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

2003 Tournament Of Towns, 4
Tags: number theory
Each term of a sequence of positive integers is obtained from the previous term by adding to it its largest digit. What is the maximal number of successive odd terms in such a sequence?

2014 Balkan MO Shortlist, N5
Tags: number theory
$\boxed{N5}$Let $a,b,c,p,q,r$ be positive integers such that $a^p+b^q+c^r=a^q+b^r+c^p=a^r+b^p+c^q.$ Prove that $a=b=c$ or $p=q=r.$

2018 Malaysia National Olympiad, B1
Tags: proof , number theory
Given two triangles with the same perimeter. Both triangles have integer side lengths. The first triangle is an equilateral triangle. The second triangle has a side with length 1 and a side with length $d$. Prove that when $d$ is divided by 3, the remainder is 1.

2013 Baltic Way, 16
Tags: inequalities , number theory
We call a positive integer $n$ [i]delightful[/i] if there exists an integer $k$, $1 < k < n$, such that \[1+2+\cdots+(k-1)=(k+1)+(k+2)+\cdots+n\] Does there exist a delightful number $N$ satisfying the inequalities \[2013^{2013}<\dfrac{N}{2013^{2013}}<2013^{2013}+4 ?\]

LMT Accuracy Rounds, 2023 S10
Tags: number theory
Positive integers $a$, $b$, and $c$ satisfy $a^2 +b^2 = c^3 -1$ where $c \le 40$. Find the sum of all distinct possible values of $c$.

2023 Olimphíada, 4
Tags: floor function , prime number , golden ratio , residue , number theory
We say that a prime $p$ is $n$-$\textit{rephinado}$ if $n | p - 1$ and all $1, 2, \ldots , \lfloor \sqrt[\delta]{p}\rfloor$ are $n$-th residuals modulo $p$, where $\delta = \varphi+1$. Are there infinitely many $n$ for which there are infinitely many $n$-$\textit{rephinado}$ primes? Notes: $\varphi =\frac{1+\sqrt{5}}{2}$. We say that an integer $a$ is a $n$-th residue modulo $p$ if there is an integer $x$ such that $$x^n \equiv a \text{ (mod } p\text{)}.$$

2003 AIME Problems, 4
Tags: 3d geometry , tetrahedron , ratio , vector , number theory , relatively prime , geometry
In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

2014 NIMO Problems, 6
Tags: binomial coefficient , probability , ratio , number theory , relatively prime
10 students are arranged in a row. Every minute, a new student is inserted in the row (which can occur in the front and in the back as well, hence $11$ possible places) with a uniform $\tfrac{1}{11}$ probability of each location. Then, either the frontmost or the backmost student is removed from the row (each with a $\tfrac{1}{2}$ probability). Suppose you are the eighth in the line from the front. The probability that you exit the row from the front rather than the back is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $100m+n$. [i]Proposed by Lewis Chen[/i]

2023 Turkey Junior National Olympiad, 3
Tags: number theory , divisibility , relatively prime
Let $m,n$ be relatively prime positive integers. Prove that the numbers $$\frac{n^4+m}{m^2+n^2} \qquad \frac{n^4-m}{m^2-n^2}$$ cannot be integer at the same time.

2001 JBMO ShortLists, 3
Tags: modular arithmetic , number theory
Find all the three-digit numbers $\overline{abc}$ such that the $6003$-digit number $\overline{abcabc\ldots abc}$ is divisible by $91$.

2010 Polish MO Finals, 2
Tags: function , algebra , polynomial , number theory
Positive rational number $a$ and $b$ satisfy the equality \[a^3 + 4a^2b = 4a^2 + b^4.\] Prove that the number $\sqrt{a}-1$ is a square of a rational number.

2016 Kosovo National Mathematical Olympiad, 1
Tags: number theory
Find all couples $(m,n)$ of positive integers such that satisfied $m^2+1=n^2+2016$ .

