Found problems: 15460
1986 All Soviet Union Mathematical Olympiad, 426
Find all the natural numbers equal to the square of its divisors number.
2008 May Olympiad, 3
In numbers $1010... 101$ Ones and zeros alternate, if there are $n$ ones, there are $n -1$ zeros ($n \ge 2$ ).Determine the values of $n$ for which the number $1010... 101$, which has $n$ ones, is prime.
2012 Dutch IMO TST, 1
For all positive integers $a$ and $b$, we de ne $a @ b = \frac{a - b}{gcd(a, b)}$ .
Show that for every integer $n > 1$, the following holds:
$n$ is a prime power if and only if for all positive integers $m$ such that $m < n$, it holds that $gcd(n, n @m) = 1$.
2010 Vietnam Team Selection Test, 1
Let $n$ be a positive integer. Let $T_n$ be a set of positive integers such that:
\[{T_n={ \{11(k+h)+10(n^k+n^h)| (1 \leq k,h \leq 10)}}\}\]
Find all $n$ for which there don't exist two distinct positive integers $a, b \in T_n$ such that $a\equiv b \pmod{110}$
1986 IMO Longlists, 67
Let $f(x) = x^n$ where $n$ is a fixed positive integer and $x =1, 2, \cdots .$ Is the decimal expansion $a = 0.f (1)f(2)f(3) . . .$ rational for any value of $n$ ?
The decimal expansion of a is defined as follows: If $f(x) = d_1(x)d_2(x) \cdots d_{r(x)}(x)$ is the decimal expansion of $f(x)$, then $a = 0.1d_1(2)d_2(2) \cdots d_{r(2)}(2)d_1(3) . . . d_{r(3)}(3)d_1(4) \cdots .$
2015 USAMO, 1
Solve in integers the equation
\[ x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3. \]
2004 IMO Shortlist, 2
The function $f$ from the set $\mathbb{N}$ of positive integers into itself is defined by the equality \[f(n)=\sum_{k=1}^{n} \gcd(k,n),\qquad n\in \mathbb{N}.\]
a) Prove that $f(mn)=f(m)f(n)$ for every two relatively prime ${m,n\in\mathbb{N}}$.
b) Prove that for each $a\in\mathbb{N}$ the equation $f(x)=ax$ has a solution.
c) Find all ${a\in\mathbb{N}}$ such that the equation $f(x)=ax$ has a unique solution.
2012 CHMMC Spring, 1
Let $a, b, c$ be positive integers. Suppose that $(a + b)(a + c) = 77$ and $(a + b)(b + c) = 56$. Find $(a + c)(b + c)$.
2012 China Team Selection Test, 1
Given an integer $n\ge 2$. Prove that there only exist a finite number of n-tuples of positive integers $(a_1,a_2,\ldots,a_n)$ which simultaneously satisfy the following three conditions:
[list]
[*] $a_1>a_2>\ldots>a_n$;
[*] $\gcd (a_1,a_2,\ldots,a_n)=1$;
[*] $a_1=\sum_{i=1}^{n}\gcd (a_i,a_{i+1})$,where $a_{n+1}=a_1$.[/list]
2015 South East Mathematical Olympiad, 8
Find all prime number $p$ such that there exists an integer-coefficient polynomial $f(x)=x^{p-1}+a_{p-2}x^{p-2}+…+a_1x+a_0$ that has $p-1$ consecutive positive integer roots and $p^2\mid f(i)f(-i)$, where $i$ is the imaginary unit.
VMEO III 2006 Shortlist, N11
Prove that the composition of the sets of one of the following two forms is finite:
(a) $2^{2^n}+1$
(b) $6^{2^n}+1$
1986 IMO Longlists, 76
Let $A, B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $ABC$ is an equilateral triangle whose side is $86$ meters long. A boy swam from $A$ directly toward $B$. After covering a distance of $x$ meters, he turned and swam westward, reaching the shore after covering a distance of $y$ meters. If $x$ and $y$ are both positive integers, determine $y.$
1981 Romania Team Selection Tests, 2.
Let $m$ be a positive integer not divisible by 3. Prove that there are infinitely many positive integers $n$ such that $s(n)$ and $s(n+1)$ are divisible by $m$, where $s(x)$ is the sum of digits of $x$.
[i]Dorel Miheț[/i]
2018 India IMO Training Camp, 3
Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both
$$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$
are integers.
2010 Malaysia National Olympiad, 8
Find the last digit of \[7^1\times 7^2\times 7^3\times \cdots \times 7^{2009}\times 7^{2010}.\]
2023 Stars of Mathematics, 1
Determine all pairs $(p,q)$ of prime numbers for which $p^2+5pq+4q^2$ is a perfect square.
1974 IMO Shortlist, 3
Let $P(x)$ be a polynomial with integer coefficients. We denote $\deg(P)$ its degree which is $\geq 1.$ Let $n(P)$ be the number of all the integers $k$ for which we have $(P(k))^{2}=1.$ Prove that $n(P)- \deg(P) \leq 2.$
2020 Malaysia IMONST 1, 17
Given a positive integer $n$. The number $2n$ has $28$ positive factors, while
the number $3n$ has $30$ positive factors.
Find the number of positive divisors of $6n$.
1994 Dutch Mathematical Olympiad, 3
$ (a)$ Prove that every multiple of $ 6$ can be written as a sum of four cubes.
$ (b)$ Prove that every integer can be written as a sum of five cubes.
2023 India National Olympiad, 1
Let $S$ be a finite set of positive integers. Assume that there are precisely 2023 ordered pairs $(x,y)$ in $S\times S$ so that the product $xy$ is a perfect square. Prove that one can find at least four distinct elements in $S$ so that none of their pairwise products is a perfect square.
[i]Note:[/i] As an example, if $S=\{1,2,4\}$, there are exactly five such ordered pairs: $(1,1)$, $(1,4)$, $(2,2)$, $(4,1)$, and $(4,4)$.
[i]Proposed by Sutanay Bhattacharya[/i]
2005 India IMO Training Camp, 2
Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying
\[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\]
for any two positive integers $ m$ and $ n$.
[i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers:
$ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$.
By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$).
[i]Proposed by Mohsen Jamali, Iran[/i]
2013 Balkan MO Shortlist, N7
Two distinct positive integers are called [i]close [/i] if their greatest common divisor equals their difference. Show that for any $n$, there exists a set $S$ of $n$ elements such that any two elements of $S$ are close.
1995 ITAMO, 6
Find all pairs of positive integers $x,y$ such that $x^2 +615 = 2^y$
2017 Pan-African Shortlist, N2
For which prime numbers $p$ can we find three positive integers $n$, $x$ and $y$ such that $p^n = x^3 + y^3$?
2015 Ukraine Team Selection Test, 3
Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$.
[i]Proposed by Belgium[/i]