Found problems: 15460
2010 IFYM, Sozopol, 8
Let $m, n,$ and $k$ be natural numbers, where $n$ is odd. Prove that
$\frac{1}{m}+\frac{1}{m+n}+...+\frac{1}{m+kn}$
is not a natural number.
2010 Purple Comet Problems, 25
Let $x_1$, $x_2$, and $x_3$ be the roots of the polynomial $x^3+3x+1$. There are relatively prime positive integers $m$ and $n$ such that $\tfrac{m}{n}=\tfrac{x_1^2}{(5x_2+1)(5x_3+1)}+\tfrac{x_2^2}{(5x_1+1)(5x_3+1)}+\tfrac{x_3^2}{(5x_1+1)(5x_2+1)}$. Find $m+n$.
2024 Bangladesh Mathematical Olympiad, P1
Find all non-negative integers $x, y$ such that\[x^3y+x+y=xy+2xy^2\]
2015 Czech-Polish-Slovak Junior Match, 4
Determine all such pairs pf positive integers $(a, b)$ such that $a + b + (gcd (a, b))^ 2 = lcm (a, b) = 2 \cdot lcm(a -1, b)$, where $lcm (a, b)$ denotes the smallest common multiple, and $gcd (a, b)$ denotes the greatest common divisor of numbers $a, b$.
2017 Romania Team Selection Test, P3
Consider the sequence of rational numbers defined by $x_1=\frac{4}{3}$, and $x_{n+1}=\frac{x_n^2}{x_n^2-x_n+1}$. Show that the nu,erator of the lowest term expression of each sum $x_1+x_2+...+x_k$ is a perfect square.
2013 Peru MO (ONEM), 1
We define the polynomial $$P (x) = 2014x^{2013} + 2013x^{2012} +... + 4x^3 + 3x^2 + 2x.$$ Find the largest prime divisor of $P (2)$.
2021 Puerto Rico Team Selection Test, 6
Two positive integers $n,m\ge 2$ are called [i]allies[/i] if when written as a product of primes (not necessarily different): $n=p_1p_2...p_s$ and $m=q_1q_2...q_t$, turns out that: $$p_1 + p_2 + ... + p_s = q_1 + q_2 + ... + q_t$$
(a) Show that the biggest ally of any positive integer has to have only $2$ and $3$ in its prime factorization.
(b) Find the biggest number which is allied of $2021$ .
2010 Germany Team Selection Test, 3
A positive integer $N$ is called [i]balanced[/i], if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$.
(a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P(1)$, $P(2)$,$\ldots$, $P(50)$ are balanced.
(b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$.
[i]Proposed by Jorge Tipe, Peru[/i]
2021 HMNT, 9
Let $N$ be the smallest positive integer for which $x^2 + x + 1$ divides $166 -\sum_{d|N, d>0} x^d$. Find the remainder when $N$ is divided by $1000$.
2011 Junior Balkan Team Selection Tests - Romania, 1
It is said that a positive integer $n > 1$ has the property ($p$) if in its prime factorization $n = p_1^{a_1} \cdot ... \cdot p_j^{a_j}$ at least one of the prime factors $p_1, ... , p_j$ has the exponent equal to $2$.
a) Find the largest number $k$ for which there exist $k$ consecutive positive integers that do not have the property ($p$).
b) Prove that there is an infinite number of positive integers $n$ such that $n, n + 1$ and $n + 2$ have the property ($p$).
2018 EGMO, 6
[list=a]
[*]Prove that for every real number $t$ such that $0 < t < \tfrac{1}{2}$ there exists a positive integer $n$ with the following property: for every set $S$ of $n$ positive integers there exist two different elements $x$ and $y$ of $S$, and a non-negative integer $m$ (i.e. $m \ge 0 $), such that \[ |x-my|\leq ty.\]
[*]Determine whether for every real number $t$ such that $0 < t < \tfrac{1}{2} $ there exists an infinite set $S$ of positive integers such that \[|x-my| > ty\] for every pair of different elements $x$ and $y$ of $S$ and every positive integer $m$ (i.e. $m > 0$).
2008 Brazil Team Selection Test, 1
Find all pairs of natural numbers $ (a, b)$ such that $ 7^a \minus{} 3^b$ divides $ a^4 \plus{} b^2$.
[i]Author: Stephan Wagner, Austria[/i]
2013 Baltic Way, 19
Let $a_0$ be a positive integer and $a_n=5a_{n-1}+4$ for all $n\ge 1$. Can $a_0$ be chosen so that $a_{54}$ is a multiple of $2013$?