2008 China Team Selection Test, 2
Tags: induction , number theory , algebra , sequence , recurrence relation
The sequence $ \{x_{n}\}$ is defined by $ x_{1} \equal{} 2,x_{2} \equal{} 12$, and $ x_{n \plus{} 2} \equal{} 6x_{n \plus{} 1} \minus{} x_{n}$, $ (n \equal{} 1,2,\ldots)$. Let $ p$ be an odd prime number, let $ q$ be a prime divisor of $ x_{p}$. Prove that if $ q\neq2,3,$ then $ q\geq 2p \minus{} 1$.

2022 Regional Olympiad of Mexico West, 5
Tags: floor function , number theory , divide
Determine all positive integers $n$ such that $\lfloor \sqrt{n} \rfloor - 1$ divides $n + 1$ and $\lfloor \sqrt{n} \rfloor +2$ divides $ n + 4$.

1999 AIME Problems, 14
Tags: trigonometry , geometry , number theory , relatively prime , trig identities
Point $P$ is located inside traingle $ABC$ so that angles $PAB, PBC,$ and $PCA$ are all congruent. The sides of the triangle have lengths $AB=13, BC=14,$ and $CA=15,$ and the tangent of angle $PAB$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

1998 India Regional Mathematical Olympiad, 5
Tags: number theory , least common multiple
Find the minimum possible least common multiple of twenty natural numbers whose sum is $801$.

2018 Chile National Olympiad, 4
Tags: floor function , algebra , number theory , function
Find all postitive integers n such that $$\left\lfloor \frac{n}{2} \right\rfloor \cdot \left\lfloor \frac{n}{3} \right\rfloor \cdot \left\lfloor \frac{n}{4} \right\rfloor=n^2$$ where $\lfloor x \rfloor$ represents the largest integer less than the real number $x$.

2009 Bosnia Herzegovina Team Selection Test, 2
Tags: number theory
Find all pairs $\left(a,b\right)$ of posive integers such that $\frac{a^{2}\left(b-a\right)}{b+a}$ is square of prime.

2020 Regional Olympiad of Mexico Center Zone, 5
Tags: number theory
Find all positive integers $m,n$ such that $m^2+5n$ and $n^2+5m$ are perfect squares.

2022 Silk Road, 2
Tags: number theory
Distinct positive integers $A$ and $B$ are given$.$ Prove that there exist infinitely many positive integers that can be represented both as $x_{1}^2+Ay_{1}^2$ for some positive coprime integers $x_{1}$ and $y_{1},$ and as $x_{2}^2+By_{2}^2$ for some positive coprime integers $x_{2}$ and $y_{2}.$ [i](Golovanov A.S.)[/i]

2023 Romania EGMO TST, P2
Tags: number theory , prime number , divisibility , power of 2 , prime divisor
Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.

2013 Costa Rica - Final Round, 6
Tags: number theory , digit , divisible
Let $a$ and $ b$ be positive integers (of one or more digits) such that $ b$ is divisible by $a$, and if we write $a$ and $ b$, one after the other in this order, we get the number $(a + b)^2$. Prove that $\frac{b}{a}= 6$.

2007 Cono Sur Olympiad, 3
Tags: number theory
Show that for each positive integer $n$, there is a positive integer $k$ such that the decimal representation of each of the numbers $k, 2k,\ldots, nk$ contains all of the digits $0, 1, 2,\ldots, 9$.

2012 Flanders Math Olympiad, 2
Tags: perfect square , number theory
Let $n$ be a natural number. Call $a$ the smallest natural number you need to subtract from $n$ to get a perfect square. Call $b$ the smallest natural number that you must add to $n$ to get a perfect square. Prove that $n - ab$ is a perfect square.

2019 Saudi Arabia JBMO TST, 3
Tags: number theory
Are there positive integers $a, b, c$, such that the numbers $a^2bc+2, b^2ca+2, c^2ab+2$ be perfect squares?