Found problems: 15460
2006 Taiwan TST Round 1, 1
Find the largest integer that is a factor of
$(a-b)(b-c)(c-d)(d-a)(a-c)(b-d)$
for all integers $a,b,c,d$.
2009 Puerto Rico Team Selection Test, 2
The last three digits of $ N$ are $ x25$. For how many values of $ x$ can $ N$ be the square of an integer?
LMT Accuracy Rounds, 2023 S5
Let $$N = \sum^{512}_{i=0}i {512 \choose i}.$$ What is the greatest integer $a$ such that $2^a$ is a divisor of $N$?
2017 ISI Entrance Examination, 7
Let $A=\{1,2,\ldots,n\}$. For a permutation $P=(P(1), P(2), \ldots, P(n))$ of the elements of $A$, let $P(1)$ denote the first element of $P$. Find the number of all such permutations $P$ so that for all $i,j \in A$:
(a) if $i < j<P(1)$, then $j$ appears before $i$ in $P$; and
(b) if $P(1)<i<j$, then $i$ appears before $j$ in $P$.
2021 All-Russian Olympiad, 2
Let $n$ be a natural number. An integer $a>2$ is called $n$-decomposable, if $a^n-2^n$ is divisible by all the numbers of the form $a^d+2^d$, where $d\neq n$ is a natural divisor of $n$. Find all composite $n\in \mathbb{N}$, for which there's an $n$-decomposable number.
1978 IMO Longlists, 10
Show that for any natural number $n$ there exist two prime numbers $p$ and $q, p \neq q$, such that $n$ divides their difference.
2015 Greece National Olympiad, 1
Find all triplets $(x,y,p)$ of positive integers such that $p$ be a prime number and $\frac{xy^3}{x+y}=p$
2018 Brazil EGMO TST, 1
(a) Let $m$ and $n$ be positive integers and $p$ a positive rational number, with $m > n$, such that $\sqrt{m} -\sqrt{n}= p$. Prove that $m$ and $n$ are perfect squares.
(b) Find all four-digit numbers $\overline{abcd}$, where each letter $a, b, c$ and $d$ represents a digit, such that $\sqrt{\overline{abcd}} -\sqrt{\overline{acd}}= \overline{bb}$.
2024 Korea Junior Math Olympiad (First Round), 3.
Find the number of positive integers (m,n) which follows the following:
1) m<n
2) The sum of even numbers between 2m and 2n is 100 greater than the sum of odd numbers between 2m and 2n.
2017 Bulgaria JBMO TST, 2
Solve the following equation over the integers
$$ 25x^2y^2+10x^2y+25xy^2+x^2+30xy+2y^2+5x+7y+6= 0.$$
1998 Estonia National Olympiad, 2
Find all prime numbers of the form $10101...01$.
2013 JBMO TST - Macedonia, 5
Let $ p, r $ be prime numbers, and $ q $ natural. Solve the equation $ (p+q+r)^2=2p^2+2q^2+r^2 $.
2005 Switzerland - Final Round, 7
Let $n\ge 1$ be a natural number. Determine all positive integer solutions of the equation
$$7 \cdot 4^n = a^2 + b^2 + c^2 + d^2.$$
2015 Purple Comet Problems, 9
Find the sum of all positive integers n with the property that the digits of n add up to 2015−n.
2000 AIME Problems, 4
What is the smallest positive integer with six positive odd integer divisors and twelve positive even integer divisors?
2013 Tuymaada Olympiad, 6
Solve the equation $p^2-pq-q^3=1$ in prime numbers.
[i]A. Golovanov[/i]
2024 Iberoamerican, 1
For each positive integer $n$, let $d(n)$ be the number of positive integer divisors of $n$.
Prove that for all pairs of positive integers $(a,b)$ we have that:
\[ d(a)+d(b) \le d(\gcd(a,b))+d(\text{lcm}(a,b)) \]
and determine all pairs of positive integers $(a,b)$ where we have equality case.
2014 China Team Selection Test, 3
Let the function $f:N^*\to N^*$ such that
[b](1)[/b] $(f(m),f(n))\le (m,n)^{2014} , \forall m,n\in N^*$;
[b](2)[/b] $n\le f(n)\le n+2014 , \forall n\in N^*$
Show that: there exists the positive integers $N$ such that $ f(n)=n $, for each integer $n \ge N$.
(High School Affiliated to Nanjing Normal University )
1989 Austrian-Polish Competition, 3
Find all natural numbers $N$ (in decimal system) with the following properties:
(i) $N =\overline{aabb}$, where $\overline{aab}$ and $\overline{abb}$ are primes,
(ii) $N = P_1P_2P_3$, where $P_k (k = 1,2,3)$ is a prime consisting of $k$ (decimal) digits.
2024 China Western Mathematical Olympiad, 1
For positive integer $n$, note $S_n=1^{2024}+2^{2024}+ \cdots +n^{2024}$.
Prove that there exists infinitely many positive integers $n$, such that $S_n$ isn’t divisible by $1865$ but $S_{n+1}$ is divisible by $1865$
2009 Jozsef Wildt International Math Competition, W. 3
Let $\Phi$ and $\Psi$ denote the Euler totient and Dedekind‘s totient respectively. Determine all $n$ such that $\Phi(n)$ divides $n +\Psi (n)$.
2024 ELMO Shortlist, N4
Find all pairs $(a,b)$ of positive integers such that $a^2\mid b^3+1$ and $b^2\mid a^3+1$.
[i]Linus Tang[/i]
2012 AMC 8, 13
Jamar bought some pencils costing more than a penny each at the school bookstore and paid $\$1.43$. Sharona bought some of the same pencils and paid $\$1.87$. How many more pencils did Sharona buy than Jamar?
$\textbf{(A)}\hspace{.05in}2 \qquad \textbf{(B)}\hspace{.05in}3 \qquad \textbf{(C)}\hspace{.05in}4 \qquad \textbf{(D)}\hspace{.05in}5 \qquad \textbf{(E)}\hspace{.05in}6 $
1985 Poland - Second Round, 5
Prove that for a natural number $n$ greater than 1, the following conditions are equivalent:
a) $ n $ is an even number,
b) there is a permutation $ (a_0, a_1, a_2, \ldots, a_{n-1}) $ of the set $ \{0,1,2,\ldots,n—1\} $ with the property that the sequence of residues from dividing by $ n $ the numbers $ a_0, a_0 + a_1, a_0 + a_1 + a_2, \ldots, a_0 + a_1 + a_2 + \ldots a_{n-1} $ is also a permutation of this set.
2022 Harvard-MIT Mathematics Tournament, 2
Compute the number of positive integers that divide at least two of the integers in the set $\{1^1,2^2,3^3,4^4,5^5,6^6,7^7,8^8,9^9,10^{10}\}$.