Found problems: 15460
2012 IMAR Test, 2
Given an integer $n \ge 2$, evaluate $\Sigma \frac{1}{pq}$ ,where the summation is over all coprime integers $p$ and $q$ such that $1 \le p < q \le n$ and $p + q > n$.
2016 India Regional Mathematical Olympiad, 6
Let $(a_1,a_2,\dots)$ be a strictly increasing sequence of positive integers in arithmetic progression. Prove that there is an infinite sub-sequence of the given sequence whose terms are in a geometric progression.
1979 Poland - Second Round, 5
Prove that among every ten consecutive natural numbers there is one that is coprime to each of the other nine.
2011 Dutch IMO TST, 1
Find all pairs $(x, y)$ of integers that satisfy $x^2 + y^2 + 3^3 = 456\sqrt{x - y}$.
2022 Iran MO (3rd Round), 2
Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that for all $x,y\in\mathbb{N}$:
$$0\le y+f(x)-f^{f(y)}(x)\le1$$
that here
$$f^n(x)=\underbrace{f(f(\ldots(f}_{n}(x))\ldots)$$
2019 Slovenia Team Selection Test, 4
Let $P$ be the set of all prime numbers. Let $A$ be some subset of $P$ that has at least two elements. Let's say that for every positive integer $n$ the following statement holds: If we take $n$ different elements $p_1,p_2...p_n \in A$, every prime number that divides $p_1 p_2 \cdots p_n-1$ is also an element of $A$. Prove, that $A$ contains all prime numbers.
2005 iTest, 25
Consider the set $\{1!, 2!, 3!, 4!, …, 2004!, 2005!\}$. How many elements of this set are divisible by $2005$?
2019 China Northern MO, 8
For positive intenger $n$, define $f(n)$: the smallest positive intenger that does not divide $n$.
Consider sequence $(a_n): a_1=a_2=1, a_n=a_{f(n)}+1(n\geq3)$.
For example, $a_3=a_2+1=2,a_4=a_3+1=3$.
[b](a)[/b] Prove that there exists a positive intenger $C$, for any positive intenger $n$, $a_n\leq C$.
[b](b)[/b] Are there positive intengers $M$ and $T$, satisfying that for any positive intenger $n\geq M$, $a_n=a_{n+T}$.
2008 Hanoi Open Mathematics Competitions, 7
Find all triples $(a, b,c)$ of consecutive odd positive integers such that $a < b < c$ and $a^2 + b^2 + c^2$ is a four digit
number with all digits equal.
2024 Taiwan Mathematics Olympiad, 2
A positive integer is [b]superb[/b] if it is the least common multiple of $1,2,\ldots, n$ for some positive integer $n$.
Find all superb $x,y,z$ such that $x+y=z$.
[i]
Proposed by usjl[/i]
2023 Malaysian IMO Training Camp, 5
Let $n\ge 3$, $d$ be positive integers. For an integer $x$, denote $r(x)$ be the remainder of $x$ when divided by $n$ such that $0\le r(x)\le n-1$. Let $c$ be a positive integer with $1<c<n$ and $\gcd(c,n)=1$, and suppose $a_1, \cdots, a_d$ are positive integers with $a_1+\cdots+a_d\le n-1$. \\
(a) Prove that if $n<2d$, then $\displaystyle\sum_{i=1}^d r(ca_i)\ge n.$ \\
(b) For each $n$, find the smallest $d$ such that $\displaystyle\sum_{i=1}^d r(ca_i)\ge n$ always holds.
[i]Proposed by Yeoh Zi Song & Anzo Teh Zhao Yang[/i]
PEN A Problems, 14
Let $n$ be an integer with $n \ge 2$. Show that $n$ does not divide $2^{n}-1$.
2023 LMT Spring, 6
Find the least positive integer $m$ such that $105| 9^{(p^2)} -29^p +m$ for all prime numbers $p > 3$.
2013 ELMO Shortlist, 5
Let $m_1,m_2,...,m_{2013} > 1$ be 2013 pairwise relatively prime positive integers and $A_1,A_2,...,A_{2013}$ be 2013 (possibly empty) sets with $A_i\subseteq \{1,2,...,m_i-1\}$ for $i=1,2,...,2013$. Prove that there is a positive integer $N$ such that
\[ N \le \left( 2\left\lvert A_1 \right\rvert + 1 \right)\left( 2\left\lvert A_2 \right\rvert + 1 \right)\cdots\left( 2\left\lvert A_{2013} \right\rvert + 1 \right) \]
and for each $i = 1, 2, ..., 2013$, there does [i]not[/i] exist $a \in A_i$ such that $m_i$ divides $N-a$.
