Found problems: 1766
2007 Kyiv Mathematical Festival, 1
Find all pairs of positive integers $(a,b)$ such that $\sqrt{a-1}+\sqrt{b-1}=\sqrt{ab-1}.$
2012 Middle European Mathematical Olympiad, 7
Find all triplets $ (x,y,z) $ of positive integers such that
\[ x^y + y^x = z^y \]\[ x^y + 2012 = y^{z+1} \]
1995 Romania Team Selection Test, 1
The sequence $ (x_n)$ is defined by $ x_1\equal{}1,x_2\equal{}a$ and $ x_n\equal{}(2n\plus{}1)x_{n\minus{}1}\minus{}(n^2\minus{}1)x_{n\minus{}2}$ $ \forall n \geq 3$, where $ a \in N^*$.For which value of $ a$ does the sequence have the property that $ x_i|x_j$ whenever $ i<j$.
1996 Romania Team Selection Test, 11
Find all primes $ p,q $ such that $ \alpha^{3pq} -\alpha \equiv 0 \pmod {3pq} $ for all integers $ \alpha $.
2007 Irish Math Olympiad, 4
Find the number of zeros in which the decimal expansion of $ 2007!$ ends. Also find its last non-zero digit.
2009 International Zhautykov Olympiad, 1
Find all pairs of integers $ (x,y)$, such that
\[ x^2 \minus{} 2009y \plus{} 2y^2 \equal{} 0
\]
2012 Turkey Team Selection Test, 2
A positive integer $n$ is called [i]good[/i] if for all positive integers $a$ which can be written as $a=n^2 \sum_{i=1}^n {x_i}^2$ where $x_1, x_2, \ldots ,x_n$ are integers, it is possible to express $a$ as $a=\sum_{i=1}^n {y_i}^2$ where $y_1, y_2, \ldots, y_n$ are integers with none of them is divisible by $n.$ Find all good numbers.
2004 Singapore Team Selection Test, 1
Let $x_0, x_1, x_2, \ldots$ be the sequence defined by
$x_i= 2^i$ if $0 \leq i \leq 2003$
$x_i=\sum_{j=1}^{2004} x_{i-j}$ if $i \geq 2004$
Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by 2004.
1973 IMO Longlists, 8
Let $a$ be a non-zero real number. For each integer $n$, we define $S_n = a^n + a^{-n}$. Prove that if for some integer $k$, the sums $S_k$ and $S_{k+1}$ are integers, then the sums $S_n$ are integers for all integers $n$.
2004 Iran Team Selection Test, 2
Suppose that $ p$ is a prime number. Prove that the equation $ x^2\minus{}py^2\equal{}\minus{}1$ has a solution if and only if $ p\equiv1\pmod 4$.
1950 Miklós Schweitzer, 5
Prove that for every positive integer $ k$ there exists a sequence of $ k$ consecutive positive integers none of which can be represented as the sum of two squares.
2011 Canada National Olympiad, 5
Let $d$ be a positive integer. Show that for every integer $S$, there exists an integer $n>0$ and a sequence of $n$ integers $\epsilon_1, \epsilon_2,..., \epsilon_n$, where $\epsilon_i = \pm 1$ (not necessarily dependent on each other) for all integers $1\le i\le n$, such that $S=\sum_{i=1}^{n}{\epsilon_i(1+id)^2}$.
2012 China Team Selection Test, 1
Given an integer $n\ge 4$. $S=\{1,2,\ldots,n\}$. $A,B$ are two subsets of $S$ such that for every pair of $(a,b),a\in A,b\in B, ab+1$ is a perfect square. Prove that
\[\min \{|A|,|B|\}\le\log _2n.\]
2006 India IMO Training Camp, 2
the positive divisors $d_1,d_2,\cdots,d_k$ of a positive integer $n$ are ordered
\[1=d_1<d_2<\cdots<d_k=n\]
Suppose $d_7^2+d_{15}^2=d_{16}^2$. Find all possible values of $d_{17}$.
1995 All-Russian Olympiad, 5
We call natural numbers [i]similar[/i] if they are written with the same (decimal) digits. For example, 112, 121, 211 are similar numbers having the digits 1, 1, 2. Show that there exist three similar 1995-digit numbers with no zero digits, such that the sum of two of them equals the third.
