Found problems: 19
Find all pairs of positive integers $(a,b)$ such that
$$a!+b!=a^b + b^a.$$
Find all pairs of natural numbers $(a, b)$ that fulfill $a^b=(a+b)^a$.
Find all natural numbers $x$ and $y$ such that $x^y-y^x=1$ .
The natural numbers $a$ and $b$ satis fy the inequalities $a > b > 1$ . It is also known that the equation
$\frac{a^x - 1}{a - 1}=\frac{b^y - 1}{b - 1}$ has at least two solutions in natural numbers, when $x > 1$ and $y > 1$.
Prove that the numbers $a$ and $b$ are coprime (their greatest common divisor is $1$).
In nonnegative set of integers solve the equation:
$$(2^{2015}+1)^x + 2^{2015}=2^y+1$$
Determine all positive roots of the equation $ x^x = \frac{1}{\sqrt{2}}.$
Solve in $ \mathbb{Z} $ the following system of equations:
$$ \left\{\begin{matrix} 5^x-\log_2 (y+3) = 3^y\\ 5^y -\log_2 (x+3)=3^x\end{matrix}\right. . $$
Let $m, n$ be positive integers and $a, b$ positive real numbers different from $1$ such thath $m > n$ and
$$\frac{a^{m+1}-1}{a^m-1} = \frac{b^{n+1}-1}{b^n-1} = c$$. Prove that $a^m c^n > b^n c^{m}$
(Turkey)
Find the real numbers $x$ such that $$3^x + 3^{\lfloor x\rfloor} + 3^{\{x\}}=4.$$
Solve in positive integers:
\[x^{y^2+1}+y^{x^2+1}=2^z\]
Suppose that the natural number $a, b, c, d$ satisfy the equation $a^ab^{a + b} = c^cd^{c + d}$.
(a) If gcd $(a, b) = $ gcd $(c, d) = 1$, prove that $a = c$ and $b = d$.
(b) Does the conclusion $a = c$ and $b = d$ apply, without the condition gcd $(a, b) = $ gcd $(c, d) = 1$?
Determine all pairs of positive integers $(x, y)$ such that:
$$x^4 - 6x^2 + 1 = 7\cdot 2^y$$
If $2016 = 2^5 + 2^6 + ...+ 2^m$ then $m$ is equal to
(A): $8$ (B): $9$ (C): $10$ (D): $11$ (E): None of the above.
Find all integer triples $(p,m,n)$ that satisfy:
$p^m-n^3=27$ where $p$ is a prime number.
Solve the following equation for real values of $x$:
\[
2 \left( 5^x + 6^x - 3^x \right) = 7^x + 9^x.
\]
Determine all positive roots of the equation $ x^x = \frac{1}{\sqrt{2}}.$
Find all real solutions to the equation $(x^2-9x+19)^{x^2-3x+2} = 1$.
Determine the natural numbers $m,n$ such as $85^m-n^4=4$
In the set of all positive real numbers define the operation $a * b = a^b$ .
Find all positive rational numbers for which $a * b = b * a$.