Found problems: 15460
2002 Baltic Way, 18
Find all integers $n>1$ such that any prime divisor of $n^6-1$ is a divisor of $(n^3-1)(n^2-1)$.
2004 Austria Beginners' Competition, 1
Find the smallest four-digit number that when divided by $3$ gives a four-digit number with the same digits.
(Note: Four digits means that the thousand Unit digit must not be $0$.)
1981 Austrian-Polish Competition, 1
Find the smallest $n$ for which we can find $15$ distinct elements $a_{1},a_{2},...,a_{15}$ of $\{16,17,...,n\}$ such that $a_{k}$ is a multiple of $k$.
2023 Romania Team Selection Test, P3
Given a positive integer $a,$ prove that $n!$ is divisible by $n^2 + n + a$ for infinitely many positive integers $n.{}$
[i]Proposed by Andrei Bâra[/i]
2021 CMIMC, 2.3 1.1
How many multiples of $12$ divide $12!$ and have exactly $12$ divisors?
[i]Proposed by Adam Bertelli[/i]
2020 China Girls Math Olympiad, 6
Let $p, q$ be integers and $p, q > 1$ , $gcd(p, \,6q)=1$. Prove that:$$\sum_{k=1}^{q-1}\left \lfloor \frac{pk}{q}\right\rfloor^2 \equiv 2p \sum_{k=1}^{q-1}k\left\lfloor \frac{pk}{q} \right\rfloor (mod \, q-1)$$
2000 AIME Problems, 4
The diagram shows a rectangle that has been dissected into nine non-overlapping squares. Given that the width and the height of the rectangle are relatively prime positive integers, find the perimeter of the rectangle.
[asy]
defaultpen(linewidth(0.7));
draw((0,0)--(69,0)--(69,61)--(0,61)--(0,0));draw((36,0)--(36,36)--(0,36));
draw((36,33)--(69,33));draw((41,33)--(41,61));draw((25,36)--(25,61));
draw((34,36)--(34,45)--(25,45));
draw((36,36)--(36,38)--(34,38));
draw((36,38)--(41,38));
draw((34,45)--(41,45));[/asy]
2014 Macedonia National Olympiad, 2
Solve the following equation in $\mathbb{Z}$:
\[3^{2a + 1}b^2 + 1 = 2^c\]
2015 All-Russian Olympiad, 5
An immortal flea jumps on whole points of the number line, beginning with $0$. The length of the first jump is $3$, the second $5$, the third $9$, and so on. The length of $k^{\text{th}}$ jump is equal to $2^k + 1$. The flea decides whether to jump left or right on its own. Is it possible that sooner or later the flee will have been on every natural point, perhaps having visited some of the points more than once?
2000 Tournament Of Towns, 2
Positive integers $a, b, c, d$ satisfy the inequality $ad - bc > 1$. Prove that at least one of the numbers $a, b, c, d$ is not divisible by $ad - bc$.
(A Spivak)
1999 Greece JBMO TST, 1
A circle is divided in $100$ equal parts and the points of this division are colored green or yellow, such that when between two points of division $A,B$ there are exactly $4$ division points and the point $A$ is green, then the point $B$ shall be yellow. Which points are more, the green or the yellow ones?
PEN C Problems, 4
Let $M$ be an integer, and let $p$ be a prime with $p>25$. Show that the set $\{M, M+1, \cdots, M+ 3\lfloor \sqrt{p} \rfloor -1\}$ contains a quadratic non-residue to modulus $p$.
2014 AIME Problems, 8
The positive integers $N$ and $N^2$ both end in the same sequence of four digits $abcd$ when written in base 10, where digit $a$ is not zero. Find the three-digit number $abc$.
2022 May Olympiad, 2
Bob chose six of the nine digits from $1$ to $9$ and wrote the list, ordered from smallest to largest, of all three-digit numbers that can be formed using the digits you chose. At Bob's list, the number $317$ appears at position $22$. What number appears at position $60$ in the list from Bob? Find all possibilities.
BIMO 2022, 2
Given a four digit string $ k=\overline{abcd} $, $ a, b, c, d\in \{0, 1, \cdots, 9\} $, prove that there exist a $n<20000$ such that $2^n$ contains $k$ as a substring when written in base $10$.
[Extra: Can you give a better bound? Mine is $12517$]
2019 BMT Spring, 10
Compute the remainder when the product of all positive integers less than and relatively prime to $2019$ is divided by $2019$.
2021 China Team Selection Test, 4
Proof that
$$ \sum_{m=1}^n5^{\omega (m)} \le \sum_{k=1}^n\lfloor \frac{n}{k} \rfloor \tau (k)^2 \le \sum_{m=1}^n5^{\Omega (m)} .$$
2021 China Team Selection Test, 2
Given distinct positive integer $ a_1,a_2,…,a_{2020} $. For $ n \ge 2021 $, $a_n$ is the smallest number different from $a_1,a_2,…,a_{n-1}$ which doesn't divide $a_{n-2020}...a_{n-2}a_{n-1}$. Proof that every number large enough appears in the sequence.
2016 Japan MO Preliminary, 11
How many pairs $(a, b)$ for integers $a, b \ge 2$ which exist the sequence $x_1, x_2, . . . , x_{1000}$ which satisfy conditions as below?
1.Terms $x_1, x_2, . . . , x_{1000}$ are sorting of $1, 2, . . . , 1000$.
2.For each integers $1 \le i < 1000$, the sequence forms $x_{i+1} = x_i + a$ or $x_{i+1} = x_i - b$.
2023 BMT, 11
Compute the sum of all positive integers $n$ for which there exists a real number $x$ satisfying
$$\left(x +\frac{n}{x} \right)^n= 2^{20}.$$
2010 Brazil Team Selection Test, 2
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]
2018 Belarusian National Olympiad, 9.2
For every integer $n\geqslant2$ prove the inequality
$$
\frac{1}{2!}+\frac{2}{3!}+\ldots+\frac{2^{n-2}}{n!}\leqslant\frac{3}{2},
$$
where $k!=1\cdot2\cdot\ldots\cdot k$.
1984 IMO, 2
Find one pair of positive integers $a,b$ such that $ab(a+b)$ is not divisible by $7$, but $(a+b)^7-a^7-b^7$ is divisible by $7^7$.
2022 Junior Balkan Team Selection Tests - Moldova, 5
Determine all nonzero natural numbers $n$, for which the number $\sqrt{n! + 5}$ is a natural number.
2011 Saudi Arabia Pre-TST, 3.4
Find all quadruples $(x,y,z,w)$ of integers satisfying the system of equations
$$x + y + z + w = xy + yz + zx + w^2 - w = xyz - w^3 = - 1$$