Found problems: 916
Find all positive integers $a, b, c$ such that $ab+1$ and $c$ are coprimes and:
$$a(ba+1)(ca^2+ba+1)=2021^{2021}$$
Let $n$ be a constant positive integer. Show that for only non-negative integers $k$, the Diophantine equation $\sum_{i=1 }^{n}{ x_i ^3}=y^{3k+2}$ has infinitely many solutions in the positive integers $x_i, y$.
Find all positive integer $n$ satisfying the conditions
$a)n^2=(a+1)^3-a^3$
$b)2n+119$ is a perfect square.
Prove without using modulo that there are no integers $a,b,c$ such that $$a^2+b^2-8c = 6$$
[i](Metamorphosis)[/i]
Prove that the equation $a^2 +b^2 =c^2 +3$ has infinitely many integer solutions $(a, b, c)$.
Suppose $r$ is a real number for which \[ \left\lfloor r + \frac{19}{100} \right\rfloor + \left\lfloor r + \frac{20}{100} \right\rfloor + \left\lfloor r + \frac{21}{100} \right\rfloor + \cdots + \left\lfloor r + \frac{91}{100} \right\rfloor = 546. \] Find $\lfloor 100r \rfloor$. (For real $x$, $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.)
FIx positive integer $n$. Prove: For any positive integers $a,b,c$ not exceeding $3n^2+4n$, there exist integers $x,y,z$ with absolute value not exceeding $2n$ and not all $0$, such that $ax+by+cz=0$
Are there integers $m$ and $n$ such that $5m^2 -6mn+7n^2 =1985$?
Find infinitely many triples $(a, b, c)$ of positive integers such that $a$, $b$, $c$ are in arithmetic progression and such that $ab+1$, $bc+1$, and $ca+1$ are perfect squares.
Determine all pairs $(x, y)$ of integers such that \[1+2^{x}+2^{2x+1}= y^{2}.\]
. Find all triplets $(x, y, z)$, $x > y > z$ of positive integers such that $\frac{1}{x}+\frac{2}{y}+\frac{3}{z}= 1$
An integer $n\ge 1$ is called balanced if it has an even number of distinct prime divisors. Prove that there exist infinitely many positive integers $n$ such that there are exactly two balanced numbers among $n,n+1,n+2$ and $n+3$.
Determine all positive integers $n$ for which there exist positive integers $a_1,a_2, ..., a_n$
with $a_1 + 2a_2 + 3a_3 +... + na_n = 6n$ and $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$
Prove that there are infinitely many quadruples of integers $(a,b,c,d)$ such that
\begin{align*}
a^2 + b^2 + 3 &= 4ab\\
c^2 + d^2 + 3 &= 4cd\\
4c^3 - 3c &= a
\end{align*}
[i]Travis Hance.[/i]
Find all triples $(x,y,z)$ of positive integers such that $3^x + 4^y = 5^z$.
Find all pairs of natural numbers $(m,n)$ bigger than $1$ for which $2^m+3^n$ is the square of whole number.
[i]I. Tonov[/i]
Show that the equation \[x^{3}+y^{3}+z^{3}+t^{3}=1999\] has infinitely many integral solutions.
Find all positive integer $n$ satisfying the conditions
$a)n^2=(a+1)^3-a^3$
$b)2n+119$ is a perfect square.
Find all prime numbers $p$ for which one can find a positive integer $m$ and nonnegative integers $a_0,a_1,...,a_m$ less than $p$ such that $$\begin{cases} a_0+a_1p+...+a_{m-1}p^{m-1}+a_{m}p^{m} = 2013 \\
a_0+a_1+...+a_{m-1}+a_{m} = 11\end{cases}$$
Find all natural numbers $x,y$ such that $$x^5=y^5+10y^2+20y+1.$$
Let $n$ be a positive integer. Prove that there is no positive integer solution to thxe equation $(x + 2)^n - x^n = 1 + 7^n$.
Find the roots $ r_i \in \mathbb{R}$ of the polynomial \[ p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n\] satisfying \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]
Number of solutions of the equation $3^x+4^x=8^x$ in reals is
[list=1]
[*] 0
[*] 1
[*] 2
[*] $\infty$
[/list]
Let $r$ be a positive integer and let $a_r$ be the number of solutions to the equation $3^x-2^y=r$ ,such that $0\leq x,y\leq 5781$ are integers. What is the maximal value of $a_r$?
Find all ordered pairs of integers $x,y$ such that $$xy(x^2y^2 - 12xy- 12x- 12y+2) = (2x + 2y)^2.$$
[i]Proposed by Henry Jiang[/i]