Found problems: 15460
1986 All Soviet Union Mathematical Olympiad, 430
The decimal notation of three natural numbers consists of equal digits: $n$ digits $x$ for $a$, $n$ digits $y$ for $b$ and $2n$ digits $z$ for $c$. For every $n > 1$ find all the possible triples of digits $x,y,z$ such, that $a^2 + b = c$
2013 All-Russian Olympiad, 1
Does exist natural $n$, such that for any non-zero digits $a$ and $b$ \[\overline {ab}\ |\ \overline {anb}\ ?\]
(Here by $ \overline {x \ldots y} $ denotes the number obtained by concatenation decimal digits $x$, $\dots$, $y$.)
[i]V. Senderov[/i]
2002 Flanders Math Olympiad, 3
show that $\frac1{15} < \frac12\cdot\frac34\cdots\frac{99}{100} < \frac1{10}$
2018 Cono Sur Olympiad, 2
Prove that every positive integer can be formed by the sums of powers of 3, 4 and 7, where do not appear two powers of the same number and with the same exponent.
Example: $2= 7^0 + 7^0$ and $22=3^2 + 3^2+4^1$ are not valid representations, but $2=3^0+7^0$ and $22=3^2+3^0+4^1+4^0+7^1$ are valid representations.
2022 Cono Sur, 3
Prove that for every positive integer $n$ there exists a positive integer $k$, such that each of the numbers $k, k^2, \dots, k^n$ have at least one block of $2022$ in their decimal representation.
For example, the numbers 4[b]2022[/b]13 and 544[b]2022[/b]1[b]2022[/b] have at least one block of $2022$ in their decimal representation.
2023 BMT, 22
Let $d_n(x)$ be the $n$-th decimal digit (after the decimal point) of $x$. For example, $d_3(\pi) = 1$ because $\pi = 3.14\underline{1}5...$ For a positive integer $k$, let $f(k) = p^4_k$, where $p_k$ is the $k$-th prime number. Compute the value of $$\sum^{2023}_{i=1} d_{f(i)} \left( \frac{1}{1275}\right).$$
1965 Czech and Slovak Olympiad III A, 1
Show that the number $5^{2n+1}2^{n+2}+3^{n+2}2^{2n+1}$ is divisible by $19$ for every non-negative integer $n$.
2019-2020 Fall SDPC, 7
Find all pairs of positive integers $a,b$ with $$a^a+b^b \mid (ab)^{|a-b|}-1.$$
2001 VJIMC, Problem 1
Let $n\ge2$ be an integer and let $x_1,x_2,\ldots,x_n$ be real numbers. Consider $N=\binom n2$ sums $x_i+x_j$, $1\le i<j\le n$, and denote them by $y_1,y_2,\ldots,y_N$ (in an arbitrary order). For which $n$ are the numbers $x_1,x_2,\ldots,x_n$ uniquely determined by the numbers $y_1,y_2,\ldots,y_N$?
2010 VJIMC, Problem 4
For every positive integer $n$ let $\sigma(n)$ denote the sum of all its positive divisors. A number $n$ is called weird if $\sigma(n)\ge2n$ and there exists no representation
$$n=d_1+d_2+\ldots+d_r,$$where $r>1$ and $d_1,\ldots,d_r$ are pairwise distinct positive divisors of $n$.
Prove that there are infinitely many weird numbers.
2021 Francophone Mathematical Olympiad, 4
Let $\mathbb{N}_{\geqslant 1}$ be the set of positive integers.
Find all functions $f \colon \mathbb{N}_{\geqslant 1} \to \mathbb{N}_{\geqslant 1}$ such that, for all positive integers $m$ and $n$:
\[\mathrm{GCD}\left(f(m),n\right) + \mathrm{LCM}\left(m,f(n)\right) =
\mathrm{GCD}\left(m,f(n)\right) + \mathrm{LCM}\left(f(m),n\right).\]
Note: if $a$ and $b$ are positive integers, $\mathrm{GCD}(a,b)$ is the largest positive integer that divides both $a$ and $b$, and $\mathrm{LCM}(a,b)$ is the smallest positive integer that is a multiple of both $a$ and $b$.
