Found problems: 15460
2013 Balkan MO, 2
Determine all positive integers $x$, $y$ and $z$ such that $x^5 + 4^y = 2013^z$.
([i]Serbia[/i])
1969 All Soviet Union Mathematical Olympiad, 122
Find four different three-digit decimal numbers starting with the same digit, such that their sum is divisible by three of them.
2001 Austrian-Polish Competition, 7
Consider the set $A$ containing all positive integers whose decimal expansion contains no $0$, and whose sum $S(N)$ of the digits divides $N$.
(a) Prove that there exist infinitely many elements in $A$ whose decimal expansion contains each digit the same number of times as each other digit.
(b) Explain that for each positive integer $k$ there exist an element in $A$ having exactly $k$ digits.
2005 CHKMO, 4
Let $S=\{1,2,...,100\}$ . Find number of functions $f: S\to S$ satisfying the following conditions
a)$f(1)=1$
b)$f$ is bijective
c)$f(n)=f(g(n))f(h(n))\forall n\in S$, where $g(n),h(n)$ are positive integer numbers such that $g(n)\leq h(n),n=g(n)h(n)$ that minimize $h(n)-g(n)$.
OMMC POTM, 2023 3
Three natural numbers are such that the product of any two of them is divisible by the sum of those two numbers. Prove that these numbers have a common divisor greater than $1$.
[i]Proposed by Evan Chang (squareman), USA[/i]
2013 NIMO Summer Contest, 11
Find $100m+n$ if $m$ and $n$ are relatively prime positive integers such that \[ \sum_{\substack{i,j \ge 0 \\ i+j \text{ odd}}} \frac{1}{2^i3^j} = \frac{m}{n}. \][i]Proposed by Aaron Lin[/i]
2000 Abels Math Contest (Norwegian MO), 1b
Determine if there is an infinite sequence $a_1,a_2,a_3,...,a_n$ of positive integers such that for all $n\ge 1$ the sum $a_1^2+a_2^2+a_3^2+...^2+a_n^2$ is a perfect square
1993 Austrian-Polish Competition, 3
Define $f (n) = n + 1$ if $n = p^k > 1$ is a power of a prime number, and $f (n) =p_1^{k_1}+... + p_r^{k_r}$ for natural numbers $n = p_1^{k_1}... p_r^{k_r}$ ($r > 1, k_i > 0$). Given $m > 1$, we construct the sequence $a_0 = m, a_{j+1} = f (a_j)$ for $j \ge 0$ and denote by $g(m)$ the smallest term in this sequence. For each $m > 1$, determine $g(m)$.
2015 BMT Spring, P1
Find two disjoint sets $N_1$ and $N_2$ with $N_1\cup N_2=\mathbb N$, so that neither set contains an infinite arithmetic progression.
1996 VJIMC, Problem 3
Prove that the equation
$$\frac x{1+x^2}+\frac y{1+y^2}+\frac z{1+z^2}=\frac1{1996}$$has finitely many solutions in positive integers.
2015 Saudi Arabia Pre-TST, 2.3
Find all integer solutions of the equation $14^x - 3^y = 2015$.
(Malik Talbi)
2008 Baltic Way, 8
Consider a set $ A$ of positive integers such that the least element of $ A$ equals $ 1001$ and the product of all elements of $ A$ is a perfect square. What is the least possible value of the greatest element of $ A$?
1998 Belarus Team Selection Test, 1
Let $n\ge 2$ be positive integer. Find the least possible number of elements of tile set $A =\{1,2,...,2n-1,2n\}$ that should be deleted in order to the sum of any two different elements remained be a composite number.
1992 IMO Longlists, 22
For each positive integer $\,n,\;S(n)\,$ is defined to be the greatest integer such that, for every positive integer $\,k\leq S(n),\;n^{2}\,$ can be written as the sum of $\,k\,$ positive squares.
[b]a.)[/b] Prove that $\,S(n)\leq n^{2}-14\,$ for each $\,n\geq 4$.
[b]b.)[/b] Find an integer $\,n\,$ such that $\,S(n)=n^{2}-14$.
[b]c.)[/b] Prove that there are infintely many integers $\,n\,$ such that $S(n)=n^{2}-14.$
2018 Bosnia and Herzegovina EGMO TST, 2
Prove that for every pair of positive integers $(m,n)$, bigger than $2$, there exists positive integer $k$ and numbers $a_0,a_1,...,a_k$, which are bigger than $2$, such that $a_0=m$, $a_1=n$ and for all $i=0,1,...,k-1$ holds
$$ a_i+a_{i+1} \mid a_ia_{i+1}+1$$
2018 BMT Spring, 1
How many multiples of $20$ are also divisors of $17!$?
2022 Kosovo National Mathematical Olympiad, 3
Find all positive integers $n$ such that $10^n+3^n+2$ is a palindrome number.
2016 All-Russian Olympiad, 6
There are $n>1$ cities in the country, some pairs of cities linked two-way through straight flight. For every pair of cities there is exactly one aviaroute (can have interchanges).
Major of every city X counted amount of such numberings of all cities from $1$ to $n$ , such that on every aviaroute with the beginning in X, numbers of cities are in ascending order. Every major, except one, noticed that results of counting are multiple of $2016$.
Prove, that result of last major is multiple of $2016$ too.
2024-IMOC, N4
Given a set of integers $S$ satisfies that: for any $a,b,c\in S$ ($a,b,c$ can be the same), $ab+c\in S$\\
Find all pairs of integers $(x,y)$ such that if $x,y\in S$, then $S=\mathbb{Z}$.
2018 Finnish National High School Mathematics Comp, 1
Eve and Martti have a whole number of euros.
Martti said to Eve: ''If you give give me three euros, so I have $n$ times the money compared to you. '' Eve in turn said to Martti: ''If you give me $n$ euros then I have triple the amount of money compared to you'' . Suppose, that both claims are valid.
What values can a positive integer $n$ get?
2005 QEDMO 1st, 11 (Z3)
Let $a,b,c$ be positive integers such that $a^2+b^2+c^2$ is divisble by $a+b+c$.
Prove that at least two of the numbers $a^3,b^3,c^3$ leave the same remainder by division through $a+b+c$.
2024 Kyiv City MO Round 1, Problem 4
For a positive integer $n$, does there exist a permutation of all its positive integer divisors $(d_1 , d_2 , \ldots, d_k)$ such that the equation $d_kx^{k-1} + \ldots + d_2x + d_1 = 0$ has a rational root, if:
a) $n = 2024$;
b) $n = 2025$?
[i]Proposed by Mykyta Kharin[/i]
2005 Croatia National Olympiad, 1
Find all positive integer solutions of the equation $k!l! = k!+l!+m!.$
1994 All-Russian Olympiad, 5
Prove that, for any natural numbers $k,m,n$: $[k,m] \cdot [m,n] \cdot [n,k] \ge [k,m,n]^2$
1969 Kurschak Competition, 1
Show that if $2 + 2\sqrt{28n^2 + 1}$ is an integer, then it is a square (for $n$ an integer).