Found problems: 15460
2019 Middle European Mathematical Olympiad, 4
Determine the smallest positive integer $n$ for which the following statement holds true: From any $n$ consecutive integers one can select a non-empty set of consecutive integers such that their sum is divisible by $2019$.
[i]Proposed by Kartal Nagy, Hungary[/i]
2007 Korea Junior Math Olympiad, 1
A sequence $a_1,a_2,...,a_{2007}$ where $a_i \in\{2,3\}$ for $i = 1,2,...,2007$ and an integer sequence $x_1,x_2,...,x_{2007}$ satis fies the following: $a_ix_i + x_{i+2 }\equiv 0$ ($mod 5$) , where the indices are taken modulo $2007$. Prove that $x_1,x_2,...,x_{2007}$ are all multiples of $5$.
2017 Greece Junior Math Olympiad, 3
Find all triplets $(a,b,p)$ where $a,b$ are positive integers and $p$ is a prime number satisfying: $\frac{1}{p}=\frac{1}{a^2}+\frac{1}{b^2}$
2003 Polish MO Finals, 6
Let $n$ be an even positive integer. Show that there exists a permutation $(x_1, x_2, \ldots, x_n)$ of the set $\{1, 2, \ldots, n\}$, such that for each $i \in \{1, 2, \ldots, n\}, x_{i+1}$ is one of the numbers $2x_i, 2x_{i}-1, 2x_i - n, 2x_i - n - 1$, where $x_{n+1} = x_1.$
2023 Cono Sur Olympiad, 4
Consider a sequence $\{a_n\}$ of integers, satisfying $a_1=1, a_2=2$ and $a_{n+1}$ is the largest prime divisor of $a_1+a_2+\ldots+a_n$. Find $a_{100}$.
2012 Indonesia TST, 4
Let $\mathbb{N}$ be the set of positive integers. For every $n \in \mathbb{N}$, define $d(n)$ as the number of positive divisors of $n$. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that:
a) $d(f(x)) = x$ for all $x \in \mathbb{N}$
b) $f(xy)$ divides $(x-1)y^{xy-1}f(x)$ for all $x,y \in \mathbb{N}$
2009 IMO Shortlist, 1
Let $ n$ be a positive integer and let $ a_1,a_2,a_3,\ldots,a_k$ $ ( k\ge 2)$ be distinct integers in the set $ { 1,2,\ldots,n}$ such that $ n$ divides $ a_i(a_{i + 1} - 1)$ for $ i = 1,2,\ldots,k - 1$. Prove that $ n$ does not divide $ a_k(a_1 - 1).$
[i]Proposed by Ross Atkins, Australia [/i]
2014 Romania National Olympiad, 3
Find the smallest integer $n$ for which the set $A = \{n, n +1, n +2,...,2n\}$ contains five elements $a<b<c<d<e$ so that
$$\frac{a}{c}=\frac{b}{d}=\frac{c}{e}$$
2019 All-Russian Olympiad, 8
Let $P(x)$ be a non-constant polynomial with integer coefficients and let $n$ be a positive integer. The sequence $a_0,a_1,\ldots$ is defined as follows: $a_0=n$ and $a_k=P(a_{k-1})$ for all positive integers $k.$ Assume that for every positive integer $b$ the sequence contains a $b$th power of an integer greater than $1.$ Show that $P(x)$ is linear.
2012 Purple Comet Problems, 14
A circle in the first quadrant with center on the curve $y=2x^2-27$ is tangent to the $y$-axis and the line $4x=3y$. The radius of the circle is $\frac{m}{n}$ where $M$ and $n$ are relatively prime positive integers. Find $m+n$.
2023 Indonesia TST, N
Find all triplets natural numbers $(a, b, c)$ satisfied
\[GCD(a, b) + LCM(a,b) = 2021^c\]
with $|a - b|$ and $(a+b)^2 + 4$ are both prime number
2019 Irish Math Olympiad, 1
De fine the [i]quasi-primes[/i] as follows.
$\bullet$ The first quasi-prime is $q_1 = 2$
$\bullet$ For $n \ge 2$, the $n^{th}$ quasi-prime $q_n$ is the smallest integer greater than $q_{n_1}$ and not of the form $q_iq_j$ for some $1 \le i \le j \le n - 1$.
Determine, with proof, whether or not $1000$ is a quasi-prime.
2016 PUMaC Number Theory B, 2
For a positive integer $n$, let $s(n)$ be the sum of the digits of $n$. If $n$ is a two-digit positive integer such that $\frac{n}{s(n)}$ is a multiple of $3$, compute the sum of all possible values of $n$.
2017 India PRMO, 29
For each positive integer $n$, consider the highest common factor $h_n$ of the two numbers $n!+1$ and $(n+1)!$. For $n<100$, find the largest value of $h_n$.
1984 Czech And Slovak Olympiad IIIA, 4
Let $r$ be a natural number greater than $1$. Then there exist positive irrational numbers $x, y$ such that $x^y = r$ . Prove it.
2020 Thailand TST, 5
We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.
1992 IMO Longlists, 53
Find all integers $\,a,b,c\,$ with $\,1<a<b<c\,$ such that \[ (a-1)(b-1)(c-1) \] is a divisor of $abc-1.$
1998 Tournament Of Towns, 2
For every four-digit number, we take the product of its four digits. Then we add all of these products together . What is the result?
( G Galperin)
MathLinks Contest 6th, 2.2
Let $a_1, a_2, ..., a_{n-1}$ be $n - 1$ consecutive positive integers in increasing order such that $k$ ${n \choose k}$ $\equiv 0$ (mod $a_k$), for all $k \in \{1, 2, ... , n - 1\}$. Find the possible values of $a_1$.
2021 Pan-American Girls' Math Olympiad, Problem 1
There are $n \geq 2$ coins numbered from $1$ to $n$. These coins are placed around a circle, not necesarily in order.
In each turn, if we are on the coin numbered $i$, we will jump to the one $i$ places from it, always in a clockwise order, beginning with coin number 1. For an example, see the figure below.
Find all values of $n$ for which there exists an arrangement of the coins in which every coin will be visited.
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.
2022 Middle European Mathematical Olympiad, 8
We call a positive integer $\textit{cheesy}$ if we can obtain the average of the digits in its decimal representation by putting a decimal separator after the leftmost digit. Prove that there are only finitely many $\textit{cheesy}$ numbers.
2015 Middle European Mathematical Olympiad, 4
Find all pairs of positive integers $(m,n)$ for which there exist relatively prime integers $a$ and $b$ greater than $1$ such that
$$\frac{a^m+b^m}{a^n+b^n}$$
is an integer.
2002 Polish MO Finals, 1
Find all the natural numbers $a,b,c$ such that:
1) $a^2+1$ and $b^2+1$ are primes
2) $(a^2+1)(b^2+1)=(c^2+1)$
2018 Iran Team Selection Test, 4
We say distinct positive integers $a_1,a_2,\ldots ,a_n $ are "good" if their sum is equal to the sum of all pairwise $\gcd $'s among them. Prove that there are infinitely many $n$ s such that $n$ good numbers exist.
[i]Proposed by Morteza Saghafian[/i]