Olympiad problems

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 $(c, d) = 1$?

2009 IMAC Arhimede, 5

Tags: diophantine equation , diophantine , exponential equation , exponential , number theory

Find all natural numbers $x$ and $y$ such that $x^y-y^x=1$ .

2009 Bulgaria National Olympiad, 1

Tags: diophantine equation , exponential equation , coprime , number theory

The natural numbers $a$ and $b$ satisfy 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 $1$).

2025 District Olympiad, P2

Tags: floor function , exponential equation

Find the real numbers $x$ such that $$3^x + 3^{\lfloor x\rfloor} + 3^{\{x\}}=4.$$

2023 Romania National Olympiad, 1

Tags: algebra , exponential , exponential equation

Solve the following equation for real values of $x$: \[ 2 \left( 5^x + 6^x - 3^x \right) = 7^x + 9^x. \]

2020 OMpD, 1

Tags: polynomial , algebra , diophantine equation , exponential equation , number theory

Determine all pairs of positive integers $(x, y)$ such that: $$x^4 - 6x^2 + 1 = 7\cdot 2^y$$

2015 Balkan MO Shortlist, A5

Tags: algebra , inequalities , exponential inequality , exponential , exponential equation

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)

1996 Tuymaada Olympiad, 7

Tags: equation , algebra , exponential , rational , exponential equation

In the set of all positive real numbers define the operation $a * b = a^b$ . Find all positive rational numbers for which $a * b = b * a$.

2016 Hanoi Open Mathematics Competitions, 1

Tags: algebra , exponential equation

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.

2015 District Olympiad, 2

Tags: algebra , exponential equation , system of equations

Solve in $ \mathbb{Z} $ the following system of equations: $$ \left\{\begin{matrix} 5^x-\log_2 (y+3) = 3^y\\ 5^y -\log_2 (x+3)=3^x\end{matrix}\right. . $$

2014 Indonesia MO Shortlist, N3

Tags: number theory , diophantine equation , exponential equation

Find all pairs of natural numbers $(a, b)$ that fulfill $a^b=(a+b)^a$.

2018 Bosnia and Herzegovina Junior BMO TST, 2

Tags: number theory , diophantine equation , exponential equation , prime number

Find all integer triples $(p,m,n)$ that satisfy: $p^m-n^3=27$ where $p$ is a prime number.