Found problems: 15460
2011 USA TSTST, 3
Prove that there exists a real constant $c$ such that for any pair $(x,y)$ of real numbers, there exist relatively prime integers $m$ and $n$ satisfying the relation
\[
\sqrt{(x-m)^2 + (y-n)^2} < c\log (x^2 + y^2 + 2).
\]
2016 IFYM, Sozopol, 5
Find all pairs of integers $(x,y)$ for which $x^z+z^x=(x+z)!$.
2014 German National Olympiad, 2
For a positive integer $n$, let $y_n$ be the number of $n$-digit positive integers containing only the digits $2,3,5, 7$ and which do not have a $5$ directly to the right of a $2.$ If $r\geq 1$ and $m\geq 2$ are integers, prove that $y_{m-1}$ divides $y_{rm-1}.$
2021 Malaysia IMONST 1, 7
Sofia has forgotten the passcode of her phone. She only remembers that it has four digits and that the product of its digits is $18$. How many passcodes satisfy these conditions?
2021 JHMT HS, 3
Let $B=\{2^1,2^2,2^3,\dots,2^{21}\}.$ Find the remainder when
\[ \sum_{m, n \in B: \ m<n}\gcd(m,n) \]
is divided by $1000,$ where the sum is taken over all pairs of elements $(m,n)$ of $B$ such that $m<n.$
2023 Singapore Senior Math Olympiad, 4
Find all positive integers $m, n$ satisfying $n!+2^{n-1}=2^m$.
2022 Balkan MO Shortlist, N4
A hare and a tortoise run in the same direction, at constant but different speeds, around the base of a tall square tower. They start together at the same vertex, and the run ends when both return to the initial vertex simultaneously for the first time. Suppose the hare runs with speed 1, and the tortoise with speed less than 1. For what rational numbers $q{}$ is it true that, if the tortoise runs with speed $q{}$, the fraction of the entire run for which the tortoise can see the hare is also $q{}$?
2023 LMT Fall, 1
George has $150$ cups of flour and $200$ eggs. He can make a cupcake with $3$ cups of flour and $2$ eggs, or he can make an omelet with $4$ eggs. What is the maximum number of treats (both omelets and cupcakes) he canmake?
2021 China Team Selection Test, 3
Given positive integers $a,b,c$ which are pairwise coprime. Let $f(n)$ denotes the number of the non-negative integer solution $(x,y,z)$ to the equation
$$ax+by+cz=n.$$
Prove that there exists constants $\alpha, \beta, \gamma \in \mathbb{R}$ such that for any non-negative integer $n$,
$$|f(n)- \left( \alpha n^2+ \beta n + \gamma \right) | < \frac{1}{12} \left( a+b+c \right).$$
1908 Eotvos Mathematical Competition, 2
Let $n$ be an integer greater than $2$. Prove that the $n$th power of the length of the hypotenuse of a right triangle is greater than the sum of the $n$th powers of the lengths of the legs.
1970 All Soviet Union Mathematical Olympiad, 137
Prove that from every set of $200$ integers you can choose a subset of $100$ with the total sum divisible by $100$.
2023 India National Olympiad, 3
Let $\mathbb N$ denote the set of all positive integers. Find all real numbers $c$ for which there exists a function $f:\mathbb N\to \mathbb N$ satisfying:
[list]
[*] for any $x,a\in\mathbb N$, the quantity $\frac{f(x+a)-f(x)}{a}$ is an integer if and only if $a=1$;
[*] for all $x\in \mathbb N$, we have $|f(x)-cx|<2023$.
[/list]
[i]Proposed by Sutanay Bhattacharya[/i]
2014 Contests, 1
Determine all triples $(a,b,c)$, where $a, b$, and $c$ are positive integers that satisfy
$a \le b \le c$ and $abc = 2(a + b + c)$.
2011 Philippine MO, 3
The $2011$th prime number is $17483$ and the next prime is $17489$.
Does there exist a sequence of $2011^{2011}$ consecutive positive integers that contain exactly $2011$ prime numbers?
2006 Junior Balkan Team Selection Tests - Romania, 2
Consider the integers $a_1, a_2, a_3, a_4, b_1, b_2, b_3, b_4$ with $a_k \ne b_k$ for all $k = 1, 2, 3, 4$. If
$\{a_1, b_1\} + \{a_2, b_2\} = \{a_3, b_3\} + \{a_4, b_4\}$, show that the number $|(a_1 - b_1)(a_2 - b_2)(a_3 - b_3)(a_4 - b_4)|$ is a square.
Note. For any sets $A$ and $B$, we denote $A + B = \{x + y | x \in A, y \in B\}$.
2024 Greece Junior Math Olympiad, 4
Prove that there are infinite triples of positive integers $(x,y,z)$ such that
$$x^2+y^2+z^2+xy+yz+zx=6xyz.$$
1991 Nordic, 1
Determine the last two digits of the number $2^5 + 2^{5^{2}} + 2^{5^{3}} +... + 2^{5^{1991}}$ , written in decimal notation.
1991 ITAMO, 3
We consider the sums of the form $\pm 1 \pm 4 \pm 9\pm ... \pm n^2$. Show that every integer can be represented in this form for some $n$. (For example, $3 = -1 + 4$ and $8 = 1-4-9+16+25-36-49+64$.)
2022 LMT Fall, 1
Let $x$ be the positive integer satisfying $5^2 +28^2 +39^2 = 24^2 +35^2 + x^2$. Find $x$.
2002 Estonia National Olympiad, 3
The teacher writes a $2002$-digit number consisting only of digits $9$ on the blackboard.
The first student factors this number as $ab$ with $a > 1$ and $b > 1$ and replaces it on the blackboard by two numbers $a'$ and $b'$ with $|a-a'| = |b-b'| = 2$. The second student chooses one of the numbers on the blackboard, factors it as $cd$ with $c > 1$ and $d > 1$ and replaces the chosen number by two numbers $c'$ and $d'$ with $|c-c'| = |d-d'| = 2$, etc. Is it possible that after a certain number of students have been to the blackboard all numbers written there are equal to $9$?
2010 Postal Coaching, 2
Find all non-negative integers $m,n,p,q$ such that \[ p^mq^n = (p+q)^2 +1 . \]
2019 Czech and Slovak Olympiad III A, 6
Assume we can fill a table $n\times n$ with all numbers $1,2,\ldots,n^2-1,n^2$ in such way that arithmetic means of numbers in every row and every column is an integer. Determine all such positive integers $n$.
2012 Uzbekistan National Olympiad, 2
For any positive integers $n$ and $m$ satisfying the equation $n^3+(n+1)^3+(n+2)^3=m^3$, prove that $4\mid n+1$.
1976 Swedish Mathematical Competition, 6
Show that there are only finitely many integral solutions to
\[
3^m - 1 = 2^n
\]
and find them.
II Soros Olympiad 1995 - 96 (Russia), 9.7
$300$ people took part in the drawing for the main prize of the television lottery. They lined up in a circle, then, starting with someone who received number $1$, they began to count them. Moreover, every third person dropped out every time. (So, in the first round, everyone with numbers divisible by $3$ dropped out). The counting continued until there was only one person left. (It is clear that more than one circle was made). This person received the main prize. (It “accidentally” turned out to be the TV director’s mother-in-law). What number did this person have in the initial lineup?