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: 1766

2015 Brazil National Olympiad, 4
Tags: number theory proposed , number theory
Let $n$ be a integer and let $n=d_1>d_2>\cdots>d_k=1$ its positive divisors. a) Prove that $$d_1-d_2+d_3-\cdots+(-1)^{k-1}d_k=n-1$$ iff $n$ is prime or $n=4$. b) Determine the three positive integers such that $$d_1-d_2+d_3-...+(-1)^{k-1}d_k=n-4.$$

2007 IMAR Test, 3
Tags: absolute value , number theory proposed , number theory
Prove that $ N\geq 2n \minus{} 2$ integers, of absolute value not higher than $ n > 2$, and of absolute value of their sum $ S$ less than $ n \minus{} 1,$ there exist some of sum $ 0.$ Show that for $ |S| \equal{} n \minus{} 1$ this is not anymore true, and neither for $ N \equal{} 2n \minus{} 3$ (when even for $ |S| \equal{} 1$ this is not anymore true).

2024 Baltic Way, 19
Tags: number theory proposed , Divisors , construction , Fractions
Does there exist a positive integer $N$ which is divisible by at least $2024$ distinct primes and whose positive divisors $1 = d_1 < d_2 < \ldots < d_k = N$ are such that the number \[ \frac{d_2}{d_1}+\frac{d_3}{d_2}+\ldots+\frac{d_k}{d_{k-1}} \] is an integer?

2006 Balkan MO, 3
Tags: number theory proposed , number theory
Find all triplets of positive rational numbers $(m,n,p)$ such that the numbers $m+\frac 1{np}$, $n+\frac 1{pm}$, $p+\frac 1{mn}$ are integers. [i]Valentin Vornicu, Romania[/i]

2012 Benelux, 1
Tags: floor function , arithmetic sequence , number theory proposed , number theory
A sequence $a_1,a_2,\ldots ,a_n,\ldots$ of natural numbers is defined by the rule \[a_{n+1}=a_n+b_n\ (n=1,2,\ldots)\] where $b_n$ is the last digit of $a_n$. Prove that such a sequence contains infinitely many powers of $2$ if and only if $a_1$ is not divisible by $5$.

2007 Junior Balkan Team Selection Tests - Romania, 1
Tags: function , geometry , 3D geometry , number theory proposed , number theory
Find the positive integers $n$ with $n \geq 4$ such that $[\sqrt{n}]+1$ divides $n-1$ and $[\sqrt{n}]-1$ divides $n+1$. [hide="Remark"]This problem can be solved in a similar way with the one given at [url=http://www.mathlinks.ro/Forum/resources.php?c=1&cid=97&year=2006]Cono Sur Olympiad 2006[/url], problem 5.[/hide]

2002 France Team Selection Test, 3
Tags: number theory proposed , number theory
Let $p\ge 3$ be a prime number. Show that there exist $p$ positive integers $a_1,a_2,\ldots ,a_p$ not exceeding $2p^2$ such that the $\frac{p(p-1)}{2}$ sums $a_i+a_j\ (i<j)$ are all distinct.

2014 District Olympiad, 2
Tags: number theory proposed , number theory
For each positive integer $n$ we denote by $p(n)$ the greatest square less than or equal to $n$. [list=a] [*]Find all pairs of positive integers $( m,n)$, with $m\leq n$, for which \[ p( 2m+1) \cdot p( 2n+1) =400 \] [*]Determine the set $\mathcal{P}=\{ n\in\mathbb{N}^{\ast}\vert n\leq100\text{ and }\dfrac{p(n+1)}{p(n)}\notin\mathbb{N}^{\ast}\}$[/list]

2004 Iran MO (3rd Round), 19
Tags: Euler , number theory proposed , number theory
Find all integer solutions of $ p^3\equal{}p^2\plus{}q^2\plus{}r^2$ where $ p,q,r$ are primes.

2011 ISI B.Math Entrance Exam, 7
Tags: geometry , algorithm , number theory proposed , number theory
If $a_1, a_2, \cdots, a_7$ are not necessarily distinct real numbers such that $1 < a_i < 13$ for all $i$, then show that we can choose three of them such that they are the lengths of the sides of a triangle.

2005 Germany Team Selection Test, 3
Tags: induction , number theory proposed , number theory
Let $b$ and $c$ be any two positive integers. Define an integer sequence $a_n$, for $n\geq 1$, by $a_1=1$, $a_2=1$, $a_3=b$ and $a_{n+3}=ba_{n+2}a_{n+1}+ca_n$. Find all positive integers $r$ for which there exists a positive integer $n$ such that the number $a_n$ is divisible by $r$.

1987 Romania Team Selection Test, 10
Tags: modular arithmetic , induction , function , algebra , polynomial , number theory proposed , number theory
Let $a,b,c$ be integer numbers such that $(a+b+c) \mid (a^{2}+b^{2}+c^{2})$. Show that there exist infinitely many positive integers $n$ such that $(a+b+c) \mid (a^{n}+b^{n}+c^{n})$. [i]Laurentiu Panaitopol[/i]

2006 Mexico National Olympiad, 1
Tags: number theory proposed , number theory
Let $ab$ be a two digit number. A positive integer $n$ is a [i]relative[/i] of $ab$ if: [list] [*] The units digit of $n$ is $b$. [*] The remaining digits of $n$ are nonzero and add up to $a$.[/list] Find all two digit numbers which divide all of their relatives.

2008 ITAMO, 1
Tags: number theory proposed , number theory
Find all triples $ (a,b,c)$ of positive integers such that $ a^2\plus{}2^{b\plus{}1}\equal{}3^c$.

2007 Polish MO Finals, 6
Tags: number theory proposed , number theory
6. Sequence $a_{0}, a_{1}, a_{2},...$ is determined by $a_{0}=-1$ and $a_{n}+\frac{a_{n-1}}{2}+\frac{a_{n-2}}{3}+...+\frac{a_{1}}{n}+\frac{a_{0}}{n+1}=0$ for $n\geq 1$ Prove that $a_{n}>0$ for $n\geq 1$

2005 Junior Balkan MO, 4
Tags: number theory proposed , number theory
Find all 3-digit positive integers $\overline{abc}$ such that \[ \overline{abc} = abc(a+b+c) , \] where $\overline{abc}$ is the decimal representation of the number.