2024 Alborz Mathematical Olympiad, P1
Find all positive integers $n$ such that if $S=\{d_1,d_2,\cdots,d_k\}$ is the set of positive integer divisors of $n$, then $S$ is a complete residue system modulo $k$. (In other words, for every pair of distinct indices $i$ and $j$, we have $d_i\not\equiv d_j \pmod{k}$).
Proposed by Heidar Shushtari
2016 Puerto Rico Team Selection Test, 2
Determine all $6$-digit numbers $(abcdef)$ such that $(abcdef) = (def)^2$ where $(x_1x_2...x_n)$ is not a multiplication but a number of $n$ digits.
2007 Greece JBMO TST, 2
Let $n$ be a positive integer such that $n(n+3)$ is a perfect square of an integer, prove that $n$ is not a multiple of $3$.
2021 Azerbaijan Senior NMO, 2
Find all triples of natural numbers $(a, b, c)$ for which the number $$2^a + 2^b + 2^c + 3$$ is the square of an integer.
Kvant 2021, M2666
Let $x{}$ and $y{}$ be natural numbers greater than 1. It turns out that $x^2+y^2-1$ is divisible by $x+y-1$. Prove that $x+y-1$ is composite.
[i]From the folklore[/i]
2020 Simon Marais Mathematics Competition, B2
For each positive integer $k$, let $S_k$ be the set of real numbers that can be expressed in the form
\[\frac{1}{n_1}+\frac{1}{n_2}+\dots+\frac{1}{n_k},\]
where $n_1,n_2\dots,n_k$ are positive integers.
Prove that $S_k$ does not contain an infinite strictly increasing sequence.
2023 Brazil National Olympiad, 3
Let $n$ be a positive integer. Show that there are integers $x_1, x_2, \ldots , x_n$, [i]not all equal[/i], satisfying $$\begin{cases} x_1^2+x_2+x_3+\ldots+x_n=0 \\ x_1+x_2^2+x_3+\ldots+x_n=0 \\ x_1+x_2+x_3^2+\ldots+x_n=0 \\ \vdots \\ x_1+x_2+x_3+\ldots+x_n^2=0 \end{cases}$$ if, and only if, $2n-1$ is not prime.
Oliforum Contest II 2009, 4
Let $ m$ a positive integer and $ p$ a prime number, both fixed. Define $ S$ the set of all $ m$-uple of positive integers $ \vec{v} \equal{} (v_1,v_2,\ldots,v_m)$ such that $ 1 \le v_i \le p$ for all $ 1 \le i \le m$. Define also the function $ f(\cdot): \mathbb{N}^m \to \mathbb{N}$, that associates every $ m$-upla of non negative integers $ (a_1,a_2,\ldots,a_m)$ to the integer $ \displaystyle f(a_1,a_2,\ldots,a_m) \equal{} \sum_{\vec{v} \in S} \left(\prod_{1 \le i \le m}{v_i^{a_i}} \right)$.
Find all $ m$-uple of non negative integers $ (a_1,a_2,\ldots,a_m)$ such that $ p \mid f(a_1,a_2,\ldots,a_m)$.
[i](Pierfrancesco Carlucci)[/i]
2025 All-Russian Olympiad, 9.7
The numbers \( 1, 2, 3, \ldots, 60 \) are written in a row in that exact order. Igor and Ruslan take turns inserting the signs \( +, -, \times \) between them, starting with Igor. Each turn consists of placing one sign. Once all signs are placed, the value of the resulting expression is computed. If the value is divisible by $3$, Igor wins; otherwise, Ruslan wins. Which player has a winning strategy regardless of the opponent’s moves? \\
1970 All Soviet Union Mathematical Olympiad, 142
All natural numbers containing not more than $n$ digits are divided onto two groups. The first contains the numbers with the even sum of the digits, the second -- with the odd sum. Prove that if $0<k<n$ than the sum of the $k$-th powers of the numbers in the first group equals to the sum of the $k$-th powers of the numbers in the second group.
2016 Iran MO (3rd Round), 1
Let $p,q$ be prime numbers ($q$ is odd). Prove that there exists an integer $x$ such that:
$$q |(x+1)^p-x^p$$
If and only if $$q \equiv 1 \pmod p$$
2021 CMIMC, 1.7
As a gift, Dilhan was given the number $n=1^1\cdot2^2\cdots2021^{2021}$, and each day, he has been dividing $n$ by $2021!$ exactly once. One day, when he did this, he discovered that, for the first time, $n$ was no longer an integer, but instead a reduced fraction of the form $\frac{a}b$. What is the sum of all distinct prime factors of $b$?
[i]Proposed by Adam Bertelli[/i]