Found problems: 15460
2022 IFYM, Sozopol, 1
Are there natural numbers $n$ and $N$ such that $n > 10^{10}$, $$n^n < 2^{2^{\frac{8N}{\omega (N)}}}$$ and $n$ is divisible by $p^{2022(v_p(N)-1)}(p-1)$ for every prime divisor $p$ of $N$?
(For a natural number $N$, we denote by $\omega (N)$ the number of its different prime divisors and with $v_p(N)$ the power of the prime number $p$ in its canonical representation.)
2010 Czech And Slovak Olympiad III A, 1
Determine all pairs of integers $a, b$ for which they apply $4^a + 4a^2 + 4 = b^2$ .
2018 Middle European Mathematical Olympiad, 8
An integer $n $ is called silesian if there exist positive integers $a,b$ and $c$ such that $$n=\frac{a^2+b^2+c^2}{ab+bc+ca}.$$
$(a)$ prove that there are infinitely many silesian integers.
$(b)$ prove that not every positive integer is silesian.
1997 Croatia National Olympiad, Problem 3
Function $f$ is defined on the positive integers by $f(1)=1$, $f(2)=2$ and
$$f(n+2)=f(n+2-f(n+1))+f(n+1-f(n))\enspace\text{for }n\ge1.$$
(a) Prove that $f(n+1)-f(n)\in\{0,1\}$ for each $n\ge1$.
(b) Show that if $f(n)$ is odd then $f(n+1)=f(n)+1$.
(c) For each positive integer $k$ find all $n$ for which $f(n)=2^{k-1}+1$.
2019 LIMIT Category A, Problem 8
There are $168$ primes below $1000$. Then sum of all primes below $1000$ is,
$\textbf{(A)}~11555$
$\textbf{(B)}~76127$
$\textbf{(C)}~57298$
$\textbf{(D)}~81722$
2022 Caucasus Mathematical Olympiad, 3
Pete wrote down $21$ pairwise distinct positive integers, each not greater than $1,000,000$. For every pair $(a, b)$ of numbers written down by Pete, Nick wrote the number
$$F(a;b)=a+b -\gcd(a;b)$$
on his piece of paper. Prove that one of Nick’s numbers differs from all of Pete’s numbers.
2017 Azerbaijan EGMO TST, 4
Find all positive integers $m$ and $n$ such that $(2^{2^{n}}+1)(2^{2^{m}}+1) $ is divisible by $m\cdot n $ .
2014 Contests, 2
Let $a_1,a_2,a_3,\ldots$ be a sequence of integers, with the property that every consecutive group of $a_i$'s averages to a perfect square. More precisely, for every positive integers $n$ and $k$, the quantity \[\frac{a_n+a_{n+1}+\cdots+a_{n+k-1}}{k}\] is always the square of an integer. Prove that the sequence must be constant (all $a_i$ are equal to the same perfect square).
[i]Evan O'Dorney and Victor Wang[/i]
2009 International Zhautykov Olympiad, 1
Find all pairs of integers $ (x,y)$, such that
\[ x^2 \minus{} 2009y \plus{} 2y^2 \equal{} 0
\]
1983 Federal Competition For Advanced Students, P2, 4
The sequence $ (x_n)_{n \in \mathbb{N}}$ is defined by $ x_1\equal{}2, x_2\equal{}3,$ and
$ x_{2m\plus{}1}\equal{}x_{2m}\plus{}x_{2m\minus{}1}$ for $ m \ge 1;$
$ x_{2m}\equal{}x_{2m\minus{}1}\plus{}2x_{2m\minus{}2}$ for $ m \ge 2.$
Determine $ x_n$ as a function of $ n$.
1996 Greece Junior Math Olympiad, 4b
Determine whether exist a prime number $p$ and natural number $n$ such that $n^2 + n + p = 1996$.
1954 Moscow Mathematical Olympiad, 261
Find a four-digit number whose division by two given distinct one-digit numbers goes along the following lines:
[img]https://cdn.artofproblemsolving.com/attachments/2/a/e1d3c68ec52e11ad59de755c3dbdc2cf54a81f.png[/img]
2022 Macedonian Mathematical Olympiad, Problem 3
The sequence $(a_n)_{n \ge 1}^\infty$ is given by: $a_1=2$ and $a_{n+1}=a_n^2+a_n$ for all $n \ge 1$.
For an integer $m \ge 2$, $L(m)$ denotes the greatest prime divisor of $m$. Prove that there exists some $k$, for which $L(a_k) > 1000^{1000}$.
[i]Proposed by Nikola Velov[/i]
2006 Baltic Way, 16
Are there $4$ distinct positive integers such that adding the product of any two of them to $2006$ yields a perfect square?
2003 Spain Mathematical Olympiad, Problem 1
Prove that for any prime ${p}$, different than ${2}$ and ${5}$, there exists such a multiple of ${p}$ whose digits are all nines. For example, if ${p = 13}$, such a multiple is ${999999 = 13 * 76923}$.
1989 IMO Longlists, 93
Prove that for each positive integer $ n$ there exist $ n$ consecutive positive integers none of which is an integral power of a prime number.
2021 Bulgaria EGMO TST, 2
Determine all positive integers $n$ such that $\frac{a^2+n^2}{b^2-n^2}$ is a positive integer for some $a,b\in \mathbb{N}$.
$Turkey$
2003 May Olympiad, 4
Celia chooses a number $n$ and writes the list of natural numbers from $1$ to $n$: $1, 2, 3, 4, ..., n-1, n.$ At each step, it changes the list: it copies the first number to the end and deletes the first two. After $n-1$ steps a single number will be written.
For example, for $n=6$ the five steps are: $$ 1,2,3,4,5,6 \to 3,4,5,6,1 \to 5,6,1,3 \to 1,3,5 \to 5,1 \to 5$$
and the number $5$ is written.
Celia chose a number $n$ between $1000$ and $3000$ and after $n-1$ steps the number $1$ was written.
Determine all the values of $n$ that Celia could have chosen.
Justify why those values work, and the others do not.
2019 Turkey Team SeIection Test, 4
For an integer $n$ with $b$ digits, let a [i]subdivisor[/i] of $n$ be a positive number which divides a number obtained by removing the $r$ leftmost digits and the $l$ rightmost digits of $n$ for nonnegative integers $r,l$ with $r+l<b$ (For example, the subdivisors of $143$ are $1$, $2$, $3$, $4$, $7$, $11$, $13$, $14$, $43$, and $143$). For an integer $d$, let $A_d$ be the set of numbers that don't have $d$ as a subdivisor. Find all $d$, such that $A_d$ is finite.
2010 Philippine MO, 5
Determine, with proof, the smallest positive integer $n$ with the following property: For every choice of $n$ integers, there exist at least two whose sum or difference is divisible by $2009$.
1999 IMO Shortlist, 2
Prove that every positive rational number can be represented in the form $\dfrac{a^{3}+b^{3}}{c^{3}+d^{3}}$ where a,b,c,d are positive integers.
2015 USAJMO, 2
Solve in integers the equation
\[ x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3. \]
1993 Iran MO (3rd Round), 1
Prove that there exist infinitely many positive integers which can't be represented as sum of less than $10$ odd positive integers' perfect squares.
2019 Bangladesh Mathematical Olympiad, 1
Find all prime numbers such that the square of the prime number can be written as the sum of cubes of two positive integers.
2012 Greece JBMO TST, 2
Find all pairs of coprime positive integers $(p,q)$ such that $p^2+2q^2+334=[p^2,q^2]$ where $[p^2,q^2]$ is the leact common multiple of $p^2,q^2$ .