Found problems: 436
Find the solutions of positive integers for the system $xy + x + y = 71$ and $x^2y + xy^2 = 880$.
Find all triples of integers $(x, y, z)$ such that $x^4 + 5y^4 = z^4$.
(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)].
Do there exist integers $x$ and $y$ such that $19^{19} = x^3 +y^4$ ? Justify your answer.
Find all positive integers $x,y,z$ such that $7^x + 13^y = 8^z$
Solve the following equation in positive integers $x, y$: $x^{2017} - 1 = (x - 1)(y^{2015}- 1)$
Show that the only integral solution to
\[\left\{ \begin{array}{l}
xy + yz + zx = 3n^2 - 1\\
x + y + z = 3n \\
\end{array} \right.
\]
with $x \geq y \geq z$ is $x=n+1$, $y=n$, $z=n-1$.
Given a natural prime $ p, $ find the number of integer solutions of the equation $ p+xy=p(x+y). $
Find all primes $p, q$ satisfying the equation $2p^q - q^p = 7.$
Determine the smallest positive integer $k$ so that the equation $$2002x+273y=200201+k$$ has integer solutions, and for that value of $k$, find the number of solutions $\left (x,y\right )$ with $x$, $y$ positive integers that have the equation.
Find all integers $a, b, c, d$ such that $$\begin{cases} ab - 2cd = 3 \\ ac + bd = 1\end{cases}$$
Prove that the equation $ 2x^2 - 215y^2 = 1 $ has no integer solutions.
Find all primes $p,q, r$ such that $\frac{p^{2q}+q^{2p}}{p^3-pq+q^3} = r$.
Titu Andreescu, Mathematics Department, College of Texas, USA
Let $p$ be a prime number. Find all pairs $(x, y)$ of positive integers such that $x^3 + y^3 - 3xy = p -1$.
Find all positive integer triples $(x, y, z)$ such that $1 + 2^x \cdot 3^y=5^z$ is true.
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 all non-negative integers $x, y$ and $z$ such that $x^3 + 2y^3 + 4z^3 = 9!$
Determine all three primes $(a, b, c)$ that satisfied the equality $a^2+ab+b^2=c^2+3$.
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.
Solve in $N$:
$$\begin{cases} a^3=b^3+c^3+12a \\ a^2=5(b+c) \end{cases}$$
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 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 $m^3-n^3=2mn +8$