binomial coefficients)

2006 China Team Selection Test, 3
Tags: number theory
For a positive integer $M$, if there exist integers $a$, $b$, $c$ and $d$ so that: \[ M \leq a < b \leq c < d \leq M+49, \qquad ad=bc \] then we call $M$ a GOOD number, if not then $M$ is BAD. Please find the greatest GOOD number and the smallest BAD number.

I Soros Olympiad 1994-95 (Rus + Ukr), 9.10
Tags: number theory , multiple , digit
For which natural $n$ there exists a natural number multiple of $n$, whose decimal notation consists only of the digits $8$ and $9$ (possibly only from numbers $8$ or only from numbers $9$)?

2019 Belarus Team Selection Test, 7.2
Tags: number theory
Define the sequence $a_0,a_1,a_2,\hdots$ by $a_n=2^n+2^{\lfloor n/2\rfloor}$. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way.

2004 All-Russian Olympiad, 1
Tags: 3d geometry , number theory , geometry
Are there such pairwise distinct natural numbers $ m, n, p, q$ satisfying $ m \plus{} n \equal{} p \plus{} q$ and $ \sqrt{m} \plus{} \sqrt[3]{n} \equal{} \sqrt{p} \plus{} \sqrt[3]{q} > 2004$ ?

2013 Israel National Olympiad, 2
Tags: combinatorics , number theory , maximum , set
Let $A=\{n\in\mathbb{Z}\mid 0<n<2013\}$. A subset $B\subseteq A$ is called [b]reduced[/b] if for any two numbers $x,y\in B$, we must have $x\cdot y \notin B$. For example, any subset containing the numbers $3,5,15$ cannot be reduced, and same for a subset containing $4,16$. [list=a] [*] Find the maximal size of a reduced subset of $A$. [*] How many reduced subsets are there with that maximal size? [/list]

2017 Rioplatense Mathematical Olympiad, Level 3, 1
Tags: number theory , diophantine , diophantine equation
Let $a$ be a fixed positive integer. Find the largest integer $b$ such that $(x+a)(x+b)=x+a+b$, for some integer $x$.

2019 Cono Sur Olympiad, 2
Tags: number theory , digit
We say that a positive integer $M$ with $2n$ digits is [i]hypersquared[/i] if the following three conditions are met: [list] [*]$M$ is a perfect square. [*]The number formed by the first $n$ digits of $M$ is a perfect square. [*]The number formed by the last $n$ digits of $M$ is a perfect square and has exactly $n$ digits (its first digit is not zero). [/list] Find a hypersquared number with $2000$ digits.

2010 Contests, 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 $n(S(n)-1)=2010.$

2014 Middle European Mathematical Olympiad, 8
Tags: modular arithmetic , quadratic , inequalities , geometry , 3d geometry , number theory
Determine all quadruples $(x,y,z,t)$ of positive integers such that \[ 20^x + 14^{2y} = (x + 2y + z)^{zt}.\]

2015 Nordic, 2
Tags: prime number , number theory
Find the primes ${p, q, r}$, given that one of the numbers ${pqr}$ and ${p + q + r}$ is ${101}$ times the other.

2021 Kyiv City MO Round 1, 8.2
Tags: number theory , digit
Oleksiy writes all the digits from $0$ to $9$ on the board, after which Vlada erases one of them. Then he writes $10$ nine-digit numbers on the board, each consisting of all the nine digits written on the board (they don't have to be distinct). It turned out that the sum of these $10$ numbers is a ten-digit number, all of whose digits are distinct. Which digit could have been erased by Vlada? [i]Proposed by Oleksii Masalitin[/i]

2022 Grand Duchy of Lithuania, 4
Tags: number theory , perfect square
Find all triples of natural numbers $(a, b, c)$ for which the number $$2^a + 2^b + 2^c + 3$$ is the square of an integer.

2024 Romanian Master of Mathematics, 6
Tags: analytic number theory , complex roots , algebra , number theory , polynomial
A polynomial $P$ with integer coefficients is [i]square-free[/i] if it is not expressible in the form $P = Q^2R$, where $Q$ and $R$ are polynomials with integer coefficients and $Q$ is not constant. For a positive integer $n$, let $P_n$ be the set of polynomials of the form $$1 + a_1x + a_2x^2 + \cdots + a_nx^n$$ with $a_1,a_2,\ldots, a_n \in \{0,1\}$. Prove that there exists an integer $N$ such that for all integers $n \geq N$, more than $99\%$ of the polynomials in $P_n$ are square-free. [i]Navid Safaei, Iran[/i]

2010 Putnam, A4
Tags: modular arithmetic , number theory , greatest common divisor , putnam number theory , logarithm
Prove that for each positive integer $n,$ the number $10^{10^{10^n}}+10^{10^n}+10^n-1$ is not prime.

1990 IMO Longlists, 98
Tags: number theory , decimal representation , prime number , digit
Find all natural numbers $ n$ for which every natural number whose decimal representation has $ n \minus{} 1$ digits $ 1$ and one digit $ 7$ is prime.

2022 Durer Math Competition Finals, 3
Tags: number theory , algebra
Three palaces, each rotating on a duck leg, make a full round in $30$, $50$, and $70$ days, respectively. Today, at noon, all three palaces face northwards. In how many days will they all face southwards?

2014 Contests, 1
Tags: modular arithmetic , number theory
Let $p$ be a prime such that $p\mid 2a^2-1$ for some integer $a$. Show that there exist integers $b,c$ such that $p=2b^2-c^2$.

2025 China National Olympiad, 3
Tags: number theory
Let \(a_1, a_2, \ldots, a_n\) be integers such that \(a_1 > a_2 > \cdots > a_n > 1\). Let \(M = \operatorname{lcm} \left( a_1, a_2, \ldots, a_n \right)\). For any finite nonempty set $X$ of positive integers, define \[ f(X) = \min_{1 \leqslant i \leqslant n} \sum_{x \in X} \left\{ \frac{x}{a_i} \right\}. \] Such a set $X$ is called [i]minimal[/i] if for every proper subset $Y$ of it, $f(Y) < f(X)$ always holds. Suppose $X$ is minimal and $f(X) \geqslant \frac{2}{a_n}$. Prove that \[ |X| \leqslant f(X) \cdot M. \]

1986 IMO Longlists, 4
Tags: number theory
Find the last eight digits of the binary development of $27^{1986}.$

1997 Romania Team Selection Test, 2
Tags: modular arithmetic , divisibility , multiplicative nt , multiplicative order , number theory
Suppose that $A$ be the set of all positive integer that can write in form $a^2+2b^2$ (where $a,b\in\mathbb {Z}$ and $b$ is not equal to $0$). Show that if $p$ be a prime number and $p^2\in A$ then $p\in A$. [i]Marcel Tena[/i]

2008 District Olympiad, 2
Tags: algebra , polynomial , number theory
Determine $ x$ irrational so that $ x^2\plus{}2x$ and $ x^3\minus{}6x$ are both rational.