[i]Proposed by Victor Wang[/i]
1995 Czech and Slovak Match, 6
Find all triples $(x; y; p)$ of two non-negative integers $x, y$ and a prime number p such that $ p^x-y^p=1 $
2024 Indonesia TST, 3
Let $a_1<a_2<a_3<\dots$ be positive integers such that $a_{k+1}$ divides $2(a_1+a_2+\dots+a_k)$ for every $k\geqslant 1$. Suppose that for infinitely many primes $p$, there exists $k$ such that $p$ divides $a_k$. Prove that for every positive integer $n$, there exists $k$ such that $n$ divides $a_k$.
2000 May Olympiad, 3
To write all consecutive natural numbers from $1ab$ to $ab2$ inclusive, $1ab1$ digits have been used. Determine how many more digits are needed to write the natural numbers up to $aab$ inclusive. Give all chances. ($a$ and $b$ represent digits)
2010 China Team Selection Test, 3
Fine all positive integers $m,n\geq 2$, such that
(1) $m+1$ is a prime number of type $4k-1$;
(2) there is a (positive) prime number $p$ and nonnegative integer $a$, such that
\[\frac{m^{2^n-1}-1}{m-1}=m^n+p^a.\]
2006 APMO, 1
Let $n$ be a positive integer. Find the largest nonnegative real number $f(n)$ (depending on $n$) with the following property: whenever $a_1,a_2,...,a_n$ are real numbers such that $a_1+a_2+\cdots +a_n$ is an integer, there exists some $i$ such that $\left|a_i-\frac{1}{2}\right|\ge f(n)$.
2013 Iran MO (3rd Round), 3
Let $p>3$ a prime number. Prove that there exist $x,y \in \mathbb Z$ such that $p = 2x^2 + 3y^2$ if and only if $p \equiv 5, 11 \; (\mod 24)$
(20 points)
2021 Switzerland - Final Round, 3
Find all finite sets $S$ of positive integers with at least $2$ elements, such that if $m>n$ are two elements of $S$, then
$$ \frac{n^2}{m-n} $$
is also an element of $S$.
1996 Romania Team Selection Test, 8
Let $ p_1,p_2,\ldots,p_k $ be the distinct prime divisors of $ n $ and let $ a_n=\frac {1}{p_1}+\frac {1}{p_2}+\cdots+\frac {1}{p_k} $ for $ n\geq 2 $. Show that for every positive integer $ N\geq 2 $ the following inequality holds: $ \sum_{k=2}^{N} a_2a_3 \cdots a_k <1 $
[i]Laurentiu Panaitopol[/i]
1993 Taiwan National Olympiad, 5
Assume $A=\{a_{1},a_{2},...,a_{12}\}$ is a set of positive integers such that for each positive integer $n \leq 2500$ there is a subset $S$ of $A$ whose sum of elements is $n$. If $a_{1}<a_{2}<...<a_{12}$ , what is the smallest possible value of $a_{1}$?
2021 JBMO Shortlist, N6
Given a positive integer $n \ge 2$, we define $f(n)$ to be the sum of all remainders obtained by dividing $n$ by all positive integers less than $n$. For example dividing $5$ with $1, 2, 3$ and $4$ we have remainders equal to $0, 1, 2$ and $1$ respectively. Therefore $f(5) = 0 + 1 + 2 + 1 = 4$. Find all positive integers $n \ge 3$ such that $f(n) = f(n - 1) + (n - 2)$.
2012 Mathcenter Contest + Longlist, 4 sl12
Given a natural $n>2$, let $\{ a_1,a_2,...,a_{\phi (n)} \} \subset \mathbb{Z}$ is the Reduced Residue System (RRS) set of modulo $n$ (also known as the set of integers $k$ where $(k,n)=1$ and no pairs are congruent in modulo $n$ ).
if write $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_{\phi (n)}}=\frac{a}{b}$$
where $a,b \in \mathbb{N}$ and $(a,b)=1$ , then prove that $n|a$.
[i](PP-nine)[/i]