Found problems: 15460
1998 South africa National Olympiad, 1
Is $\log_{10}{8}$ rational?
2015 USA Team Selection Test, 2
Prove that for every $n\in \mathbb N$, there exists a set $S$ of $n$ positive integers such that for any two distinct $a,b\in S$, $a-b$ divides $a$ and $b$ but none of the other elements of $S$.
[i]Proposed by Iurie Boreico[/i]
2002 Tournament Of Towns, 2
John and Mary select a natural number each and tell that to Bill. Bill wrote their sum and product in two papers hid one paper and showed the other to John and Mary.
John looked at the number (which was $2002$ ) and declared he couldn't determine Mary's number. Knowing this Mary also said she couldn't determine John's number as well.
What was Mary's Number?
2005 MOP Homework, 4
Let $p$ be an odd prime. Prove that \[\sum^{p-1}_{k=1} k^{2p-1} \equiv \frac{p(p+1)}{2}\pmod{p^2}.\]
2004 Estonia Team Selection Test, 5
Find all natural numbers $n$ for which the number of all positive divisors of the number lcm $(1,2,..., n)$ is equal to $2^k$ for some non-negative integer $k$.
2004 Romania Team Selection Test, 13
Let $m\geq 2$ be an integer. A positive integer $n$ has the property that for any positive integer $a$ coprime with $n$, we have $a^m - 1\equiv 0 \pmod n$.
Prove that $n \leq 4m(2^m-1)$.
Created by Harazi, modified by Marian Andronache.
2011 JBMO Shortlist, 1
Solve in positive integers the equation $1005^x + 2011^y = 1006^z$.
1980 Yugoslav Team Selection Test, Problem 3
A sequence $(x_n)$ satisfies $x_{n+1}=\frac{x_n^2+a}{x_{n-1}}$ for all $n\in\mathbb N$. Prove that if $x_0,x_1$, and $\frac{x_0^2+x_1^2+a}{x_0x_1}$ are integers, then all the terms of sequence $(x_n)$ are integers.
2017 NIMO Problems, 4
For how many positive integers $100 < n \le 10000$ does $\lfloor \sqrt{n-100} \rfloor$ divide $n$?
[i]Proposed by Michael Tang[/i]
1949 Moscow Mathematical Olympiad, 158
a) Prove that $x^2 + y^2 + z^2 = 2xyz$ for integer $x, y, z$ only if $x = y = z = 0$.
b) Find integers $x, y, z, u$ such that $x^2 + y^2 + z^2 + u^2 = 2xyzu$.
2007 IMO Shortlist, 2
Let $b,n > 1$ be integers. Suppose that for each $k > 1$ there exists an integer $a_k$ such that $b - a^n_k$ is divisible by $k$. Prove that $b = A^n$ for some integer $A$.
[i]Author: Dan Brown, Canada[/i]
1991 IMTS, 4
Let $n$ points with integer coordinates be given in the $xy$-plane. What is the minimum value of $n$ which will ensure that three of the points are the vertices of a triangel with integer (possibly, 0) area?
2024 Kyiv City MO Round 1, Problem 2
Write the numbers from $1$ to $16$ in the cells of a of a $4 \times 4$ square so that:
1. Each cell contains exactly one number;
2. Each number is written exactly once;
3. For any two cells that are symmetrical with respect to any of the perpendicular bisectors of sides of the
original $4 \times 4$ square, the sum of numbers in them is a prime number
The figure below shows examples of such pairs of cells, sums of numbers in which have to be prime.
[img]https://i.ibb.co/fqX05dY/Kyiv-MO-2024-Round-1-8-2.png[/img]
[i]Proposed by Mykhailo Shtandenko[/i]
2017 Germany, Landesrunde - Grade 11/12, 4
Find the smallest positive integer $n$ that is divisible by $100$ and has exactly $100$ divisors.
2014 Contests, 3
Give are the integers $a_{1}=11 , a_{2}=1111, a_{3}=111111, ... , a_{n}= 1111...111$( with $2n$ digits) with $n > 8$ .
Let $q_{i}= \frac{a_{i}}{11} , i= 1,2,3, ... , n$ the remainder of the division of $a_{i}$ by$ 11$ .
Prove that the sum of nine consecutive quotients: $s_{i}=q_{i}+q_{i+1}+q_{i+2}+ ... +q_{i+8}$ is a multiple of $9$ for any $i= 1,2,3, ... , (n-8)$
2014 Contests, 1
Four consecutive three-digit numbers are divided respectively by four consecutive two-digit numbers. What minimum number of different remainders can be obtained?
[i](A. Golovanov)[/i]
2016 VJIMC, 2
Find all positive integers $n$ such that $\varphi(n)$ divides $n^2 + 3$.
1994 IMO Shortlist, 1
$ M$ is a subset of $ \{1, 2, 3, \ldots, 15\}$ such that the product of any three distinct elements of $ M$ is not a square. Determine the maximum number of elements in $ M.$
2016 CMIMC, 3
Let $\{x\}$ denote the fractional part of $x$. For example, $\{5.5\}=0.5$. Find the smallest prime $p$ such that the inequality \[\sum_{n=1}^{p^2}\left\{\dfrac{n^p}{p^2}\right\}>2016\] holds.
2018 PUMaC Live Round, 4.1
The number $400000001$ can be written as $p\cdot q$, where $p$ and $q$ are prime numbers. Find the sum of the prime factors of $p+q-1$.
1986 Canada National Olympiad, 4
For all positive integers $n$ and $k$, define $F(n,k) = \sum_{r = 1}^n r^{2k - 1}$. Prove that $F(n,1)$ divides $F(n,k)$.
2019 Swedish Mathematical Competition, 6
Is there an infinite sequence of positive integers $\{a_n\}_{n = 1}^{\infty}$ which contains each positive integer exactly once and is such that the number $a_n + a_{n + 1} $ is a perfect square for each $n$?
2020 Purple Comet Problems, 19
Find the least prime number greater than $1000$ that divides $2^{1010} \cdot 23^{2020} + 1$.
2021 CHMMC Winter (2021-22), 2
For any positive integer $n$, let $p(n)$ be the product of its digits in base-$10$ representation. Find the maximum possible value of $\frac{p(n)}{n}$ over all integers $n \ge 10$.
2017 Azerbaijan Junior National Olympiad, P2
For all $n>1$ let $f(n)$ be the sum of the smallest factor of $n$ that is not 1 and $n$ . The computer prints $f(2),f(3),f(4),...$ with order:$4,6,6,...$ ( Because $f(2)=2+2=4,f(3)=3+3=6,f(4)=4+2=6$ etc.). In this infinite sequence, how many times will be $ 2015$ and $ 2016$ written? (Explain your answer)