2018 Argentina National Olympiad Level 2, 5
A positive integer is called [i]pretty[/i] if it is equal to the sum of the fourth powers of five distinct divisors.
[list=a]
[*]Prove that every pretty number is divisible by $5$.
[*]Determine if there are infinitely many beautiful numbers.
[/list]
2016 Junior Regional Olympiad - FBH, 3
Prove that when dividing a prime number with $30$, remainder is always not a composite number
1990 Romania Team Selection Test, 10
Let $p,q$ be positive prime numbers and suppose $q>5$. Prove that if $q \mid 2^{p}+3^{p}$, then $q>p$.
[i]Laurentiu Panaitopol[/i]
1959 Miklós Schweitzer, 1
[b]1.[/b] Let $p_n$ be the $n$th prime number. Prove that
$\sum_{n=2}^{\infty} \frac{1}{np_n-(n-1)p_{n-1}}= \infty$
[b](N.17)[/b]
2015 JBMO TST - Turkey, 1
Let $p,q$ be prime numbers such that their sum isn't divisible by $3$. Find the all $(p,q,r,n)$ positive integer quadruples satisfy:
$$p+q=r(p-q)^n$$
[i]Proposed by Şahin Emrah[/i]
1997 All-Russian Olympiad Regional Round, 8.7
Find all pairs of prime numbers $p$ and $q$ such that $p^3-q^5 = (p+q)^2$.
1994 IMO, 1
Let $ m$ and $ n$ be two positive integers. Let $ a_1$, $ a_2$, $ \ldots$, $ a_m$ be $ m$ different numbers from the set $ \{1, 2,\ldots, n\}$ such that for any two indices $ i$ and $ j$ with $ 1\leq i \leq j \leq m$ and $ a_i \plus{} a_j \leq n$, there exists an index $ k$ such that $ a_i \plus{} a_j \equal{} a_k$. Show that
\[ \frac {a_1 \plus{} a_2 \plus{} ... \plus{} a_m}{m} \geq \frac {n \plus{} 1}{2}.
\]
2017 German National Olympiad, 6
Prove that there exist infinitely many positive integers $m$ such that there exist $m$ consecutive perfect squares with sum $m^3$. Specify one solution with $m>1$.
2020 JBMO TST of France, 2
a) Find the minimum positive integer $k$ so that for every positive integers $(x, y) $, for which $x/y^2$ and $y/x^2$, then $xy/(x+y) ^k$
b) Find the minimum positive integer $l$ so that for every positive integers $(x, y, z) $, for which $x/y^2$, $y/z^2$ and $z/x^2$, then $xyz/(x+y+z)^l$
1978 IMO Longlists, 39
$A$ is a $2m$-digit positive integer each of whose digits is $1$. $B$ is an $m$-digit positive integer each of whose digits is $4$. Prove that $A+B +1$ is a perfect square.
2002 AMC 12/AHSME, 12
For how many integers $ n$ is $ \frac{n}{20\minus{}n}$ the square of an integer?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 10$
2021 Brazil EGMO TST, 5
Let $S$ be a set, such that for every positive integer $n$, we have $|S\cap T|=1$, where $T=\{n,2n,3n\}$. Prove that if $2\in S$, then $13824\notin S$.
2024 Singapore MO Open, Q3
Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.
2010 Mathcenter Contest, 5
The set $X$ of integers is called [i]good[/i] If for each pair $a,b\in X$ , only one of $a+b,\mid a-b\mid$ is a member of $X$ ($a,b$ may be equal). Find the total number of sets with $2008$ as member.
[i](tatari/nightmare)[/i]