Found problems: 15460
2005 Estonia National Olympiad, 3
How many such four-digit natural numbers divisible by $7$ exist such when changing the first and last number we also get a four-digit divisible by $7$?
1990 Romania Team Selection Test, 1
Let a,b,n be positive integers such that $(a,b) = 1$.
Prove that if $(x,y)$ is a solution of the equation $ax+by = a^n + b^n$ then
$$\left[\frac{x}{b}\right]+\left[\frac{y}{a}\right]=\left[\frac{a^{n-1}}{b}\right]+\left[\frac{b^{n-1}}{a}\right]$$
2018 Czech-Polish-Slovak Junior Match, 1
For natural numbers $a, b c$ it holds that $(a + b + c)^2 | ab (a + b) + bc (b + c) + ca(c + a) + 3abc$.
Prove that $(a + b + c) |(a - b)^2 + (b - c)^2 + (c - a)^2$
2004 Czech-Polish-Slovak Match, 2
Show that for each natural number $k$ there exist only finitely many triples $(p, q, r)$ of distinct primes for which $p$ divides $qr-k$, $q$ divides $pr-k$, and $r$ divides $pq - k$.
2003 Romania Team Selection Test, 10
Let $\mathcal{P}$ be the set of all primes, and let $M$ be a subset of $\mathcal{P}$, having at least three elements, and such that for any proper subset $A$ of $M$ all of the prime factors of the number $ -1+\prod_{p\in A}p$ are found in $M$. Prove that $M= \mathcal{P}$.
[i]Valentin Vornicu[/i]
2014 VTRMC, Problem 3
Find the least positive integer $n$ such that $2^{2014}$ divides $19^n-1$.
2024 ELMO Shortlist, N1
Find all pairs $(n,d)$ of positive integers such that $d\mid n^2$ and $(n-d)^2<2d$.
[i]Linus Tang[/i]
2000 AIME Problems, 11
Let $S$ be the sum of all numbers of the form $a/b,$ where $a$ and $b$ are relatively prime positive divisors of $1000.$ What is the greatest integer that does not exceed $S/10?$
2011 Greece National Olympiad, 1
Solve in integers the equation
\[{x^3}{y^2}\left( {2y - x} \right) = {x^2}{y^4} - 36\]
2008 Saint Petersburg Mathematical Olympiad, 4
There are $100$ numbers on circle, and no one number is divided by other. In same time for all numbers we make next operation:
If $(a,b)$ are two neighbors ($a$ is left neighbor) , then we write between $a,b$ number $\frac{a}{(a,b)}$ and erase $a,b$
This operation was repeated some times. What maximum number of $1$ we can receive ?
Example: If we have circle with $3$ numbers $4,5,6$ then after operation we receive circle with numbers $\frac{4}{(4,5)}=4,\frac{5}{(5,6)}=5, \frac{6}{(6,4)}=3$.
2019 Durer Math Competition Finals, 2
Anne multiplies each two-digit number by $588$ in turn, and writes down the so-obtained products. How many perfect squares does she write down?
2017 Hanoi Open Mathematics Competitions, 4
Put $S = 2^1 + 3^5 + 4^9 + 5^{13} + ... + 505^{2013} + 506^{2017}$. The last digit of $S$ is
(A): $1$ (B): $3$ (C): $5$ (D): $7$ (E): None of the above.
2016 Switzerland Team Selection Test, Problem 7
Find all positive integers $n$ such that $$\sum_{d|n, 1\leq d <n}d^2=5(n+1)$$
2013 Online Math Open Problems, 19
Let $\sigma(n)$ be the number of positive divisors of $n$, and let $\operatorname{rad} n$ be the product of the distinct prime divisors of $n$. By convention, $\operatorname{rad} 1 = 1$. Find the greatest integer not exceeding \[ 100\left(\sum_{n=1}^{\infty}\frac{\sigma(n)\sigma(n \operatorname{rad} n)}{n^2\sigma(\operatorname{rad} n)}\right)^{\frac{1}{3}}. \][i]Proposed by Michael Kural[/i]
2022 Regional Competition For Advanced Students, 4
We are given the set $$M = \{-2^{2022}, -2^{2021}, . . . , -2^{2}, -2, -1, 1, 2, 2^2, . . . , 2^{2021}, 2^{2022}\}.$$
Let $T$ be a subset of $M$, such that neighbouring numbers have the same difference when the elements are ordered by size.
(a) Determine the maximum number of elements that such a set $T$ can contain.
(b) Determine all sets $T$ with the maximum number of elements.
[i](Walther Janous)[/i]
2017 Thailand Mathematical Olympiad, 5
Does there exist $2017$ consecutive positive integers, none of which could be written as $a^2 + b^2$ for some integers $a, b$? Justify your answer.
VMEO III 2006, 12.2
Find all positive integers $(m, n)$ that satisfy $$m^2 =\sqrt{n} +\sqrt{2n + 1}.$$
2020 Taiwan TST Round 2, 5
A finite set $K$ consists of at least 3 distinct positive integers. Suppose that $K$ can be partitioned into two nonempty subsets $A,B\in K$ such that $ab+1$ is always a perfect square whenever $a\in A$ and $b\in B$. Prove that
\[\max_{k\in K}k\geq \left\lfloor (2+\sqrt{3})^{\min\{|A|,|B|\}-1}\right\rfloor+1,\]where $|X|$ stands for the cartinality of the set $X$, and for $x\in \mathbb{R}$, $\lfloor x\rfloor$ is the greatest integer that does not exceed $x$.
2021 Brazil EGMO TST, 2
Let $a,b,k$ be positive integers such that
$gcd(a,b)^2+lcm(a,b)^2+a^2b^2=2020^k$
Prove that $k$ is an even number.
2017 May Olympiad, 5
We will say that two positive integers $a$ and $b$ form a [i]suitable pair[/i] if $a+b$ divides $ab$ (its sum divides its multiplication). Find $24$ positive integers that can be distribute into $12$ suitable pairs, and so that each integer number appears in only one pair and the largest of the $24$ numbers is as small as possible.
2013 IFYM, Sozopol, 3
Let $a$ and $b$ be two distinct natural numbers. It is known that $a^2+b|b^2+a$ and that $b^2+a$ is a power of a prime number. Determine the possible values of $a$ and $b$.
2021 Romania Team Selection Test, 1
Find all pairs $(m,n)$ of positive odd integers, such that $n \mid 3m+1$ and $m \mid n^2+3$.
2021 IMO Shortlist, N4
Let $r>1$ be a rational number. Alice plays a solitaire game on a number line. Initially there is a red bead at $0$ and a blue bead at $1$. In a move, Alice chooses one of the beads and an integer $k \in \mathbb{Z}$. If the chosen bead is at $x$, and the other bead is at $y$, then the bead at $x$ is moved to the point $x'$ satisfying $x'-y=r^k(x-y)$.
Find all $r$ for which Alice can move the red bead to $1$ in at most $2021$ moves.
2019 LIMIT Category A, Problem 12
What is the smallest positive integer $n$ such that $n=x^3+y^3$ for two different positive integer tuples $(x,y)$?
2007 Postal Coaching, 3
Suppose $n$ is a natural number such that $4^n + 2^n + 1$ is a prime. Prove that $n = 3^k$ for some nonnegative integer $k$.