Found problems: 15460
2000 Moldova National Olympiad, Problem 1
Find all positive integers $a$ for which $a^{2000}-1$ is divisible by $10$.
2024 Kazakhstan National Olympiad, 4
Prove that for any positive integers $a$, $b$, $c$, at least one of the numbers $a^3b+1$, $b^3c+1$, $c^3a+1$ is not divisible by $a^2+b^2+c^2$.
2013 Nordic, 1
Let ${(a_n)_{n\ge1}} $ be a sequence with ${a_1 = 1} $ and ${a_{n+1} = \lfloor a_n +\sqrt{a_n}+\frac{1}{2}\rfloor }$ for all ${n \ge 1}$, where ${\lfloor x \rfloor}$ denotes the greatest integer less than or equal to ${x}$. Find all ${n \le 2013}$ such that ${a_n}$ is a perfect square
2023 Mongolian Mathematical Olympiad, 3
Let $m$ be a positive integer. We say that a sequence of positive integers written on a circle is [i] good [/i], if the sum of any $m$ consecutive numbers on this circle is a power of $m$.
1. Let $n \geq 2$ be a positive integer. Prove that for any [i] good [/i] sequence with $mn$ numbers, we can remove $m$ numbers such that the remaining $mn-m$ numbers form a [i] good [/i] sequence.
2. Prove that in any [i] good [/i] sequence with $m^2$ numbers, we can always find a number that was repeated at least $m$ times in the sequence.
The Golden Digits 2024, P1
Find all functions $f:\mathbb{Z}_{>0}\rightarrow\mathbb{Z}_{>0}$ with the following properties:
1) For every natural number $n\geq 3$, $\gcd(f(n),n)\neq 1$.
2) For every natural number $n\geq 3$, there exists $i_n\in\mathbb{Z}_{>0}$, $1\leq i_n\leq n-1$, such that $f(n)=f(i_n)+f(n-i_n)$.
[i]Proposed by Pavel Ciurea[/i]
2023 Canadian Mathematical Olympiad Qualification, 5
Six decks of $n$ cards, numbered from $1$ to $n$, are given. Melanie arranges each of the decks in some order, such that for any distinct numbers $x$, $y$, and $z$ in $\{1, 2, . . . , n\}$, there is exactly one deck where card $x$ is above card $y$ and card $y$ is above card $z$. Show that there is some $n$ for which Melanie cannot arrange these six decks of cards with this property.
2019 India Regional Mathematical Olympiad, 4
Consider the following $3\times 2$ array formed by using the numbers $1,2,3,4,5,6$,
$$\begin{pmatrix} a_{11}& a_{12}\\a_{21}& a_{22}\\ a_{31}& a_{32}\end{pmatrix}=\begin{pmatrix}1& 6\\2& 5\\ 3& 4\end{pmatrix}.$$
Observe that all row sums are equal, but the sum of the square of the squares is not the same for each row. Extend the above array to a $3\times k$ array $(a_{ij})_{3\times k}$ for a suitable $k$, adding more columns, using the numbers $7,8,9,\dots ,3k$ such that $$\sum_{j=1}^k a_{1j}=\sum_{j=1}^k a_{2j}=\sum_{j=1}^k a_{3j}~~\text{and}~~\sum_{j=1}^k (a_{1j})^2=\sum_{j=1}^k (a_{2j})^2=\sum_{j=1}^k (a_{3j})^2$$
2010 Saudi Arabia IMO TST, 1
Find all real numbers $x$ that can be written as $$x= \frac{a_0}{a_1a_2..a_n}+\frac{a_1}{a_2a_3..a_n}+\frac{a_2}{a_3a_4..a_n}+...+\frac{a_{n-2}}{a_{n-1}a_n}+\frac{a_{n-1}}{a_n}$$
where $n, a_1,a_2,...,a_n$ are positive integers and $1 = a_0 \le a_1 <... < a_n$
2003 German National Olympiad, 6
Prove that there are infinitely many coprime, positive integers $a,b$ such that $a$ divides $b^2 -5$ and $b$ divides $a^2 -5.$
2024 Saint Petersburg Mathematical Olympiad, 6
Polynomial $P(x)$ with integer coefficients is given. For some positive integer $n$ numbers $P(0),P(1),\dots,P(2^n+1)$ are all divisible by $2^{2^n}$. Prove that values of $P(x)$ in all integer points are divisible by $2^{2^n}$.
