Olympiad problems

2021 Brazil Undergrad MO, Problem 4

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 $$lim_{n \rightarrow \infty} \frac{\sum_{j=2}^{j=n+1}{k(j)}}{n}$$

2015 Middle European Mathematical Olympiad, 4

Tags: factorization , euclidean algorithm , exponent , number theory , relatively prime

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.

2019 AMC 10, 1

Tags: exponent

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

2018 Junior Regional Olympiad - FBH, 5

Tags: exponent , exponential , integer

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

2015 NIMO Summer Contest, 4

Tags: algebra , exponent

Let $P$ be a function defined by $P(t)=a^t+b^t$, where $a$ and $b$ are complex numbers. If $P(1)=7$ and $P(3)=28$, compute $P(2)$. [i] Proposed by Justin Stevens [/i]

2017 Macedonia JBMO TST, 5

Tags: number theory , digit , exponent , prime factor

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 $n$.

2006 Swedish Mathematical Competition, 6

Tags: algebra , exponent

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

2024 Junior Balkan MO, 3

Tags: number theory , diophantine equation , exponent

Find all triples of positive integers $(x, y, z)$ that satisfy the equation $$2020^x + 2^y = 2024^z.$$ [i]Proposed by Ognjen Tešić, Serbia[/i]

2001 Saint Petersburg Mathematical Olympiad, 9.6

Tags: exponent , diophanative equations , number theory

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: calculus , trigonometry , exponent , integration

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

2023 AMC 10, 8

Tags: exponent

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

1959 AMC 12/AHSME, 5

Tags: algebra , exponent

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$