Olympiad problems

1984 All Soviet Union Mathematical Olympiad, 379

Find integers $m$ and $n$ such that $(5 + 3 \sqrt2)^m = (3 + 5 \sqrt2)^n$.

2014 India PRMO, 7

Tags: algebra , exponential

If $x^{x^4}=4 $ what is the value of $x^{x^2}+x^{x^8} $ ?

1980 Putnam, B1

Tags: inequalities , exponential

For which real numbers $c$ is $$\frac{e^x +e^{-x} }{2} \leq e^{c x^2 }$$ for all real $x?$

2015 Finnish National High School Mathematics Comp, 3

Tags: number theory , factorial , factor , exponential , divide

Determine the largest integer $k$ for which $12^k$ is a factor of $120! $

1996 Tuymaada Olympiad, 7

Tags: algebra , equation , 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 $a * b = b * a$.

2007 Germany Team Selection Test, 3

Tags: modular arithmetic , number theory , divisibility , exponential

For all positive integers $n$, show that there exists a positive integer $m$ such that $n$ divides $2^{m} + m$. [i]Proposed by Juhan Aru, Estonia[/i]

1958 February Putnam, A3

Tags: probability , expected value , exponential

Real numbers are chosen at random from the interval $[0,1].$ If after choosing the $n$-th number the sum of the numbers so chosen first exceeds $1$, show that the expected value for $n$ is $e$.

2013 Danube Mathematical Competition, 3

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

Determine the natural numbers $m,n$ such as $85^m-n^4=4$

2023 Romania National Olympiad, 1

Tags: exponential , exponential equation , algebra

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

2006 Greece JBMO TST, 2

Tags: prime number , exponential , prime , number theory

Let $a,b,c$ be positive integers such that the numbers $k=b^c+a, l=a^b+c, m=c^a+b$ to be prime numbers. Prove that at least two of the numbers $k,l,m$ are equal.

2009 IMAC Arhimede, 5

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

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

2021 JBMO Shortlist, N7

Tags: shortlist , game , exponential , number theory

Alice chooses a prime number $p > 2$ and then Bob chooses a positive integer $n_0$. Alice, in the first move, chooses an integer $n_1 > n_0$ and calculates the expression $s_1 = n_0^{n_1} + n_1^{n_0}$; then Bob, in the second move, chooses an integer $n_2 > n_1$ and calculates the expression $s_2 = n_1^{n_2} + n_2^{n_1}$; etc. one by one. (Each player knows the numbers chosen by the other in the previous moves.) The winner is the one who first chooses the number $n_k$ such that $p$ divides $s_k(s_1 + 2s_2 + · · · + ks_k)$. Who has a winning strategy? Proposed by [i]Borche Joshevski, Macedonia[/i]

2009 Federal Competition For Advanced Students, P1, 1

Tags: factorial , exponential , exponential inequality , inequalities , algebra , diophantine , induction

Show that for all positive integer $n$ the following inequality holds $3^{n^2} > (n!)^4$ .

1982 All Soviet Union Mathematical Olympiad, 341

Tags: algebra , inequalities , exponential

Prove that the following inequality is valid for the positive $x$: $$2^{x^{1/12}}+ 2^{x^{1/4}} \ge 2^{1 + x^{1/6} }$$

2007 Dutch Mathematical Olympiad, 4

Tags: perfect square , exponential , number theory

Determine the number of integers $a$ satisfying $1 \le a \le 100$ such that $a^a$ is a perfect square. (And prove that your answer is correct.)

1988 All Soviet Union Mathematical Olympiad, 471

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

Find all positive integers $n$ satisfying $\left(1 +\frac{1}{n}\right)^{n+1} = \left(1 + \frac{1}{1998}\right)^{1998}$.

1976 IMO Longlists, 47

Tags: digit , decimal representation , exponential , number theory

Prove that $5^n$ has a block of $1976$ consecutive $0's$ in its decimal representation.

2006 Germany Team Selection Test, 2

Tags: factorial , modular arithmetic , number theory , divisibility , exponential , chinese remainder theorem

Find all positive integers $ n$ such that there exists a unique integer $ a$ such that $ 0\leq a < n!$ with the following property: \[ n!\mid a^n \plus{} 1 \] [i]Proposed by Carlos Caicedo, Colombia[/i]

2017 India PRMO, 5

Tags: geometric sequence , geometric progression , exponential , algebra

Let $u, v,w$ be real numbers in geometric progression such that $u > v > w$. Suppose $u^{40} = v^n = w^{60}$. Find the value of $n$.

2015 Hanoi Open Mathematics Competitions, 13

Tags: number theory , relatively prime , exponential

Let $m$ be given odd number, and let $a, b$ denote the roots of equation $x^2 + mx - 1 = 0$ and $c = a^{2014} + b^{2014}$ , $d =a^{2015} + b^{2015}$ . Prove that $c$ and $d$ are relatively prime numbers.

2010 Belarus Team Selection Test, 7.3

Tags: inequalities , exponential inequality , exponential , algebra

Prove that all positive real $x, y, z$ satisfy the inequality $x^y + y^z + z^x > 1$. (D. Bazylev)

1948 Moscow Mathematical Olympiad, 148

Tags: diophantine , exponential , algebra , number theory

a) Find all positive integer solutions of the equation $x^y = y^x$ ($x \ne y$). b) Find all positive rational solutions of the equation $x^y = y^x$ ($x \ne y$).

2015 Abels Math Contest (Norwegian MO) Final, 4

Tags: number theory , exponential , perfect square

a. Determine all nonnegative integers $x$ and $y$ so that $3^x + 7^y$ is a perfect square and $y$ is even. b. Determine all nonnegative integers $x$ and $y$ so that $3^x + 7^y$ is a perfect square and $y$ is odd

2019 Korea USCM, 5

Tags: series , exponential , logarithm , sequence

A sequence $\{a_n\}_{n\geq 1}$ is defined by a recurrence relation $$a_1 = 1,\quad a_{n+1} = \log \frac{e^{a_n}-1}{a_n}$$ And a sequence $\{b_n\}_{n\geq 1}$ is defined as $b_n = \prod\limits_{i=1}^n a_i$. Evaluate an infinite series $\sum\limits_{n=1}^\infty b_n$.