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

2012 Kazakhstan National Olympiad, 3
Tags: number theory
Consider the equation $ax^{2}+by^{2}=1$, where $a,b$ are fixed rational numbers. Prove that either such an equation has no solutions in rational numbers, or it has infinitely many solutions.

2023 Thailand October Camp, 2
Tags: number theory
A number is called [i]Norwegian[/i] if it has three distinct positive divisors whose sum is equal to $2022$. Determine the smallest Norwegian number. (Note: The total number of positive divisors of a Norwegian number is allowed to be larger than $3$.)

2022 IMO Shortlist, N5
Tags: number theory
For each $1\leq i\leq 9$ and $T\in\mathbb N$, define $d_i(T)$ to be the total number of times the digit $i$ appears when all the multiples of $1829$ between $1$ and $T$ inclusive are written out in base $10$. Show that there are infinitely many $T\in\mathbb N$ such that there are precisely two distinct values among $d_1(T)$, $d_2(T)$, $\dots$, $d_9(T)$.

2011 Pre-Preparation Course Examination, 5
Tags: calculus , logarithm , function , inequalities , number theory , integration , prime number
suppose that $v(x)=\sum_{p\le x,p\in \mathbb P}log(p)$ (here $\mathbb P$ denotes the set of all positive prime numbers). prove that the two statements below are equivalent: [b]a)[/b] $v(x) \sim x$ when $x \longrightarrow \infty$ [b]b)[/b] $\pi (x) \sim \frac{x}{ln(x)}$ when $x \longrightarrow \infty$. (here $\pi (x)$ is number of the prime numbers less than or equal to $x$).

2011 NIMO Summer Contest, 11
Tags: number theory , least common multiple , greatest common divisor
How many ordered pairs of positive integers $(m, n)$ satisfy the system \begin{align*} \gcd (m^3, n^2) & = 2^2 \cdot 3^2, \\ \text{LCM} [m^2, n^3] & = 2^4 \cdot 3^4 \cdot 5^6, \end{align*} where $\gcd(a, b)$ and $\text{LCM}[a, b]$ denote the greatest common divisor and least common multiple of $a$ and $b$, respectively?

2004 Thailand Mathematical Olympiad, 17
Tags: number theory , remainder , sum of powers
Compute the remainder when $1^{2547} + 2^{2547} +...+ 2547^{2547}$ is divided by $25$.

2025 CMIMC Algebra/NT, 1
Tags: number theory , algebra
Four runners are preparing to begin a $1$-mile race from the same starting line. When the race starts, runners Alice, Bob, and Charlie all travel at constant speeds of $8$ mph, $4$ mph, and $2$ mph, respectively. The fourth runner, Dave, is initially half as slow as Charlie, but Dave has a superpower where he suddenly doubles his running speed every time a runner finishes the race. How many hours does it take for Dave to finish the race?

2024 Korea National Olympiad, 6
Tags: number theory
For a positive integer $n$, let $g(n) = \left[ \displaystyle \frac{2024}{n} \right]$. Find the value of $$\sum_{n = 1}^{2024}\left(1 - (-1)^{g(n)}\right)\phi(n).$$

2003 Regional Competition For Advanced Students, 2
Tags: number theory
Find all prime numbers $ p$ with $ 5^p\plus{}4p^4$ is the square of an integer.

2022 Bulgaria JBMO TST, 3
Tags: number theory , coprime , divisibility , relatively prime
The integers $a$, $b$, $c$ and $d$ are such that $a$ and $b$ are relatively prime, $d\leq 2022$ and $a+b+c+d = ac + bd = 0$. Determine the largest possible value of $d$,

2014 District Olympiad, 4
Tags: floor function , ceiling function , number theory
Determine all positive integers $a$ for which there exist exactly $2014$ positive integers $b$ such that $\displaystyle2\leq\frac{a}{b}\leq5$.

2020-21 KVS IOQM India, 26
Tags: number theory
Let $a,b,c$ be three distinct positive integers such that the sum of any two of them is a perfect square and having minimal sum $a + b + c$. Find this sum.

