Found problems: 1766
2008 Vietnam Team Selection Test, 1
Let $ m$ and $ n$ be positive integers. Prove that $ 6m | (2m \plus{} 3)^n \plus{} 1$ if and only if $ 4m | 3^n \plus{} 1$
2013 India IMO Training Camp, 3
We define an operation $\oplus$ on the set $\{0, 1\}$ by
\[ 0 \oplus 0 = 0 \,, 0 \oplus 1 = 1 \,, 1 \oplus 0 = 1 \,, 1 \oplus 1 = 0 \,.\]
For two natural numbers $a$ and $b$, which are written in base $2$ as $a = (a_1a_2 \ldots a_k)_2$ and $b = (b_1b_2 \ldots b_k)_2$ (possibly with leading 0's), we define $a \oplus b = c$ where $c$ written in base $2$ is $(c_1c_2 \ldots c_k)_2$ with $c_i = a_i \oplus b_i$, for $1 \le i \le k$. For example, we have $7 \oplus 3 = 4$ since $ 7 = (111)_2$ and $3 = (011)_2$.
For a natural number $n$, let $f(n) = n \oplus \left[ n/2 \right]$, where $\left[ x \right]$ denotes the largest integer less than or equal to $x$. Prove that $f$ is a bijection on the set of natural numbers.
2007 Indonesia TST, 3
Find all pairs of function $ f: \mathbb{N} \rightarrow \mathbb{N}$ and polynomial with integer coefficients $ p$ such that:
(i) $ p(mn) \equal{} p(m)p(n)$ for all positive integers $ m,n > 1$ with $ \gcd(m,n) \equal{} 1$, and
(ii) $ \sum_{d|n}f(d) \equal{} p(n)$ for all positive integers $ n$.
2010 Portugal MO, 1
Giraldo wrote five distinct natural numbers on the vertices of a pentagon. And next he wrote on each side of the pentagon the least common multiple of the numbers written of the two vertices who were on that side and noticed that the five numbers written on the sides were equal. What is the smallest number Giraldo could have written on the sides?
2011 Middle European Mathematical Olympiad, 4
Let $k$ and $m$, with $k > m$, be positive integers such that the number $km(k^2 - m^2)$ is divisible by $k^3 - m^3$. Prove that $(k - m)^3 > 3km$.
1998 All-Russian Olympiad, 8
Two distinct positive integers $a,b$ are written on the board. The smaller of them is erased and replaced with the number $\frac{ab}{|a-b|}$. This process is repeated as long as the two numbers are not equal. Prove that eventually the two numbers on the board will be equal.
2024 Middle European Mathematical Olympiad, 8
Let $k$ be a positive integer and $a_1,a_2,\dots$ be an infinite sequence of positive integers such that
\[a_ia_{i+1} \mid k-a_i^2\]
for all integers $i \ge 1$. Prove that there exists a positive integer $M$ such that $a_n=a_{n+1}$ for all
integers $n \ge M$.
2010 Indonesia TST, 3
Let $ \mathbb{Z}$ be the set of all integers. Define the set $ \mathbb{H}$ as follows:
(1). $ \dfrac{1}{2} \in \mathbb{H}$,
(2). if $ x \in \mathbb{H}$, then $ \dfrac{1}{1\plus{}x} \in \mathbb{H}$ and also $ \dfrac{x}{1\plus{}x} \in \mathbb{H}$.
Prove that there exists a bijective function $ f: \mathbb{Z} \rightarrow \mathbb{H}$.
1974 IMO Longlists, 6
Prove that the product of two natural numbers with their sum cannot be the third power of a natural number.
2012 ELMO Shortlist, 4
Do there exist positive integers $b,n>1$ such that when $n$ is expressed in base $b$, there are more than $n$ distinct permutations of its digits? For example, when $b=4$ and $n=18$, $18 = 102_4$, but $102$ only has $6$ digit arrangements. (Leading zeros are allowed in the permutations.)
