Found problems: 15460
2015 Argentina National Olympiad Level 2, 1
Find all natural numbers $a$ such that for every positive integer $n$ the number $n(a+n)$ is not a perfect square.
2007 Hanoi Open Mathematics Competitions, 2
Which is largest positive integer n satisfying the following inequality: $n^{2007} > (2007)^n$.
2014 India National Olympiad, 3
Let $a,b$ be natural numbers with $ab>2$. Suppose that the sum of their greatest common divisor and least common multiple is divisble by $a+b$. Prove that the quotient is at most $\frac{a+b}{4}$. When is this quotient exactly equal to $\frac{a+b}{4}$
2009 Vietnam Team Selection Test, 3
Let a, b be positive integers. a, b and a.b are not perfect squares.
Prove that at most one of following equations
$ ax^2 \minus{} by^2 \equal{} 1$ and $ ax^2 \minus{} by^2 \equal{} \minus{} 1$
has solutions in positive integers.
2019 China Team Selection Test, 2
Fix a positive integer $n\geq 3$. Does there exist infinitely many sets $S$ of positive integers $\lbrace a_1,a_2,\ldots, a_n$, $b_1,b_2,\ldots,b_n\rbrace$, such that $\gcd (a_1,a_2,\ldots, a_n$, $b_1,b_2,\ldots,b_n)=1$, $\lbrace a_i\rbrace _{i=1}^n$, $\lbrace b_i\rbrace _{i=1}^n$ are arithmetic progressions, and $\prod_{i=1}^n a_i = \prod_{i=1}^n b_i$?
2014 NIMO Problems, 11
Consider real numbers $A$, $B$, \dots, $Z$ such that \[
EVIL = \frac{5}{31}, \;
LOVE = \frac{6}{29}, \text{ and }
IMO = \frac{7}{3}.
\] If $OMO = \tfrac mn$ for relatively prime positive integers $m$ and $n$, find the value of $m+n$.
[i]Proposed by Evan Chen[/i]
2012 India PRMO, 1
Rama was asked by her teacher to subtract $3$ from a certain number and then divide the result by $9$. Instead, she subtracted $9$ and then divided the result by $3$. She got $43$ as the answer. What should have been her answer if she had solved the problem correctly?
1994 Romania TST for IMO, 1:
Let $p$ be a (positive) prime number. Suppose that real numbers $a_1, a_2, . . ., a_{p+1}$ have the property that, whenever one of the numbers is deleted, the remaining numbers can be partitioned into two classes with the same arithmetic mean. Show that these numbers must be equal.
2012 Tournament of Towns, 7
Peter and Paul play the following game. First, Peter chooses some positive integer $a$ with the sum of its digits equal to $2012$. Paul wants to determine this number, he knows only that the sum of the digits of Peter’s number is $2012$. On each of his moves Paul chooses a positive integer $x$ and Peter tells him the sum of the digits of $|x - a|$. What is the minimal number of moves in which Paul can determine Peter’s number for sure?
1992 Hungary-Israel Binational, 3
We examine the following two sequences: The Fibonacci sequence: $F_{0}= 0, F_{1}= 1, F_{n}= F_{n-1}+F_{n-2 }$ for $n \geq 2$; The Lucas sequence: $L_{0}= 2, L_{1}= 1, L_{n}= L_{n-1}+L_{n-2}$ for $n \geq 2$. It is known that for all $n \geq 0$ \[F_{n}=\frac{\alpha^{n}-\beta^{n}}{\sqrt{5}},L_{n}=\alpha^{n}+\beta^{n},\] where $\alpha=\frac{1+\sqrt{5}}{2},\beta=\frac{1-\sqrt{5}}{2}$. These formulae can be used without proof.
We call a nonnegative integer $r$-Fibonacci number if it is a sum of $r$ (not necessarily distinct) Fibonacci numbers. Show that there infinitely many positive integers that are not $r$-Fibonacci numbers for any $r, 1 \leq r\leq 5.$
2016 PUMaC Team, 6
Compute the sum of all positive integers less than $100$ that do not have consecutive $1$s in their binary representation.
