Found problems: 15460
2006 All-Russian Olympiad Regional Round, 8.5
The product $a_1 \cdot a_2 \cdot ... \cdot a_{100}$ is written on the board , where $a_1$, $a_2$, $ ... $, $a_{100}$, are natural numbers. Let's consider $99$ expressions, each of which is obtained by replacing one of the multiplication signs with an addition sign. It is known that the values of exactly $32$ of these expressions are even. What is the largest number of even numbers among $a_1$, $a_2$, $ ... $, $a_{100}$ could it be?
2014 Postal Coaching, 2
Let $A=\{1,2,3,\ldots,40\}$. Find the least positive integer $k$ for which it is possible to partition $A$ into $k$ disjoint subsets with the property that if $a,b,c$ (not necessarily distinct) are in the same subset, then $a\ne b+c$.
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.
2003 IMO Shortlist, 8
Let $p$ be a prime number and let $A$ be a set of positive integers that satisfies the following conditions:
(i) the set of prime divisors of the elements in $A$ consists of $p-1$ elements;
(ii) for any nonempty subset of $A$, the product of its elements is not a perfect $p$-th power.
What is the largest possible number of elements in $A$ ?
2010 All-Russian Olympiad Regional Round, 10.4
We call a natural number $b$ [i]lucky [/i] if for any natural number $a$ such that $a^5$ is divisible by $b^2$, the number $a^2$ is divisible by $b$. Find the number of [i]lucky [/i] natural numbers less than $2010$.
2020 Ecuador NMO (OMEC), 2
Find all pairs $(n, q)$ such that $n$ is a positive integer, $q$ is a not integer rational and
$$n^q-q$$
is an integer.
1999 Israel Grosman Mathematical Olympiad, 1
For any $16$ positive integers $n,a_1,a_2,...,a_{15}$ we define $T(n,a_1,a_2,...,a_{15}) = (a_1^n+a_2^n+ ...+a_{15}^n)a_1a_2...a_{15}$.
Find the smallest $n$ such that $T(n,a_1,a_2,...,a_{15})$ is divisible by $15$ for any choice of $a_1,a_2,...,a_{15}$.
2010 South africa National Olympiad, 1
For a positive integer $n$, $S(n)$ denotes the sum of its digits and $U(n)$ its unit digit. Determine all positive integers $n$ with the property that
\[n = S(n) + U(n)^2.\]
2011 Brazil Team Selection Test, 2
Let $n\ge 3$ be an integer such that for every prime factor $q$ of $n-1$ exists an integer $a > 1$ such that $a^{n-1} \equiv 1 \,(\mod n \, )$ and $a^{\frac{n-1} {q}}\not\equiv 1 \,(\mod n \, )$. Prove that $n$ is not prime.
2022 Israel National Olympiad, P4
Find all triples $(a,b,c)$ of integers for which the equation
\[x^3-a^2x^2+b^2x-ab+3c=0\]
has three distinct integer roots $x_1,x_2,x_3$ which are pairwise coprime.
2005 Spain Mathematical Olympiad, 1
Prove that for every positive integer $n$, the decimal expression of $\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}$ is periodic .
2022 Polish Junior Math Olympiad First Round, 7.
None of the $n$ (not necessarily distinct) digits selected are equal to $0$ or $7$. It turns out that every $n$-digit number formed using these digits is divisible by $7$. Prove that $n$ is divisible by $6$.
2023 Quang Nam Province Math Contest (Grade 11), Problem 4
a) Find all integer pairs $(x,y)$ satisfying $x^4+(y+2)^3=(x+2)^4.$
b) Prove that: if $p$ is a prime of the form $p=4k+3$ $(k$ is a non-negative number$),$ then there doesn's exist $p-1$ consecutive non-negative integers such that we can divide the set of these numbers into $2$ distinct subsets so that the product of all the numbers in one subset is equal to that in the remained subset.
2023 CMIMC Algebra/NT, 4
An arithmetic sequence of exactly $10$ positive integers has the property that any two elements are relatively prime. Compute the smallest possible sum of the $10$ numbers.
[i]Proposed by Kyle Lee[/i]
LMT Team Rounds 2021+, 1
Let $x$ be the positive integer satisfying $5^2 +28^2 +39^2 = 24^2 +35^2 + x^2$. Find $x$.
2004 Austrian-Polish Competition, 4
Determine all $n \in \mathbb{N}$ for which $n^{10} + n^5 + 1$ is prime.
2015 Junior Balkan Team Selection Tests - Romania, 1
Define the set $M_q=\{x \in \mathbb{Q} \mid x^3-2015x=q \}$ , where $q$ is an arbitrary rational number.
[b]a)[/b] Show that there exists values for $q$ such that the set is null as well as values for which it has exactly one element.
[b]b)[/b] Determine all the possible values for the cardinality of $M_q$
2018 Swedish Mathematical Competition, 4
Find the least positive integer $n$ with the property:
Among arbitrarily $n$ selected consecutive positive integers, all smaller than $2018$, there is at least one that is divisible by its sum of digits .
2025 Belarusian National Olympiad, 11.4
A finite set $S$ consists of primes, and $3$ is not in $S$. Prove that there exists a positive integer $M$ such that for every $p \in S$ one can shuffle the digits of $M$ to get a number divisible by $p$ and not divisible by all other numbers in $S$. (Note: the first digit of a positive integer can not be zero).
[i]A. Voidelevich[/i]
2025 China Team Selection Test, 20
Let \( n \) be an odd integer, \( m = \frac{n+1}{2} \). Consider \( 2m \) integers \( a_1, a_2, \ldots, a_m, b_1, b_2, \ldots, b_m \) such that for any \( 1 \leq i < j \leq m \), \( a_i \not\equiv a_j \pmod{n} \) and \( b_i \not\equiv b_j \pmod{n} \). Prove that the number of \( k \in \{0, 1, \ldots, n-1\} \) for which satisfy \( a_i + b_j \equiv k \pmod{n} \) for some \( i \neq j \), $i, j \in \left \{ 1,2,\cdots,m \right \} $ is greater than \( n - \sqrt{n} - \frac{1}{2} \).
2020 Romanian Master of Mathematics, 6
For each integer $n \geq 2$, let $F(n)$ denote the greatest prime factor of $n$. A [i]strange pair[/i] is a pair of distinct primes $p$ and $q$ such that there is no integer $n \geq 2$ for which $F(n)F(n+1)=pq$.
Prove that there exist infinitely many strange pairs.
2022 Caucasus Mathematical Olympiad, 1
Positive integers $a$, $b$, $c$ are given. It is known that $\frac{c}{b}=\frac{b}{a}$, and the number $b^2-a-c+1$ is a prime. Prove that $a$ and $c$ are double of a squares of positive integers.
2010 Tournament Of Towns, 7
A multi-digit number is written on the blackboard. Susan puts in a number of plus signs between some pairs of adjacent digits. The addition is performed and the process is repeated with the sum. Prove that regardless of what number was initially on the blackboard, Susan can always obtain a single-digit number in at most ten steps.
2010 Contests, 2
Determine all triples $(x, y, z)$ of positive integers $x > y > z > 0$, such that $x^2 = y \cdot 2^z + 1$
2020 Caucasus Mathematical Olympiad, 3
Let $a_n$ be a sequence given by $a_1 = 18$, and $a_n = a_{n-1}^2+6a_{n-1}$, for $n>1$. Prove that this sequence contains no perfect powers.