Found problems: 916
Let $a$, $b$, $c$ be given integers, $a>0$, $ac-b^2=p$ a squarefree positive integer. Let $M(n)$ denote the number of pairs of integers $(x, y)$ for which $ax^2 +bxy+cy^2=n$. Prove that $M(n)$ is finite and $M(n)=M(p^{k} \cdot n)$ for every integer $k \ge 0$.
How many pairs of positive integers $(x, y)$ are there, those satisfy the identity $2^x - y^2 = 4$?
(A): $1$ (B): $2$ (C): $3$ (D): $4$ (E): None of the above.
We say that a subset $A$ of $\mathbb N$ is good if for some positive integer $n$, the equation $x-y=n$ admits infinitely many solutions with $x,y\in A$. If $A_1,A_2,\ldots,A_{100}$ are sets whose union is $\mathbb N$, prove that at least one of the $A_i$s is good.
Determine if there is any triple of nonnegative integers, not necessarily different, $(a, b, c)$ such that:
$$a^3 + b^3 + c^3 = 2016$$
Provided the equation $xyz = p^n(x + y + z)$ where $p \geq 3$ is a prime and $n \in \mathbb{N}$. Prove that the equation has at least $3n + 3$ different solutions $(x,y,z)$ with natural numbers $x,y,z$ and $x < y < z$. Prove the same for $p > 3$ being an odd integer.
Determine all positive integers $a$, $b$ and $c$ which satisfy the equation
$$a^2+b^2+1=c!.$$
[i]Proposed by Nikola Velov[/i]
Find all the integer solutions of the equation $ m^4 + 2n^2 = 9mn$.
For all integral values of parameter $t$, find all integral solutions $(x,y)$ of the equation
$$ y^2 = x^4-22x^3+43x^2+858x+t^2+10452(t+39)$$ .
Find the natural numbers $x, y, z$ that verify the equation: $$2^x + 3 \cdot 11^y =7^z$$
Determine all pairs of integer numbers $(x, y)$ such that:
$$\frac{(x - y)^2}{x + y} = x - y + 6$$
Let $A, B, C, D, E$ be integers, $B \neq 0$ and $F=AD^{2}-BCD+B^{2}E \neq 0$. Prove that the number $N$ of pairs of integers $(x, y)$ such that \[Ax^{2}+Bxy+Cx+Dy+E=0,\] satisfies $N \le 2 d( \vert F \vert )$, where $d(n)$ denotes the number of positive divisors of positive integer $n$.
For every positive integer $ n$, $ b(n)$ denote the number of positive divisors of $ n$ and $ p(n)$ denote the sum of all positive divisors of $ n$. For example, $ b(14)\equal{}4$ and $ p(14)\equal{}24$. Let $ k$ be a positive integer greater than $ 1$.
(a) Prove that there are infinitely many positive integers $ n$ which satisfy $ b(n)\equal{}k^2\minus{}k\plus{}1$.
(b) Prove that there are finitely many positive integers $ n$ which satisfy $ p(n)\equal{}k^2\minus{}k\plus{}1$.
Solve the following equation in real numbers:
$\frac{(x^2-1)(|x|+1)}{x+sgnx}=[x+1].$
Find all pairs $(m,n)$ of positive integers such that $m+n$ and $mn+1$ are both powers of $2$.
Find all pairs of positive integers $x,y$ such that $x^2 +615 = 2^y$
Prove there exist infinitely many pairs $(x,y)$ of integers $1<x<y$, such that $x^3+y \mid x+y^3$.
([i]Dan Schwarz[/i])
If $x$ is the number of solutions to the equation $a^2 + b^2 + c^2 = d^2$ of the form $(a,b,c,d)$ such that $\{a,b,c\}$ are three consecutive square numbers and $d$ is also a square number, find $x$.
Show that there are no integers $x$ and $y$ satisfying $x^2 + 5 = y^3$.
Daniel Harrer
Find all integer solutions of the equation $xy+5x-3y=27$.
The equation
\[x^5 + 5 \lambda x^4 - x^3 + (\lambda \alpha - 4)x^2 - (8 \lambda + 3)x + \lambda \alpha - 2 = 0\]
is given. Determine $\alpha$ so that the given equation has exactly (i) one root or (ii) two roots, respectively, independent from $\lambda.$
Show that there are infinitely many pairs $(x, y)$ of rational numbers such that $x^3 +y^3 =9$.
Find the triplets of natural numbers $(p,q,r)$ that satisfy the equality $$\frac{1}{p}+\frac{q}{q^r -1}=1.$$
$(BUL 1)$ Prove that the equation $\sqrt{x^3 + y^3 + z^3}=1969$ has no integral solutions.
For a given natural number $ n $, find the number of solutions to the equation $ \sqrt{x} + \sqrt{y} = n $ in natural numbers $ x, y $.
For prime numbers $p$ and $q$, natural numbers $n$, $k$, $r$, the equality $p^{2k}+q^{2n}=r^2$ holds. Prove that the number $r$ is prime.