Found problems: 988
Find all positive integers $a$, $b$ and $c$ such that $ab$ is a square, and
\[a+b+c-3\sqrt[3]{abc}=1.\]
[i]Proposed by usjl[/i]
Prove that the equation
\[ 2^x + 5^y - 31^z = n! \]
has only a finite number of non-negative integer solutions $x,y,z,n$.
Find all pairs of integers $(m, n)$ such that $(m - n)^2 =\frac{4mn}{m + n - 1}$
Consider the system \[x+y=z+u,\] \[2xy=zu.\] Find the greatest value of the real constant $m$ such that $m \le \frac{x}{y}$ for any positive integer solution $(x, y, z, u)$ of the system, with $x \ge y$.
Find all integers $x$ and $y$ satisfying the following equation $x^2 - 2xy + 5y^2 + 2x - 6y - 3 = 0$.
Find all pairs of integers $x, y$ such that $y^5 + 2xy = x^2 + 2y^4.$
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Find all pairs of non-negative integers $m$ and $n$ that satisfy $$3 \cdot 2^m + 1 = n^2.$$
Note that $8^3 - 7^3 = 169 = 13^2$ and $13 = 2^2 + 3^2.$ Prove that if the difference between two consecutive cubes is a square, then it is the square of the sum of two consecutive squares.
Are there any integral solutions to $n^2 + (n+1)^2 + (n+2)^2 = m^2$ ?
Determine all the integer solutions of the equation $3x^4-2024y+1= 0$.
Find the solutions of positive integers for the system $xy + x + y = 71$ and $x^2y + xy^2 = 880$.
Find all triples $(a, b, c)$ of positive integers to the equation \[a! b! = a!+b!+c!.\]
Find all pairs of relatively prime integers $(x, y)$ that satisfy equality $2 (x^3 - x) = 5 (y^3 - y)$.
Find all triples of integers $(x, y, z)$ such that $x^4 + 5y^4 = z^4$.
Determine all prime numbers $p$ and all positive integers $x$ and $y$ satisfying $$x^3+y^3=p(xy+p).$$
Find the number of different positive integer triples $(x, y,z)$ satisfying the equations
$x^2 + y -z = 100$ and $x + y^2 - z = 124$:
(a) Find all pairs $(x,k)$ of positive integers such that $3^k -1 = x^3$ .
(b) Prove that if $n > 1$ is an integer, $n \ne 3$, then there are no pairs $(x,k)$ of positive integers such that $3^k -1 = x^n$.
If a is a natural number, show that the number of positive integral solutions of the indeterminate equation
$$x_1 + 2x_2 + 3x_3 + ... + nx_n = a \ \ (1) $$
is equal to the number of non-negative integral solutions of
$$y_1 + 2y_2 + 3y_3 + ... + ny_n = a - \frac{n(n + 1)}{2} \ \ (2)$$
[By a solution of equation (1), we mean a set of numbers $\{x_1, x_2,..., x_n\}$ which satisfies equation (1)].
Prove there aren't positive integers $a, b, c, d$ forming an arithmetic progression such that $ ab + 1, ac + 1, ad + 1, bc + 1, bd + 1, cd + 1 $ are all perfect squares.
Do there exist integers $x$ and $y$ such that $19^{19} = x^3 +y^4$ ? Justify your answer.
Find all pairs $(a,b)$ of positive integers that satisfy the equation \[a^{a^{a}}= b^{b}.\]
Find all positive integers $x,y,z$ such that $7^x + 13^y = 8^z$
Find all positive integers $p,q,r,s>1$ such that $$p!+q!+r!=2^s.$$
Show that there are no triples of positive integers $(x,y,z)$ satisfying the equation \[x^2= 5^y+3^z\]
Prove that there exist infinitely many positive integers $n$ such that the number $\frac{1^2+2^2+\cdots+n^2}{n}$ is a perfect square. Obviously, $1$ is the least integer having this property. Find the next two least integers having this property.