Found problems: 15460
2010 Dutch IMO TST, 3
Let $n\ge 2$ be a positive integer and $p $ a prime such that $n|p-1$ and $p | n^3-1$. Show $ 4p-3$ is a square.
2011 Regional Competition For Advanced Students, 1
Let $p_1, p_2, \ldots, p_{42}$ be $42$ pairwise distinct prime numbers. Show that the sum \[\sum_{j=1}^{42}\frac{1}{p_j^2+1}\] is not a unit fraction $\frac{1}{n^2}$ of some integer square number.
2007 Korea National Olympiad, 4
For all positive integer $ n\geq 2$, prove that product of all prime numbers less or equal than $ n$ is smaller than $ 4^{n}$.
2005 MOP Homework, 4
Let $p$ be an odd prime. Prove that \[\sum^{p-1}_{k=1} k^{2p-1} \equiv \frac{p(p+1)}{2}\pmod{p^2}.\]
Kvant 2021, M2558
We have $n>2$ non-zero integers such that each one of them is divisible by the sum of the other $n-1$ numbers. Prove that the sum of all the given numbers is zero.
2022 Brazil Team Selection Test, 4
Let $a_1,a_2,a_3,\ldots$ be an infinite sequence of positive integers such that $a_{n+2m}$ divides $a_{n}+a_{n+m}$ for all positive integers $n$ and $m.$ Prove that this sequence is eventually periodic, i.e. there exist positive integers $N$ and $d$ such that $a_n=a_{n+d}$ for all $n>N.$
2016 Israel Team Selection Test, 4
Find the greatest common divisor of all numbers of the form $(2^{a^2}\cdot 19^{b^2} \cdot 53^{c^2} + 8)^{16} - 1$ where $a,b,c$ are integers.
2006 Junior Balkan Team Selection Tests - Romania, 3
For any positive integer $n$ let $s(n)$ be the sum of its digits in decimal representation. Find all numbers $n$ for which $s(n)$ is the largest proper divisor of $n$.
2022 Durer Math Competition Finals, 5
$n$ people sitting at a round table. In the beginning, everyone writes down a positive number $n$ on piece of paper in front of them. From now on, in every minute, they write down the number that they get if they subtract the number of their right-hand neighbour from their own number. They write down the new number and erase the original. Give those number $n$ that there exists an integer $k$ in a way that regardless of the starting numbers, after $k$ minutes, everyone will have a number that is divisible by $n$.
2010 Vietnam Team Selection Test, 1
Let $n$ be a positive integer. Let $T_n$ be a set of positive integers such that:
\[{T_n={ \{11(k+h)+10(n^k+n^h)| (1 \leq k,h \leq 10)}}\}\]
Find all $n$ for which there don't exist two distinct positive integers $a, b \in T_n$ such that $a\equiv b \pmod{110}$
2009 Junior Balkan Team Selection Tests - Moldova, 6
Prove that there are no pairs of nonnegative integers $(x,y)$ that satisfy the equality $$x^3-y^3=x-y+2^{x-y}.$$
1979 Canada National Olympiad, 3
Let $a$, $b$, $c$, $d$, $e$ be integers such that $1 \le a < b < c < d < e$. Prove that
\[\frac{1}{[a,b]} + \frac{1}{[b,c]} + \frac{1}{[c,d]} + \frac{1}{[d,e]} \le \frac{15}{16},\]
where $[m,n]$ denotes the least common multiple of $m$ and $n$ (e.g. $[4,6] = 12$).
2009 Spain Mathematical Olympiad, 1
Find all the finite sequences with $ n$ consecutive natural numbers $ a_1, a_2,..., a_n$, with $ n\geq3$ such that $ a_1\plus{} a_2\plus{}...\plus{} a_n\equal{}2009$.
2020 Czech-Austrian-Polish-Slovak Match, 5
Let $n$ be a positive integer and let $d(n)$ denote the number of ordered pairs of positive integers $(x,y)$ such that
$(x+1)^2-xy(2x-xy+2y)+(y+1)^2=n$. Find the smallest positive integer $n$ satisfying $d(n) = 61$.