2019 Olympic Revenge, 5
Tags: function , modular arithmetic , number theory , prime number
Define $f: \mathbb{N} \rightarrow \mathbb{N}$ by $$f(n) = \sum \frac{(1+\sum_{i=1}^{n} t_i)!}{(1+t_1) \cdot \prod_{i=1}^{n} (t_i!) }$$ where the sum runs through all $n$-tuples such that $\sum_{j=1}^{n}j \cdot t_j=n$ and $t_j \ge 0$ for all $1 \le j \le n$. Given a prime $p$ greater than $3$, prove that $$\sum_{1 \le i < j <k \le p-1 } \frac{f(i)}{i \cdot j \cdot k} \equiv \sum_{1 \le i < j <k \le p-1 } \frac{2^i}{i \cdot j \cdot k} \pmod{p}.$$

2000 Saint Petersburg Mathematical Olympiad, 10.7
Tags: number theory , prime number , divisibility
We'll call a positive integer "almost prime", if it is not divisible by any prime from the interval $[3,19]$. We'll call a number "very non-prime", if it has at least 2 primes from interval $[3,19]$ dividing it. What is the greatest amount of almost prime numbers can be selected, such that the sum of any two of them is a very non-prime number? [I]Proposed by S. Berlov, S. Ivanov[/i]

2009 All-Russian Olympiad Regional Round, 10.2
Tags: number theory , consecutive , product
Prove that there is a natural number $n > 1$ such that the product of some $n$ consecutive natural numbers is equal to the product of some $n + 100$ consecutive natural numbers.

2017 Iran Team Selection Test, 4
Tags: number theory , prime number , modular arithmetic
We arranged all the prime numbers in the ascending order: $p_1=2<p_2<p_3<\cdots$. Also assume that $n_1<n_2<\cdots$ is a sequence of positive integers that for all $i=1,2,3,\cdots$ the equation $x^{n_i} \equiv 2 \pmod {p_i}$ has a solution for $x$. Is there always a number $x$ that satisfies all the equations? [i]Proposed by Mahyar Sefidgaran , Yahya Motevasel[/i]

1980 IMO Longlists, 9
Tags: number theory , modular arithmetic , representation , pascal triangle
Let $p$ be a prime number. Prove that there is no number divisible by $p$ in the $n-th$ row of Pascal's triangle if and only if $n$ can be represented in the form $n = p^sq - 1$, where $s$ and $q$ are integers with $s \geq 0, 0 < q < p$.

2022 Durer Math Competition Finals, 1
Tags: number theory , digit
How many $10$-digit sequences are there, made up of $1$ four, $2$ threes, $3$ twos, and $4$ ones, in which there is a two in between any two ones, a three in between any two twos, and a four in between any two threes?

2015 Tuymaada Olympiad, 2
Tags: number theory
We call number as funny if it divisible by sum its digits $+1$.(for example $ 1+2+1|12$ ,so $12$ is funny) What is maximum number of consecutive funny numbers ? [i] O. Podlipski [/i]

1998 Tournament Of Towns, 2
Tags: number theory , sum , product , digit , 4-digit
For every four-digit number, we take the product of its four digits. Then we add all of these products together . What is the result? ( G Galperin)

2020 Saint Petersburg Mathematical Olympiad, 6.
Tags: invariant , algebra , number theory , analytic geometry , combinatorics
The points $(1,1),(2,3),(4,5)$ and $(999,111)$ are marked in the coordinate system. We continue to mark points in the following way : [list] [*]If points $(a,b)$ are marked then $(b,a)$ and $(a-b,a+b)$ can be marked [*]If points $(a,b)$ and $(c,d)$ are marked then so can be $(ad+bc, 4ac-4bd)$. [/list] Can we, after some finite number of these steps, mark a point belonging to the line $y=2x$.

2003 All-Russian Olympiad Regional Round, 9.7
Tags: number theory , coprime , relatively prime
Prove that of any six four-digit numbers, mutual prime in total, you can always choose five numbers that are also relatively prime in total. [hide=original wording]Докажите, что из любых шести четырехзначных чисел, взаимно простых в совокупности, всегда можно выбратьпя ть чисел, также взаимно простых в совокупности.[/hide]

2019 Tournament Of Towns, 5
Tags: number theory , perfect square
Let us say that the pair $(m, n)$ of two positive different integers m and n is [i]nice [/i] if $mn$ and $(m + 1)(n + 1)$ are perfect squares. Prove that for each positive integer m there exists at least one $n > m$ such that the pair $(m, n)$ is nice. (Yury Markelov)

2003 Turkey Team Selection Test, 4
Tags: number theory , function , algebra
Find the least a. positive real number b. positive integer $t$ such that the equation $(x^2+y^2)^2 + 2tx(x^2 + y^2) = t^2y^2$ has a solution where $x,y$ are positive integers.

2023 VIASM Summer Challenge, Problem 2
Tags: number theory
Find all positive integers $n$ such that there exists positive integers $a, b, m$ satisfying$$\left( a+b\sqrt{n}\right)^{2023}=\sqrt{m}+\sqrt{m+2022}.$$