Found problems: 15460
2014 Contests, 2
Prove that among any $16$ perfect cubes we can always find two cubes whose difference is divisible by $91$.
2019 Brazil Team Selection Test, 5
Four positive integers $x,y,z$ and $t$ satisfy the relations
\[ xy - zt = x + y = z + t. \]
Is it possible that both $xy$ and $zt$ are perfect squares?
2007 Korea Junior Math Olympiad, 1
A sequence $a_1,a_2,...,a_{2007}$ where $a_i \in\{2,3\}$ for $i = 1,2,...,2007$ and an integer sequence $x_1,x_2,...,x_{2007}$ satis fies the following: $a_ix_i + x_{i+2 }\equiv 0$ ($mod 5$) , where the indices are taken modulo $2007$. Prove that $x_1,x_2,...,x_{2007}$ are all multiples of $5$.
1991 Tournament Of Towns, (319) 6
An arithmetical progression (whose difference is not equal to zero) consists of natural numbers without any nines in its decimal notation.
(a) Prove that the number of its terms is less than $100$.
(b) Give an example of such a progression with $72$ terms.
(c) Prove that the number of terms in any such progression does not exceed $72$.
(V. Bugaenko and Tarasov, Moscow)
2012 NIMO Problems, 7
For how many positive integers $n \le 500$ is $n!$ divisible by $2^{n-2}$?
[i]Proposed by Eugene Chen[/i]
2000 Croatia National Olympiad, Problem 2
Find all $5$-tuples of different four-digit integers with the same initial digit such that the sum of the five numbers is divisible by four of them.
2011 Indonesia TST, 4
Prove that there exists infinitely many positive integers $n$ such that $n^2+1$ has a prime divisor greater than $2n+\sqrt{5n+2011}$.
2016 Polish MO Finals, 1
Let $p$ be a certain prime number. Find all non-negative integers $n$ for which polynomial $P(x)=x^4-2(n+p)x^2+(n-p)^2$ may be rewritten as product of two quadratic polynomials $P_1, \ P_2 \in \mathbb{Z}[X]$.
2021 Malaysia IMONST 1, 9
Find the sum of (decimal) digits of the number $(10^{2021} + 2021)^2$?
2023 Belarusian National Olympiad, 8.7
A sequence $(a_n)$ positive integers is determined by equalities $a_1=20,a_2=22$ and $a_{n+1}=4a_n^2+5a_{n-1}^3$ for all $n \geq 2$.
Find the maximum power of two which divides $a_{2023}$.
2018 India IMO Training Camp, 3
Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both
$$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$
are integers.
2009 Dutch Mathematical Olympiad, 1
In this problem, we consider integers consisting of $5$ digits, of which the rst and last one are nonzero. We say that such an integer is a palindromic product if it satis es the following two conditions:
- the integer is a palindrome, (i.e. it doesn't matter if you read it from left to right, or the other way around);
- the integer is a product of two positive integers, of which the fi rst, when read from left to right, is equal to the second, when read from right to left, like $4831$ and $1384$.
For example, $20502$ is a palindromic product, since $102 \cdot 201 = 20502$, and $20502$ itself is a palindrome.
Determine all palindromic products of $5$ digits.
1966 Polish MO Finals, 1
Solve in integers the equation $$x^4 +4y^4 = 2(z^4 +4u^4)$$
1945 Moscow Mathematical Olympiad, 093
Find all two-digit numbers $\overline {ab}$ such that $\overline {ab} + \overline {ba}$ is a perfect square.
1978 IMO Shortlist, 5
For every integer $d \geq 1$, let $M_d$ be the set of all positive integers that cannot be written as a sum of an arithmetic progression with difference $d$, having at least two terms and consisting of positive integers. Let $A = M_1$, $B = M_2 \setminus \{2 \}, C = M_3$. Prove that every $c \in C$ may be written in a unique way as $c = ab$ with $a \in A, b \in B.$
2003 India IMO Training Camp, 2
Find all triples $(a,b,c)$ of positive integers such that
(i) $a \leq b \leq c$;
(ii) $\text{gcd}(a,b,c)=1$; and
(iii) $a^3+b^3+c^3$ is divisible by each of the numbers $a^2b, b^2c, c^2a$.
2017 Lusophon Mathematical Olympiad, 4
Find how many multiples of 360 are of the form $\overline{ab2017cd}$, where a, b, c, d are digits, with a > 0.
2020 CCA Math Bonanza, I13
Let $n$ be a positive integer. Compute, in terms of $n$, the number of sequences $(x_1,\ldots,x_{2n})$ with each $x_i\in\{0,1,2,3,4\}$ such that $x_1^2+\dots+x_{2n}^2$ is divisible by $5$.
[i]2020 CCA Math Bonanza Individual Round #13[/i]
2018 German National Olympiad, 5
We define a sequence of positive integers $a_1,a_2,a_3,\dots$ as follows: Let $a_1=1$ and iteratively, for $k =2,3,\dots$ let $a_k$ be the largest prime factor of $1+a_1a_2\cdots a_{k-1}$. Show that the number $11$ is not an element of this sequence.
2008 JBMO Shortlist, 10
Prove that $2^n + 3^n$ is not a perfect cube for any positive integer $n$.
2005 Croatia National Olympiad, 3
Show that there is a unique positive integer which consists of the digits $2$ and $5$, having $2005$ digits and divisible by $2^{2005}$.
2017 Greece JBMO TST, 3
Prove that for every positive integer $n$, the number $A_n = 7^{2n} -48n - 1$ is a multiple of $9$.
2017 Estonia Team Selection Test, 1
Do there exist two positive powers of $5$ such that the number obtained by writing one after the other is also a power of $5$?
2006 Thailand Mathematical Olympiad, 10
Find the remainder when $26!^{26} + 27!^{27}$ is divided by $29$.
Russian TST 2016, P1
Find all natural $n{}$ such that for every natural $a{}$ that is mutually prime with $n{}$, the number $a^n - 1$ is divisible by $2n^2$.