Found problems: 916
Find all integers $x,y$ which satisfy the equation $xy=20-3x+y$.
Determine all pairs $(m, n)$ of non-negative integers that satisfy the equation
$$
20^m - 10m^2 + 1 = 19^n.
$$
Find all positive integer triples $(x, y, z)$ such that $1 + 2^x \cdot 3^y=5^z$ is true.
Determine all $(m,n) \in \mathbb{Z}^+ \times \mathbb{Z}^+$ which satisfy $3^m-7^n=2.$
Determine all triples of prime numbers $(p, q, r)$ that satisfy
\[p2^q + r^2 = 2025.\]
Proposed by [i]Ilija Jovcevski[/i]
Find the triple of positive integers $(x,y,z)$ with $z$ least possible for which there are positive integers $a, b, c, d$ with the following properties:
(i) $x^y = a^b = c^d$ and $x > a > c$
(ii) $z = ab = cd$
(iii) $x + y = a + b$.
Find positive integers $x, y, z$ such that $x > z > 1999 \cdot 2000 \cdot 2001 > y$ and $2000x^{2}+y^{2}= 2001z^{2}.$
Determine all $(m,n) \in \mathbb{Z}^+ \times \mathbb{Z}^+$ which satisfy $3^m-7^n=2.$
Find all pairs of solutions $(x,y)$:
\[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]
Find all non-negative integers $x, y$ and $z$ such that $x^3 + 2y^3 + 4z^3 = 9!$
Show that there are infintely many pairs $(a,b)$ of relatively prime integers (not necessarily positive) such that both the equations \begin{eqnarray*} x^2 +ax +b &=& 0 \\ x^2 + 2ax + b &=& 0 \\ \end{eqnarray*} have integer roots.
Determine all three primes $(a, b, c)$ that satisfied the equality $a^2+ab+b^2=c^2+3$.
Prove that if $n$ is a positive integer such that the equation \[ x^3-3xy^2+y^3=n \] has a solution in integers $x,y$, then it has at least three such solutions. Show that the equation has no solutions in integers for $n=2891$.
If $p$ is a prime number such that there exist positive integers $a$ and $b$ such that $\frac{1}{p}=\frac{1}{a^2}+\frac{1}{b^2}$ then $p$ is
(A): $3$, (B): $5$, (C): $11$, (D): $7$, (E) None of the above.
One of Euler's conjectures was disproved in the $1980$s by three American Mathematicians when they showed that there is a positive integer $n$ such that \[n^{5}= 133^{5}+110^{5}+84^{5}+27^{5}.\] Find the value of $n$.
Determine all real values of the parameter $a$ for which the equation
\[16x^4 -ax^3 + (2a + 17)x^2 -ax + 16 = 0\]
has exactly four distinct real roots that form a geometric progression.
Show that there do not exist positive integers $m$ and $n$ such that \[ \dfrac{m}{n} + \dfrac{n+1}{m} = 4 . \]
Let $S$ be the set of positive integers $x$ for which there exist positive integers $y$ and $m$ such that $y^2-2^m=x^2$.
(a) Find all of the elements of $S$.
(b) Find all $x$ such that both $x$ and $x+1$ are in $S$.
Solve the equation for primes $p$ and $q$: $$p^3-q^3=pq^3-1.$$
Find all solutions of $2^n + 7 = x^2$ in which n and x are both integers . Prove that there are no other solutions.
Find all pairs $(x, y)$ of integers that satisfy $x^2 + y^2 + 3^3 = 456\sqrt{x - y}$.
Prove that the equation $x^8 = n! + 1$ has finitely many solutions in positive integers.
Find all integers $a, b$ such that $$a^2 + b = b^{2022}.$$
Prove that for every integer $n\ge 3$ there are such positives integers $x$ and $y$ such that $2^n = 7x^2 + y^2$
Find all pairs of integers $(m,n)$ such that $m^3-n^3=2mn +8$