Found problems: 988
Find all positive integers $k$ such that there exists positive integers $a, b$ such that
\[a^2+4=(k^2-4)b^2.\]
Let $A, B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $ABC$ is an equilateral triangle whose side is $86$ meters long. A boy swam from $A$ directly toward $B$. After covering a distance of $x$ meters, he turned and swam westward, reaching the shore after covering a distance of $y$ meters. If $x$ and $y$ are both positive integers, determine $y.$
Let $ a, b \in \mathbb{Z}$ which are not perfect squares. Prove that if \[ x^2 \minus{} ay^2 \minus{} bz^2 \plus{} abw^2 \equal{} 0\] has a nontrivial solution in integers, then so does \[ x^2 \minus{} ay^2 \minus{} bz^2 \equal{} 0.\]
Four positive integers $x,y,z$ and $t$ satisfy the relations
\[ xy - zt = x + y = z + t. \]
Is it possible that both $xy$ and $zt$ are perfect squares?
Find all rational terms of sequence defined by formula $ a_n=\sqrt{\frac{9n-2}{n+1}},
n \in N $
Find integer solutions of the equation $x^3 - 2y^3 - 4z^3 = 0$.
Four positive integers $x,y,z$ and $t$ satisfy the relations
\[ xy - zt = x + y = z + t. \]
Is it possible that both $xy$ and $zt$ are perfect squares?
let $p$and $q=p+2$ be twin primes. consider the diophantine equation $(+)$ given by
$n!+pq^2=(mp)^2$ $m\geq1$, $n\geq1$
i. if $m=p$,find the value of $p$.
ii. how many solution quadruple $(p,q,m,n)$ does $(+)$ have ?
Find all solutions of the equation $(a^a)^5 = b^b$ in positive integers.
Find all pairs $(p,n)$ so that $p$ is a prime number, $n$ is a positive integer and \[p^3-2p^2+p+1=3^n \] holds.
Find all pairs $(p,q)$ of prime numbers such that
$$ p(p^2 - p - 1) = q(2q + 3) .$$
Find all primes $p$ and positive integers $n$ satisfying
\[n \cdot 5^{n-n/p} = p! (p^2+1) + n.\]
Find all pairs of solutions $(x,y)$:
\[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]
Let $n$ be a positive integer and $a_1,a_2,...,a_{2n}$ be $2n$ distinct integers. Given that the equation $|x-a_1| |x-a_2| ... |x-a_{2n}| =(n!)^2$ has an integer solution $x = m$, find $m$ in terms of $a_1,a_2,...,a_{2n}$
Prove:
(a) There are infinitely many triples of positive integers $m, n, p$ such that $4mn - m- n = p^2 - 1.$
(b) There are no positive integers $m, n, p$ such that $4mn - m- n = p^2.$
Solve in $ \mathbb{N}^2: $
$$ x+y=\sqrt x+\sqrt y+\sqrt{xy} . $$
Find all triples $(a, b,c) $ of positive integers, such that: \[ a! + b! = c!! \] where $(2k)!! = 2 \cdot 4 \cdot \ldots \cdot (2k)$ and $ (2k + 1)!! = 1 \cdot 3 \cdot \ldots \cdot (2k+1).$
[i]Proposed by Stijn Cambie, Belgium [/i]
Show that there cannot be four squares in arithmetical progression.
Let $n$ be a positive integer and $a_1, a_2, \dots , a_{2n}$ mutually distinct integers. Find all integers $x$ satisfying
\[(x - a_1) \cdot (x - a_2) \cdots (x - a_{2n}) = (-1)^n(n!)^2.\]
Find all non-negative integer solutions of the equation $2^a+p^b=n^{p-1}$, where $p$ is a prime number.
Proposed by [i]Máté Weisz[/i], Cambridge
Determine all $6$-tuples $(x,y,z,u,v,w)$ of integers satisfying the equation
\[x^3+7y^3+49z^3=2u^3+14v^3+98w^3.\]
Find all $x, y, z \in N_0$ with $(2^x + 1) (2^y-1) = 2^z-1$.
Determine all pairs $(x, y)$ of positive integers satisfying
$x + y + 1 | 2xy$ and $ x + y - 1 | x^2 + y^2 - 1$.
Find all quadruples $(a, b, p, n)$ of positive integers, such that $p$ is a prime and
\[a^3 + b^3 = p^n\mbox{.}\]
[i](2nd Benelux Mathematical Olympiad 2010, Problem 4)[/i]
Find all integers $m$ and $n$ such that $2m^2 +n^2 = 2mn+3n$