Found problems: 916
Find all pairs of integers $a, b$ for which equality holds $\frac{a^2+1}{2b^2-3}=\frac{a-1}{2b-1}$
Solve the diophantine equation $x^{2018}-y^{2018}=(xy)^{2017}$ when $x$ and $y$ are non-negative integers.
Let $a, b, c$, and $d$ be four distinct integers.
Prove that $(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)$ is divisible by $12$.
Determine whether there exist an infinite number of positive integers $x,y $ satisfying the condition: $x^2+y \mid x+y^2.$ Please prove it.
Determine all integer solutions of the system \[2uv-xy=16,\] \[xv-yu=12.\]
Let $n$ be an odd positive integer. Prove that if the equation $\frac1x+\frac1y=\frac4n$ has a solution in positive integers $x,y$, then $n$ has at least one divisor of the form $4k-1$, $k\in\mathbb N$.
Find all triples $ \left(p,x,y\right)$ such that $ p^x\equal{}y^4\plus{}4$, where $ p$ is a prime and $ x$ and $ y$ are natural numbers.
Determine all triples $(a, b, c)$ of positive integers such that
$$a! +b! = 2^{c!} .$$
Prove that the equation $(\sqrt5 +1)^{2x}+ (\sqrt5 -1)^{2x}=2^x(y^2+2)$ has an infinite number of solutions in natural numbers.
Determine all pairs of integers $(x, y)$ that satisfy equation $(y - 2) x^2 + (y^2 - 6y + 8) x = y^2 - 5y + 62$.
Solve in positive integers the equation $1005^x + 2011^y = 1006^z$.
Find all positive integers $p,q,r,s>1$ such that $$p!+q!+r!=2^s.$$
Show that no pairs of integers $(m, n)$ satisfy $2560m^2 + 5m + 6 = n^5$.
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Determine all triples $(a, b, c)$ of nonnegative integers that satisfy: $$(c-1) (ab- b -a) = a + b-2$$
Let a, b be positive integers. a, b and a.b are not perfect squares.
Prove that at most one of following equations
$ ax^2 \minus{} by^2 \equal{} 1$ and $ ax^2 \minus{} by^2 \equal{} \minus{} 1$
has solutions in positive integers.
Let $\overline{abc}$ be a three digit number with nonzero digits such that $a^2 + b^2 = c^2$. What is the largest possible prime factor of $\overline{abc}$
Let $ a_1 \geq a_2 \geq a_3 \in \mathbb{Z}^\plus{}$ be given and let N$ (a_1, a_2, a_3)$ be the number of solutions $ (x_1, x_2, x_3)$ of the equation
\[ \sum^3_{k\equal{}1} \frac{a_k}{x_k} \equal{} 1.\]
where $ x_1, x_2,$ and $ x_3$ are positive integers. Prove that \[ N(a_1, a_2, a_3) \leq 6 a_1 a_2 (3 \plus{} ln(2 a_1)).\]
Positive integers $a$, $b$ and $c$ are positive integers with greatest common divisor equal to $1$ (i.e. they have no common divisors greater than $1$), and
$$\frac{ab}{a-b}=c$$
Prove that $a -b$ is a perfect square.
(SL Berlov)
Solve in $ \mathbb{Z}^2 $ the following equation:
$$ (x+1)(x+2)(x+3) +x(x+2)(x+3)+x(x+1)(x+3)+x(x+1)(x+2)=y^{2^x} . $$
[i]Adrian Zanoschi[/i]
Find all quadruples of positive integers $(p, q, a, b)$, where $p$ and $q$ are prime numbers and $a > 1$, such that $$p^a = 1 + 5q^b.$$
Let $p$ be a prime number. Find all triples $(a, b, c)$ of integers (not necessarily positive) such that $a^bb^cc^a = p$.
Let $a, b, c, d, e, f$ be integers such that $b^{2}-4ac>0$ is not a perfect square and $4acf+bde-ae^{2}-cd^{2}-fb^{2}\neq 0$. Let \[f(x, y)=ax^{2}+bxy+cy^{2}+dx+ey+f\] Suppose that $f(x, y)=0$ has an integral solution. Show that $f(x, y)=0$ has infinitely many integral solutions.
$p$ is a prime. Find the all $(m,n,p)$ positive integer triples satisfy $m^3+7p^2=2^n$.
Find all pairs of prime numbers $(p,q)$ for which
\[2^p = 2^{q-2} + q!.\]
Find all triples $(a,b,c)$ of integers for which the equation
\[x^3-a^2x^2+b^2x-ab+3c=0\]
has three distinct integer roots $x_1,x_2,x_3$ which are pairwise coprime.