Olympiad problems

2013 Irish Math Olympiad, 1

Find the smallest positive integer $m$ such that $5m$ is an exact 5th power, $6m$ is an exact 6th power, and $7m$ is an exact 7th power.

2006 Swedish Mathematical Competition, 6

Tags: algebra , Exponents , Sweden

Determine all positive integers $a,b,c$ satisfying $a^{(b^c)}=(b^a)^c$

2017 Macedonia JBMO TST, 5

Tags: number theory , Digits , prime factors , Exponents

Find all the positive integers $n$ so that $n$ has the same number of digits as its number of different prime factors and the sum of these different prime factors is equal to the sum of exponents of all these primes in factorization of $n$.

2019 AMC 10, 1

Tags: AMC 10 , AMC , AMC 10 A , 2019 AMC 10A , 2019 AMC , Exponents

What is the value of $$2^{\left(0^{\left(1^9\right)}\right)}+\left(\left(2^0\right)^1\right)^9\,?$$ $\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4$

2015 Middle European Mathematical Olympiad, 4

Tags: number theory , relatively prime , Exponents , factorization , Euclidean algorithm

Find all pairs of positive integers $(m,n)$ for which there exist relatively prime integers $a$ and $b$ greater than $1$ such that $$\frac{a^m+b^m}{a^n+b^n}$$ is an integer.

2023 AMC 10, 8

Tags: AMC , AMC 10 , AMC 10 B , Exponents

What is the units digit of $2022^{2023} + 2023^{2022}$? $\textbf{(A)}~7\qquad\textbf{(B)}~1\qquad\textbf{(C)}~3\qquad\textbf{(D)}~5\qquad\textbf{(E)}~9$

2021 Brazil Undergrad MO, Problem 4

Tags: real analysis , limits , mean , Exponents , Brazilian Undergrad MO 2021

For every positive integeer $n>1$, let $k(n)$ the largest positive integer $k$ such that there exists a positive integer $m$ such that $n = m^k$. Find $$lim_{n \rightarrow \infty} \frac{\sum_{j=2}^{j=n+1}{k(j)}}{n}$$

1959 AMC 12/AHSME, 5

Tags: AMC , Exponents , algebra , AMC 12

The value of $\left(256\right)^{.16}\left(256\right)^{.09}$ is: $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 256.25\qquad\textbf{(E)}\ -16$

2018 Junior Regional Olympiad - FBH, 5

Tags: Integers , exponential , Exponents

Find all integers $x$ and $y$ such that $2^x+1=y^2$

2001 Saint Petersburg Mathematical Olympiad, 9.6

Tags: number theory , diophanative equations , Exponents

Find all positive integer solution: $$k^m+m^n=k^n+1$$ [I]Proposed by V. Frank, F. Petrov[/i]

2019 Jozsef Wildt International Math Competition, W. 10

Tags: integration , trigonometry , Exponents , calculus

If ${si}(x) =- \int \limits_{x}^{\infty}\left(\frac{\sin t}{t}\right)dt; x>0$ then $$\int \limits_{e}^{e^2} \left(\frac{1}{x}\left(si\left(e^4x\right)-si\left(e^3x\right)\right)\right)\,dx=\int \limits_{3}^{e^4} \left(\frac{1}{x}\left(\operatorname{si}\left(e^2x\right)-si\left(ex\right)\right)\right)dx$$