Found problems: 15460
2022 Argentina National Olympiad, 1
For every positive integer $n$, $P(n)$ is defined as follows: For each prime divisor $p$ of $n$ is considered the largest integer $k$ such that $p^k\le n$ and all the $p^k$ are added. For example, for $n=100=2^2 \cdot 5^2$, as $2^6<100<2^7$ and $5^2<100<5^3$, it turns out that $P(100)=2^6+5^2=89$ Prove that there are infinitely many positive integers $n$ such that $P(n)>n$..
2024 Nepal Mathematics Olympiad (Pre-TST), Problem 4
Find all integer/s $n$ such that $\displaystyle{\frac{5^n-1}{3}}$ is a prime or a perfect square of an integer.
[i]Proposed by Prajit Adhikari, Nepal[/i]
2006 Hanoi Open Mathematics Competitions, 3
Find the number of different positive integer triples $(x, y,z)$ satisfying the equations
$x^2 + y -z = 100$ and $x + y^2 - z = 124$:
1999 Tournament Of Towns, 2
Prove that there exist infinitely many odd positive integers $n$ for which the number $2^n + n$ is composite.
(V Senderov)
2006 Switzerland Team Selection Test, 1
Let $n$ be natural number and $1=d_1<d_2<\ldots <d_k=n$ be the positive divisors of $n$.
Find all $n$ such that $2n = d_5^2+ d_6^2 -1$.
1974 IMO Longlists, 24
Let $a_i, b_i$ be coprime positive integers for $i = 1, 2, \ldots , k$, and $m$ the least common multiple of $b_1, \ldots , b_k$. Prove that the greatest common divisor of $a_1 \frac{m}{b_1} , \ldots, a_k \frac{m}{b_k}$ equals the greatest common divisor of $a_1, \ldots , a_k.$
2011 Saudi Arabia Pre-TST, 3.2
Find all pairs of nonnegative integers $(a, b)$ such that $a+2b-b^2=\sqrt{2a+a^2+|2a+1-2b|}$.
2022 Cyprus JBMO TST, 4
Let $A$ be a subset of $\{1, 2, 3, \ldots, 50\}$ with the property: for every $x,y\in A$ with $x\neq y$, it holds that
\[\left| \frac{1}{x}- \frac{1}{y}\right|>\frac{1}{1000}.\]
Determine the largest possible number of elements that the set $A$ can have.
1989 IMO Longlists, 24
Let $ a, b, c, d$ be positive integers such that $ ab \equal{} cd$ and $ a\plus{}b \equal{} c \minus{} d.$ Prove that there exists a right-angled triangle the measure of whose sides (in some unit) are integers and whose area measure is $ ab$ square units.
2015 India Regional MathematicaI Olympiad, 3
Find all integers \(a,b,c\) such that \(a^{2}=bc+4\) and \(b^{2}=ca+4\).
1949 Miklós Schweitzer, 3
Let $ p$ be an odd prime number and $ a_1,a_2,...,a_p$ and $ b_1,b_2,...,b_p$ two arbitrary permutations of the numbers $ 1,2,...,p$ . Show that the least positive residues modulo $ p$ of the numbers $ a_1b_1, a_2b_2,...,a_pb_p$ never form a permutation of the numbers $ 1,2,...,p$.
2019 LIMIT Category A, Problem 6
Let $d_1,d_2,\ldots,d_k$ be all factors of a positive integer $n$ including $1$ and $n$. If $d_1+d_2+\ldots+d_k=72$ then $\frac1{d_1}+\frac1{d_2}+\ldots+\frac1{d_k}$ is
$\textbf{(A)}~\frac{k^2}{72}$
$\textbf{(B)}~\frac{72}k$
$\textbf{(C)}~\frac{72}n$
$\textbf{(D)}~\text{None of the above}$
2012 Stars of Mathematics, 1
The positive integer $N$ is said[i] amiable [/i]if the set $\{1,2,\ldots,N\}$ can be partitioned into pairs of elements, each pair having the sum of its elements a perfect square. Prove there exist infinitely many amiable numbers which are themselves perfect squares.