1968 German National Olympiad, 4
Sixteen natural numbers written in the decimal system may form a geometric sequence, of which the first five members have nine digits, five further members have ten digits, four members have eleven digits and two terms have twelve digits. Prove that there is exactly one sequence with these properties.
2010 Purple Comet Problems, 26
In the coordinate plane a parabola passes through the points $(7,6)$, $(7,12)$, $(18,19)$, and $(18,48)$. The axis of symmetry of the parabola is a line with slope $\tfrac{r}{s}$ where r and s are relatively prime positive integers. Find $r + s$.
2014 Argentine National Olympiad, Level 3, 1.
$201$ positive integers are written on a line, such that both the first one and the last one are equal to $19999$. Each one of the remaining numbers is less than the average of its neighbouring numbers, and the differences between each one of the remaining numbers and the average of its neighbouring numbers are all equal to a unique integer. Find the second-to-last term on the line.
2022 Harvard-MIT Mathematics Tournament, 4
Compute the sum of all 2-digit prime numbers $p$ such that there exists a prime number $q$ for which $100q + p$ is a perfect square.
2021 Malaysia IMONST 1, 10
Determine the number of integer solutions $(x, y, z)$, with $0 \le x, y, z \le 100$, for the equation $$(x - y)^2 + (y + z)^2 = (x + y)^2 + (y - z)^2.$$
LMT Team Rounds 2021+, 7
How many $2$-digit factors does $555555$ have?
2021 Junior Balkan Team Selection Tests - Moldova, 7
Determine all pairs of integer numbers $(a, b)$ that satisfy the relation:
$$(a + b)(a + b + 6) = 34 \cdot 3^{|2a-b|}- 7$$
2006 Kyiv Mathematical Festival, 5
See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url]
Let $a, b, c, d$ be positive integers and $p$ be prime number such that $a^2+b^2=p$ and $c^2+d^2$ is divisible by $p.$ Prove that there exist positive integers $e$ and $f$ such that $e^2+f^2=\frac{c^2+d^2}{p}.$
2011 Princeton University Math Competition, A6
Let $a$ and $b$ be positive integers such that $a + bz = x^3 + y^4$ has no solutions for any integers $x, y, z$, with $b$ as small as possible, and $a$ as small as possible for the minimum $b$. Find $ab$.
2024 Iran MO (3rd Round), 4
For a given positive integer number $n$ find all subsets $\{r_0,r_1,\cdots,r_n\}\subset \mathbb{N}$ such that
$$
n^n+n^{n-1}+\cdots+1 | n^{r_n}+\cdots+ n^{r_0}.
$$
Proposed by [i]Shayan Tayefeh[/i]
2009 Croatia Team Selection Test, 4
Determine all triplets off positive integers $ (a,b,c)$ for which $ \mid2^a\minus{}b^c\mid\equal{}1$
2013 Gheorghe Vranceanu, 1
Find all natural numbers $ a,b $ such that $ a^3+b^3 $ a power of $3.$
2021 Bangladeshi National Mathematical Olympiad, 2
Let $x$ and $y$ be positive integers such that $2(x+y)=gcd(x,y)+lcm(x,y)$. Find $\frac{lcm(x,y)}{gcd(x,y)}$.
2024 Saint Petersburg Mathematical Olympiad, 5
$2 \ 000 \ 000$ points with integer coordinates are marked on the numeric axis. Segments of lengths $97$, $100$ and $103$ with ends at these points are considered. What is the largest number of such segments?
2009 China Team Selection Test, 1
Let $ a > b > 1, b$ is an odd number, let $ n$ be a positive integer. If $ b^n|a^n\minus{}1,$ then $ a^b > \frac {3^n}{n}.$