Found problems: 15460
2003 Abels Math Contest (Norwegian MO), 2b
Let $a_1,a_2,...,a_n$ be $n$ different positive integers where $n\ge 1$. Show that $$\sum_{i=1}^n a_i^3 \ge \left(\sum_{i=1}^n a_i\right)^2$$
2004 All-Russian Olympiad Regional Round, 8.7
A set of five-digit numbers $\{N_1,... ,N_k\}$ is such that any five-digit a number whose digits are all in ascending order is the same in at least one digit with at least one of the numbers $N_1$,$...$ ,$N_k$. Find the smallest possible value of $k$.
2021 HMNT, 2
Joey wrote a system of equations on a blackboard, where each of the equations was of the form $a + b = c$ or $a \cdot b = c$ for some variables or integers $a, b, c$. Then Sean came to the board and erased all of the plus signs and multiplication signs, so that the board reads:
$x\,\,\,\, z = 15$
$x\,\,\,\, y = 12$
$x\,\,\,\, x = 36$
If $x, y, z$ are integer solutions to the original system, find the sum of all possible values of $100x+10y+z$.
2017 Rioplatense Mathematical Olympiad, Level 3, 3
Show that there are infinitely many pairs of positive integers $(m,n)$, with $m<n$, such that
$m$ divides $n^{2016}+n^{2015}+\dots+n^2+n+1$ and $n$ divides $m^{2016}+m^{2015} +\dots+m^2+m+1$.
2016 Belarus Team Selection Test, 1
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.
2021 Latvia Baltic Way TST, P13
Does there exist a natural number $a$ so that:
a) $\Big ((a^2-3)^3+1\Big) ^a-1$ is a perfect square?
b) $\Big ((a^2-3)^3+1\Big) ^{a+1}-1$ is a perfect square?
2017 Saudi Arabia BMO TST, 1
Find the smallest prime $q$ such that $$q = a_1^2 + b_1^2 = a_2^2 + 2b_2^2 = a_3^2 + 3b_3^2 = ... = a_{10}^ 2 + 10b_{10}^2$$ where $a_i, b_i(i = 1, 2, ...,10)$ are positive integers
2015 May Olympiad, 4
The first $510$ positive integers are written on a blackboard: $1, 2, 3, ..., 510$. An [i]operation [/i] consists of of erasing two numbers whose sum is a prime number. What is the maximum number of operations in a row what can be done? Show how it is accomplished and explain why it can be done in no more operations.
2017 HMNT, 3
Find the number of integers $n$ with $1 \le n \le 2017$ so that $(n-2)(n-0)(n-1)(n-7)$ is an integer
multiple of $1001$.
VMEO II 2005, 3
Given positive integers $a_1$, $a_2$, $...$, $a_m$ ($m \ge 1$). Consider the sequence $\{u_n\}_{n=1}^{\infty}$, with $$u_n = a_1^n + a_2^n + ... + a_m^n.$$ We know that this sequence has a finite number of prime divisors. Prove that $a_1 = a_2 = ...= a_m$.
2018 BMT Spring, Tie 1
Compute the least positive $x$ such that $25x - 6$ is divisible by $1001$.
2000 Mexico National Olympiad, 4
Let $a$ and $b$ be positive integers not divisible by $5$. A sequence of integers is constructed as follows: the first term is $5$, and every consequent term is obtained by multiplying its precedent by $a$ and adding $b$. (For example, if $a = 2$ and $b = 4$, the first three terms are $5,14,32$.) What is the maximum possible number of primes that can occur before encoutering the first composite term?
2002 Manhattan Mathematical Olympiad, 2
Let us consider the sequence $1,2, 3, \ldots , 2002$. Somebody choses $1002$ numbers from the sequence. Prove that there are two of the chosen numbers which are relatively prime (i.e. do not have any common divisors except $1$).
Fractal Edition 1, P4
Let \( P(x) \) be a polynomial with natural coefficients. We denote by \( d(n) \) the number of positive divisors of the natural number \( n \), and by \( \sigma(n) \), the sum of these divisors. The sequence \( a_n \) is defined as follows:
\[
a_{n+1} \in \left\{
\begin{array}{ll}
\sigma(P(d(a_n))) \\
d(P(\sigma(a_n)))
\end{array}
\right.
\]
That is, \( a_{n+1} \) is one of the two terms above. Show that there exists a constant \( C \), depending on \( a_1 \) and \( P(x) \), such that for all \( i \), \( a_i < C \); in other words, show that the sequence \( a_n \) is bounded.
2022 LMT Fall, 1
Let $x$ be the positive integer satisfying $5^2 +28^2 +39^2 = 24^2 +35^2 + x^2$. Find $x$.
2021 Malaysia IMONST 1, 17
Determine the sum of all positive integers $n$ that satisfy the following condition:
when $6n + 1$ is written in base $10$, all its digits are equal.
2016 Tuymaada Olympiad, 5
The ratio of prime numbers $p$ and $q$ does not exceed 2 ($p\ne q$).
Prove that there are two consecutive positive integers such that
the largest prime divisor of one of them is $p$ and that of the other is $q$.
2011 Costa Rica - Final Round, 5
Given positive integers $a,b,c$ which are pairwise relatively prime, show that \[2abc-ab-bc-ac \] is the biggest number that can't be expressed in the form $xbc+yca+zab$ with $x,y,z$ being natural numbers.
1993 Italy TST, 2
Suppose that $p,q$ are prime numbers such that $\sqrt{p^2 +7pq+q^2}+\sqrt{p^2 +14pq+q^2}$ is an integer.
Show that $p = q$.
2004 Hong kong National Olympiad, 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 $h(n)-g(n)$.
1992 Balkan MO, 1
For all positive integers $m,n$ define $f(m,n) = m^{3^{4n}+6} - m^{3^{4n}+4} - m^5 + m^3$. Find all numbers $n$ with the property that $f(m, n)$ is divisible by 1992 for every $m$.
[i]Bulgaria[/i]
1991 Greece National Olympiad, 4
Find all positive intger solutions of $3^x+29=2^y$.
2012 Dutch BxMO/EGMO TST, 5
Let $A$ be a set of positive integers having the following property:
for each positive integer $n$ exactly one of the three numbers $n, 2n$ and $3n$ is an element of $A$.
Furthermore, it is given that $2 \in A$. Prove that $13824 \notin A$.
2022 Azerbaijan Junior National Olympiad, N2
If $x,y,z \in\mathbb{N}$ and $2x^2+3y^3=4z^4$, Prove that $6|x,y,z$
2018 Pan-African Shortlist, A5
Let $g : \mathbb{N} \to \mathbb{N}$ be a function satisfying:
[list]
[*] $g(xy) = g(x)g(y)$ for all $x, y \in \mathbb{N}$,
[*] $g(g(x)) = x$ for all $x \in \mathbb{N}$, and
[*] $g(x) \neq x$ for $2 \leq x \leq 2018$.
[/list]
Find the minimum possible value of $g(2)$.