([i]Dan Schwarz[/i])
2002 Estonia National Olympiad, 3
John takes seven positive integers $a_1,a_2,...,a_7$ and writes the numbers $a_i a_j$, $a_i+a_j$ and $|a_i -a_j |$ for all $i \ne j$ on the blackboard. Find the greatest possible number of distinct odd integers on the blackboard.
1963 Polish MO Finals, 1
Prove that two natural numbers whose digits are all ones are relatively prime if and only if the numbers of their digits are relatively prime.
1998 AMC 12/AHSME, 28
In triangle $ ABC$, angle $ C$ is a right angle and $ CB > CA$. Point $ D$ is located on $ \overline{BC}$ so that angle $ CAD$ is twice angle $ DAB$. If $ AC/AD \equal{} 2/3$, then $ CD/BD \equal{} m/n$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.
$ \textbf{(A)}\ 10\qquad
\textbf{(B)}\ 14\qquad
\textbf{(C)}\ 18\qquad
\textbf{(D)}\ 22\qquad
\textbf{(E)}\ 26$
2012 Argentina National Olympiad, 2
Determine all natural numbers $n$ for which there are $2n$ distinct positive integers $x_1,…,x_n,y_1,…,y_n$
such that the product $$(11x^2_1+12y^2_1)(11x^2_2+12y^2_2)…(11x^2_n+12y^2_n)$$ is a perfect square.
2023 BMT, 8
Define a family of functions $S_k(n)$ for positive integers $n$ and $k$ by the following two rules:
$$S_0(n) = 1,$$
$$S_k(n) = \sum_{d | n} dS_{k-1}(d).$$
Compute the remainder when $S_{30}(30)$ is divided by $1001$.
2003 All-Russian Olympiad Regional Round, 9.7
Prove that of any six four-digit numbers, mutual prime in total, you can always choose five numbers that are also relatively prime in total.
[hide=original wording]Докажите, что из любых шести четырехзначных чисел, взаимно простых в совокупности, всегда можно выбратьпя ть чисел, также взаимно простых в совокупности.[/hide]
2018 Finnish National High School Mathematics Comp, 4
Define $f : \mathbb{Z}_+ \to \mathbb{Z}_+$ such that $f(1) = 1$ and $f(n) $ is the greatest prime divisor of $n$ for $n > 1$.
Aino and Väinö play a game, where each player has a pile of stones. On each turn the player to turn with $m$ stones in his pile may remove at most $f(m)$ stones from the opponent's pile, but must remove at least one stone. (The own pile stays unchanged.) The first player to clear the opponent's pile wins the game. Prove that there exists a positive integer $n$ such that Aino loses, when both players play optimally, Aino starts, and initially both players have $n$ stones.
2015 NZMOC Camp Selection Problems, 6
In many computer languages, the division operation ignores remainders. Let’s denote this operation by $//$, so for instance $13//3 = 4$. If, for some $b$, $a//b = c$, then we say that $c$ is a [i]near factor[/i] of $a$. Thus, the near factors of $13$ are $1$, $2$, $3$, $4$, and $6$. Let $a$ be a positive integer. Prove that every positive integer less than or equal to $\sqrt{a}$ is a near factor of $a$.
2008 Mathcenter Contest, 6
For even positive integers $a>1$. Prove that there are infinite positive integers $n$ that makes $n | a^n+1$.
[i](tomoyo-jung)[/i]
2004 Korea - Final Round, 3
For prime number $p$, let $f_p(x)=x^{p-1} +x^{p-2} + \cdots + x + 1$.
(1) When $p$ divides $m$, prove that there exists a prime number that is coprime with $m(m-1)$ and divides $f_p(m)$.
(2) Prove that there are infinitely many positive integers $n$ such that $pn+1$ is prime number.
1995 Hungary-Israel Binational, 1
Let the sum of the first $ n$ primes be denoted by $ S_n$. Prove that for any positive integer $ n$, there exists a perfect square between $ S_n$ and $ S_{n\plus{}1}$.
2017 Kyiv Mathematical Festival, 5
Find all the pairs of integers $(x,y)$ for which $(x^2+y)(y^2+x)=(x+1)(y+1).$