[i]S. Dvoryaninov[/i]
2009 Moldova Team Selection Test, 4
[color=darkred]Let $ p$ be a prime divisor of $ n\ge 2$. Prove that there exists a set of natural numbers $ A \equal{} \{a_1,a_2,...,a_n\}$ such that product of any two numbers from $ A$ is divisible by the sum of any $ p$ numbers from $ A$.[/color]
2012 ELMO Shortlist, 1
Find all positive integers $n$ such that $4^n+6^n+9^n$ is a square.
[i]David Yang, Alex Zhu.[/i]
2001 JBMO ShortLists, 7
Prove that there are are no positive integers $x$ and $y$ such that $x^5+y^5+1=(x+2)^5+(y-3)^5$.
[hide="Note"]
The restriction $x,y$ are positive isn't necessary.[/hide]
2008 Irish Math Olympiad, 4
Given $ k \in [0,1,2,3]$ and a positive integer $ n$, let $ f_k(n)$ be the number of sequences $ x_1,...,x_n,$ where $ x_i \in [\minus{}1,0,1]$ for $ i\equal{}1,...,n,$ and
$ x_1\plus{}...\plus{}x_n \equiv k$ mod 4
a) Prove that $ f_1(n) \equal{} f_3(n)$ for all positive integers $ n$.
(b) Prove that
$ f_0(n) \equal{} [{3^n \plus{} 2 \plus{} [\minus{}1]^n}] / 4$
for all positive integers $ n$.
2024 Abelkonkurransen Finale, 1a
Determine all integers $n \ge 2$ such that $n \mid s_n-t_n$ where $s_n$ is the sum of all the integers in the interval $[1,n]$ that are mutually prime to $n$, and $t_n$ is the sum of the remaining integers in the same interval.
2018 JBMO Shortlist, NT3
Find all positive integers $abcd=a^{a+b+c+d} - a^{-a+b-c+d} + a$, where $abcd$ is a four-digit number
2009 Tournament Of Towns, 3
Are there positive integers $a; b; c$ and $d$ such that $a^3 + b^3 + c^3 + d^3 =100^{100}$ ?
[i](4 points)[/i]
2012 ELMO Shortlist, 7
A diabolical combination lock has $n$ dials (each with $c$ possible states), where $n,c>1$. The dials are initially set to states $d_1, d_2, \ldots, d_n$, where $0\le d_i\le c-1$ for each $1\le i\le n$. Unfortunately, the actual states of the dials (the $d_i$'s) are concealed, and the initial settings of the dials are also unknown. On a given turn, one may advance each dial by an integer amount $c_i$ ($0\le c_i\le c-1$), so that every dial is now in a state $d_i '\equiv d_i+c_i \pmod{c}$ with $0\le d_i ' \le c-1$. After each turn, the lock opens if and only if all of the dials are set to the zero state; otherwise, the lock selects a random integer $k$ and cyclically shifts the $d_i$'s by $k$ (so that for every $i$, $d_i$ is replaced by $d_{i-k}$, where indices are taken modulo $n$).
Show that the lock can always be opened, regardless of the choices of the initial configuration and the choices of $k$ (which may vary from turn to turn), if and only if $n$ and $c$ are powers of the same prime.
[i]Bobby Shen.[/i]
2014 ELMO Shortlist, 4
Let $\mathbb N$ denote the set of positive integers, and for a function $f$, let $f^k(n)$ denote the function $f$ applied $k$ times. Call a function $f : \mathbb N \to \mathbb N$ [i]saturated[/i] if \[ f^{f^{f(n)}(n)}(n) = n \] for every positive integer $n$. Find all positive integers $m$ for which the following holds: every saturated function $f$ satisfies $f^{2014}(m) = m$.
[i]Proposed by Evan Chen[/i]
2014 India National Olympiad, 3
Let $a,b$ be natural numbers with $ab>2$. Suppose that the sum of their greatest common divisor and least common multiple is divisble by $a+b$. Prove that the quotient is at most $\frac{a+b}{4}$. When is this quotient exactly equal to $\frac{a+b}{4}$