Found problems: 15460
2017 Moscow Mathematical Olympiad, 11
There is one nonzero digit in every cell of $2017\times 2017 $ table.
On the board we writes $4034$ numbers that are rows and columns of table. It is known, that $4033$ numbers are divisible by prime $p$ and last is not divisible by $p$. Find all possible values of $p$.
[hide=Example]Example for $2\times2$. If table is
|1|4|
|3|7|.
Then numbers on board are $14,37,13,47$[/hide]
2021 Peru IMO TST, P1
For any positive integer $n$, we define $S(n)$ to be the sum of its digits in the decimal representation. Prove that for any positive integer $m$, there exists a positive integer $n$ such that $S(n)-S(n^2)>m$.
2021 Purple Comet Problems, 10
Find the value of $n$ such that the two inequalities
$$|x + 47| \le n \,\,\, and \,\,\, \frac{1}{17} \le \frac{4}{3 - x} \le \frac{1}{8}$$
have the same solutions.
2023 Austrian Junior Regional Competition, 4
Determine all triples $(a, b, c)$ of positive integers such that
$$a! + b! = 2^{c!}.$$
[i](Walther Janous)[/i]
2023 Germany Team Selection Test, 2
A number is called [i]Norwegian[/i] if it has three distinct positive divisors whose sum is equal to $2022$. Determine the smallest Norwegian number.
(Note: The total number of positive divisors of a Norwegian number is allowed to be larger than $3$.)
2013 Romanian Masters In Mathematics, 1
For a positive integer $a$, define a sequence of integers $x_1,x_2,\ldots$ by letting $x_1=a$ and $x_{n+1}=2x_n+1$ for $n\geq 1$. Let $y_n=2^{x_n}-1$. Determine the largest possible $k$ such that, for some positive integer $a$, the numbers $y_1,\ldots,y_k$ are all prime.
2023 May Olympiad, 1
At Easter Day, $4$ children and their mothers participated in a game in which they had to find hidden chocolate eggs. Augustine found $4$ eggs, Bruno found $6$, Carlos found $9$ and Daniel found $12$. Mrs. Gómez found the same number of eggs as her son, Mrs. Junco found twice as many eggs as her son, Mrs. Messi found three times as many eggs as her son, and Mrs. Núñez found five times as many eggs as her son. At the end of the day, they put all the eggs in boxes, with $18$ eggs in each box, and only one egg was left over. Determine who the mother of each child is.
2024 Baltic Way, 18
An infinite sequence $a_1, a_2,\ldots$ of positive integers is such that $a_n \geq 2$ and $a_{n+2}$ divides $a_{n+1} + a_n$ for all $n \geq 1$. Prove that there exists a prime which divides infinitely many terms of the sequence.
2023 Durer Math Competition Finals, 2
[b]a)[/b] Find all solutions of the equation $p^2+q^2+r^2=pqr$, where $p,q,r$ are positive primes.\\
[b]b)[/b] Show that for every positive integer $N$, there exist three integers $a,b,c\geq N$ with $a^2+b^2+c^2=abc$.
1983 Spain Mathematical Olympiad, 6
In a cafeteria, a glass of lemonade, three sandwiches and seven biscuits have cost $1$ shilling and $2$ pence, and a glass of lemonade, four sandwiches and $10$ biscuits they are worth $1$ shilling and $5$ pence. Find the price of:
a) a glass of lemonade, a sandwich and a cake;
b) two glasses of lemonade, three sandwiches and five biscuits.
($1$ shilling = $12$ pence).
2021 BMT, 24
Suppose that $a, b, c$, and p are positive integers such that $p$ is a prime number and $$a^2 + b^2 + c^2 = ab + bc + ca + 2021p$$.
Compute the least possible value of $\max \,(a, b, c)$.
2006 Singapore MO Open, 5
Let $a,b,n$ be positive integers. Prove that $n!$ divides \[b^{n-1}a(a+b)(a+2b)...(a+(n-1)b)\]
2010 German National Olympiad, 4
Find all positive integer solutions for the equation $(3x+1)(3y+1)(3z+1)=34xyz$
Thx
1988 All Soviet Union Mathematical Olympiad, 469
If rationals $x, y$ satisfy $x^5 + y^5 = 2 x^2 y^2$, show that $1-x y$ is the square of a rational.
Kvant 2020, M2607
Let $n$ be a natural number. The set $A{}$ of natural numbers has the following property: for any natural number $m\leqslant n$ in the set $A{}$ there is a number divisible by $m{}$. What is the smallest value that the sum of all the elements of the set $A{}$ can take?
[i]Proposed by A. Kuznetsov[/i]
2024 Simon Marais Mathematical Competition, A4
Define a sequence by $s_0 = 1$ and for $d \geq 1$, $s_d = s_{d-1} + X_d$, where $X_d$ is chosen uniformly at random from the set $\{1, 2, \dots, d\}$.
What is the probability that the sequence $s_0, s_1, s_2, \dots$ contains infinitely many primes?
2014 Iran MO (3rd Round), 2
We say two sequence of natural numbers A=($a_1,...,a_n$) , B=($b_1,...,b_n$)are the exchange and we write $A\sim B$.
if $503\vert a_i - b_i$ for all $1\leq i\leq n$.
also for natural number $r$ : $A^r$ = ($a_1^r,a_2^r,...,a_n^r$).
Prove that there are natural number $k,m$ such that :
$i$)$250 \leq k $
$ii$)There are different permutations $\pi _1,...,\pi_k$ from {$1,2,3,...,502$} such that for $1\leq i \leq k-1$ we have $\pi _i^m\sim \pi _{i+1}$
(15 points)
2014 Junior Regional Olympiad - FBH, 5
How many are there $4$ digit numbers such that they have two odd digits and two even digits
2007 Cuba MO, 5
Prove that there is a unique positive integer formed only by the digits $2$ and $5$, which has $ 2007$ digits and is divisible by $2^{2007}$.
2016 Romania Team Selection Tests, 2
Determine all $f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ such that $f(m)\geq m$ and $f(m+n) \mid f(m)+f(n)$ for all $m,n\in \mathbb{Z}^+$
2022 Assara - South Russian Girl's MO, 4
Alina knows how to twist a periodic decimal fraction in the following way: she finds the minimum preperiod of the fraction, then takes the number that makes up the period and rearranges the last one in it digit to the beginning of the number. For example, from the fraction, $0.123(56708)$ she will get $0.123(85670)$. What fraction will Alina get from fraction $\frac{503}{2022}$ ?
2001 Irish Math Olympiad, 1
Find all positive integer solutions $ (a,b,c,n)$ of the equation: $ 2^n\equal{}a!\plus{}b!\plus{}c!$.
1984 Czech And Slovak Olympiad IIIA, 4
Let $r$ be a natural number greater than $1$. Then there exist positive irrational numbers $x, y$ such that $x^y = r$ . Prove it.
1978 IMO Shortlist, 3
Let $ m$ and $ n$ be positive integers such that $ 1 \le m < n$. In their decimal representations, the last three digits of $ 1978^m$ are equal, respectively, to the last three digits of $ 1978^n$. Find $ m$ and $ n$ such that $ m \plus{} n$ has its least value.
2022 Czech and Slovak Olympiad III A, 2
We say that a positive integer $k$ is [i]fair [/i] if the number of $2021$-digit palindromes that are a multiple of $k$ is the same as the number of $2022$-digit palindromes that are a multiple of $k$. Does the set $M = \{1, 2,..,35\}$ contain more numbers that are fair or those that are not fair?
(A palindrome is an integer that reads the same forward and backward.)
[i](David Hruska)[/i]