(Patrik Bak, Slovakia)
1994 Romania TST for IMO, 1:
Let $p$ be a (positive) prime number. Suppose that real numbers $a_1, a_2, . . ., a_{p+1}$ have the property that, whenever one of the numbers is deleted, the remaining numbers can be partitioned into two classes with the same arithmetic mean. Show that these numbers must be equal.
2014 BMT Spring, 1
A [i]festive [/i] number is a four-digit integer containing one of each of the digits $0, 1, 2$, and $4$ in its decimal representation. How many festive numbers are there?
VI Soros Olympiad 1999 - 2000 (Russia), 8.6
Two players take turns writing down all proper non-decreasing fractions with denominators from $1 $ to $1999$ and at the same time writing a "$+$" sign before each fraction. After all such fractions are written out, their sum is found. If this amount is an integer number, then the one who made the entry last wins, otherwise his opponent wins. Who will be able to secure a win?
1994 AIME Problems, 9
A solitaire game is played as follows. Six distinct pairs of matched tiles are placed in a bag. The player randomly draws tiles one at a time from the bag and retains them, except that matching tiles are put aside as soon as they appear in the player's hand. The game ends if the player ever holds three tiles, no two of which match; otherwise the drawing continues until the bag is empty. The probability that the bag will be emptied is $p/q,$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$
1993 Korea - Final Round, 4
An integer which is the area of a right-angled triangle with integer sides is called [i]Pythagorean[/i]. Prove that for every positive integer $n > 12$ there exists a Pythagorean number between $n$ and $2n.$
2014 Costa Rica - Final Round, 3
Find all possible pairs of integers $ a$ and $ b$ such that $ab = 160 + 90 (a,b)$, where $(a, b)$ is the greatest common divisor of $ a$ and $ b$.
2010 Mathcenter Contest, 2
Let $k$ and $d$ be integers such that $k>1$ and $0\leq d<9$. Prove that there exists some integer $n$ such that the $k$th digit from the right of $2^n$ is $d$.
[i](tatari/nightmare)[/i]
2018 CMIMC Number Theory, 2
Find all integers $n$ for which $(n-1)\cdot 2^n + 1$ is a perfect square.
1992 China National Olympiad, 3
Let sequence $\{a_1,a_2,\dots \}$ with integer terms satisfy the following conditions:
1) $a_{n+1}=3a_n-3a_{n-1}+a_{n-2}, n=2,3,\dots$ ;
2) $2a_1=a_0+a_2-2$ ;
3) for arbitrary natural number $m$, there exist $m$ consecutive terms $a_k, a_{k-1}, \dots ,a_{k+m-1}$ among the sequence such that all such $m$ terms are perfect squares.
Prove that all terms of the sequence $\{a_1,a_2,\dots \}$ are perfect squares.
2018 Latvia Baltic Way TST, P15
Determine whether there exists a positive integer $n$ such that it is possible to find at least $2018$ different quadruples $(x,y,z,t)$ of positive integers that simultaneously satisfy equations
$$\begin{cases}
x+y+z=n\\
xyz = 2t^3.
\end{cases}$$
2009 QEDMO 6th, 9
For every natural $n$ let $\phi (n)$ be the number of coprime numbers $k \in \{1,2,...,n\}$. (Example: $\phi (12) = 4$, because among the numbers $1, 2, ..., 12$ there are only the$ 4$ numbers, $1, 5, 7$ and $11$ coprime to$12.$)
If $k$ is a natural number, then one defines $\phi^k (n)=\underbrace{\strut \phi (\phi ...(\phi (n)) ...)}_{(k \, times \phi)}$ (Example: $\phi^3 (n)=\phi (\phi (\phi (n))) $)
For every whole $n> 2$ let $c(n)$ be the smallest natural number $k$ with $\phi^k (n)= 2$.
Prove that $c (ab) = c (a) + c (b)$ for odd integers $a$ and $b$, both of which are greater than $2$, .