[i]Lewis Chen.[/i]
2014 China Team Selection Test, 6
Let $k$ be a fixed even positive integer, $N$ is the product of $k$ distinct primes $p_1,...,p_k$, $a,b$ are two positive integers, $a,b\leq N$. Denote
$S_1=\{d|$ $d|N, a\leq d\leq b, d$ has even number of prime factors$\}$,
$S_2=\{d|$ $d|N, a\leq d\leq b, d$ has odd number of prime factors$\}$,
Prove: $|S_1|-|S_2|\leq C^{\frac{k}{2}}_k$
2010 Turkey Junior National Olympiad, 2
Determine the number of positive integers $n$ for which $(n+15)(n+2010)$ is a perfect square.
2011 Turkey Junior National Olympiad, 3
$m < n$ are positive integers. Let $p=\frac{n^2+m^2}{\sqrt{n^2-m^2}}$.
[b](a)[/b] Find three pairs of positive integers $(m,n)$ that make $p$ prime.
[b](b)[/b] If $p$ is prime, then show that $p \equiv 1 \pmod 8$.
2013 All-Russian Olympiad, 3
Find all positive $k$ such that product of the first $k$ odd prime numbers, reduced by 1 is exactly degree of natural number (which more than one).
2006 JBMO ShortLists, 4
Determine the biggest possible value of $ m$ for which the equation $ 2005x \plus{} 2007y \equal{} m$ has unique solution in natural numbers.
2011 Postal Coaching, 2
For a positive integer $n$, consider the set
\[S = \{0, 1, 1 + 2, 1 + 2 + 3, \ldots, 1 + 2 + 3 +\ldots + (n - 1)\}\]
Prove that the elements of $S$ are mutually incongruent modulo $n$ if and only if $n$ is a power of $2$.
2007 Iran Team Selection Test, 2
Find all monic polynomials $f(x)$ in $\mathbb Z[x]$ such that $f(\mathbb Z)$ is closed under multiplication.
[i]By Mohsen Jamali[/i]
2006 Poland - Second Round, 1
Let $c$ be fixed natural number. Sequence $(a_n)$ is defined by:
$a_1=1$, $a_{n+1}=d(a_n)+c$ for $n=1,2,...$.
where $d(m)$ is number of divisors of $m$. Prove that there exist $k$ natural such that sequence $a_k,a_{k+1},...$ is periodic.
2023 German National Olympiad, 1
Determine all pairs $(m,n)$ of integers with $n \ge m$ satisfying the equation
\[n^3+m^3-nm(n+m)=2023.\]
2014 China Northern MO, 7
Prove that there exist infinitely many positive integers $n$ such that $3^n+2$ and $5^n+2$ are all composite numbers.
2016 Korea Winter Program Practice Test, 1
Solve:
$a, b, m, n\in \mathbb{N}$
$a^2+b^2=m^2-n^2, ab=2mn$
1998 Taiwan National Olympiad, 2
Does there exist a solution $(x,y,z,u,v)$ in integers greater than $1998$ to the equation $x^{2}+y^{2}+z^{2}+u^{2}+v^{2}=xyzuv-65$?
2010 Olympic Revenge, 1
Prove that the number of ordered triples $(x, y, z)$ such that $(x+y+z)^2 \equiv axyz \mod{p}$, where $gcd(a, p) = 1$ and $p$ is prime is $p^2 + 1$.
1997 Flanders Math Olympiad, 1
Write the number 1997 as the sum of positive integers for which the product is maximal, and prove there's no better solution.
2014 China Team Selection Test, 3
Let the function $f:N^*\to N^*$ such that
[b](1)[/b] $(f(m),f(n))\le (m,n)^{2014} , \forall m,n\in N^*$;
[b](2)[/b] $n\le f(n)\le n+2014 , \forall n\in N^*$
Show that: there exists the positive integers $N$ such that $ f(n)=n $, for each integer $n \ge N$.
(High School Affiliated to Nanjing Normal University )