2019 PUMaC Individual Finals A, B, A2
Prove that for every positive integer $m$, every prime $p$ and every positive integer $j \le p^{m-1}$,
$p^m$ divides $${p^m \choose p^j }- {p^{m-1} \choose j}$$
2017 Kazakhstan National Olympiad, 6
Show that there exist infinitely many composite positive integers $n$ such that $n$ divides $2^{\frac{n-1}{2}}+1$
1980 IMO, 8
Prove that if $(a,b,c,d)$ are positive integers such that $(a+2^{\frac13}b+2^{\frac23}c)^2=d$ then $d$ is a perfect square (i.e is the square of a positive integer).
2018 EGMO, 2
Consider the set
\[A = \left\{1+\frac{1}{k} : k=1,2,3,4,\cdots \right\}.\]
[list=a]
[*]Prove that every integer $x \geq 2$ can be written as the product of one or more elements of $A$, which are not necessarily different.
[*]For every integer $x \geq 2$ let $f(x)$ denote the minimum integer such that $x$ can be written as the
product of $f(x)$ elements of $A$, which are not necessarily different.
Prove that there exist infinitely many pairs $(x,y)$ of integers with $x\geq 2$, $y \geq 2$, and \[f(xy)<f(x)+f(y).\] (Pairs $(x_1,y_1)$ and $(x_2,y_2)$ are different if $x_1 \neq x_2$ or $y_1 \neq y_2$).
[/list]
2008 Romania Team Selection Test, 5
Find the greatest common divisor of the numbers \[ 2^{561}\minus{}2, 3^{561}\minus{}3, \ldots, 561^{561}\minus{}561.\]
2011 IMO, 5
Let $f$ be a function from the set of integers to the set of positive integers. Suppose that, for any two integers $m$ and $n$, the difference $f(m) - f(n)$ is divisible by $f(m- n)$. Prove that, for all integers $m$ and $n$ with $f(m) \leq f(n)$, the number $f(n)$ is divisible by $f(m)$.
[i]Proposed by Mahyar Sefidgaran, Iran[/i]
2014 Baltic Way, 17
Do there exist pairwise distinct rational numbers $x, y$ and $z$ such that \[\frac{1}{(x - y)^2}+\frac{1}{(y - z)^2}+\frac{1}{(z - x)^2}= 2014?\]
1990 IMO Shortlist, 26
Let $ p(x)$ be a cubic polynomial with rational coefficients. $ q_1$, $ q_2$, $ q_3$, ... is a sequence of rationals such that $ q_n \equal{} p(q_{n \plus{} 1})$ for all positive $ n$. Show that for some $ k$, we have $ q_{n \plus{} k} \equal{} q_n$ for all positive $ n$.
1957 Moscow Mathematical Olympiad, 357
For which integer $n$ is $N = 20^n + 16^n - 3^n - 1$ divisible by $323$?
2014 JBMO TST - Turkey, 2
Find all triples of positive integers $(a, b, c)$ satisfying $(a^3+b)(b^3+a)=2^c$.
2012 Indonesia TST, 4
Determine all integer $n > 1$ such that
\[\gcd \left( n, \dfrac{n-m}{\gcd(n,m)} \right) = 1\]
for all integer $1 \le m < n$.
2021 AIME Problems, 9
Find the number of ordered pairs $(m, n)$ such that $m$ and $n$ are positive integers in the set $\{1, 2, ..., 30\}$ and the greatest common divisor of $2^m + 1$ and $2^n - 1$ is not $1.$
2021 China National Olympiad, 3
Let $n$ be positive integer such that there are exactly 36 different prime numbers that divides $n.$ For $k=1,2,3,4,5,$ $c_n$ be the number of integers that are mutually prime numbers to $n$ in the interval $[\frac{(k-1)n}{5},\frac{kn}{5}] .$ $c_1,c_2,c_3,c_4,c_5$ is not exactly the same.Prove that$$\sum_{1\le i<j\le 5}(c_i-c_j)^2\geq 2^{36}.$$
2011 Lusophon Mathematical Olympiad, 2
A non-negative integer $n$ is said to be [i]squaredigital[/i] if it equals the square of the sum of its digits. Find all non-negative integers which are squaredigital.