This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 49

2015 Indonesia MO Shortlist, N4
Tags: number theory , exponential equation , exponential , greatest common divisor , diophantine equation
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 $(c, d) = 1$?

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]

2006 Greece JBMO TST, 2
Tags: number theory , prime number , prime , exponential
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.

2023 Romania National Olympiad, 1
Tags: exponential , exponential equation , algebra
Solve the following equation for real values of $x$: \[ 2 \left( 5^x + 6^x - 3^x \right) = 7^x + 9^x. \]

2015 Balkan MO Shortlist, A5
Tags: inequalities , algebra , 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)

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]

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)

2017 Balkan MO Shortlist, N3
Tags: number theory , divide , exponential
Prove that for all positive integer $n$, there is a positive integer $m$ that $7^n | 3^m +5^m -1$.

1954 Putnam, B7
Tags: limit , exponential
Let $a>0$. Show that $$ \lim_{n \to \infty} \sum_{s=1}^{n} \left( \frac{a+s}{n} \right)^{n}$$ lies between $e^a$ and $e^{a+1}.$

1976 IMO Longlists, 47
Tags: number theory , digit , decimal representation , exponential
Prove that $5^n$ has a block of $1976$ consecutive $0's$ in its decimal representation.

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 $x^y = y^x$ ($x \ne y$).

1983 All Soviet Union Mathematical Olympiad, 360
Tags: number theory , divisible , exponential
Given natural $n,m,k$. It is known that $m^n$ is divisible by $n^m$, and $n^k$ is divisible by $k^n$. Prove that $m^k$ is divisible by $k^m$.

2014 Federal Competition For Advanced Students, P2, 3
Tags: inequalities , triangle inequality , geometric inequality , exponential
(i) For which triangles with side lengths $a, b$ and $c$ apply besides the triangle inequalities $a + b> c, b + c> a$ and $c + a> b$ also the inequalities $a^2 + b^2> c^2, b^2 + c^2> a^2$ and $a^2 + c^2> b^2$ ? (ii) For which triangles with side lengths $a, b$ and $c$ apply besides the triangle inequalities $a + b> c, b + c> a$ and $c + a> b$ also for all positive natural $n$ the inequalities $a^n + b^n> c^n, b^n + c^n> a^n$ and $a^n + c^n> b^n$ ?

1990 Mexico National Olympiad, 3
Tags: number theory , exponential
Show that $n^{n-1}-1$ is divisible by$ (n-1)^2$ for $n > 2$.

2013 Danube Mathematical Competition, 3
Tags: diophantine equation , diophantine , exponential equation , exponential , number theory
Determine the natural numbers $m,n$ such as $85^m-n^4=4$

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 $\sum\limits_{n=1}^\infty b_n$.

2009 Belarus Team Selection Test, 2
Tags: algebra , constant , exponential
Find all $n \in N$ for which the value of the expression $x^n+y^n+z^n$ is constant for all $x,y,z \in R$ such that $x+y+z=0$ and $xyz=1$. D. Bazylev

1988 All Soviet Union Mathematical Olympiad, 471
Tags: diophantine , diophantine equation , exponential , algebra , number theory
Find all positive integers $n$ satisfying $\left(1 +\frac{1}{n}\right)^{n+1} = \left(1 + \frac{1}{1998}\right)^{1998}$.

2024 VJIMC, 1
Tags: inequalities , function , exponential , calculus
Suppose that $f:[-1,1] \to \mathbb{R}$ is continuous and satisfies \[\left(\int_{-1}^1 e^xf(x) dx\right)^2 \ge \left(\int_{-1}^1 f(x) dx\right)\left(\int_{-1}^1 e^{2x}f(x) dx\right).\] Prove that there exists a point $c \in (-1,1)$ such that $f(c)=0$.

1957 Putnam, A2
Tags: physics , exponential
Let $a>1.$ A uniform wire is bent into a form coinciding with the portion of the curve $y=e^x$ for $x\in [0,a]$, and the line segment $y=e^a$ for $x\in [a-1,a].$ The wire is then suspended from the point $(a-1, e^a)$ and a horizontal force $F$ is applied to the point $(0,1)$ to hold the wire in coincidence with the curve and segment. Show that the force $F$ is directed to the right.

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 $x?$

1984 All Soviet Union Mathematical Olympiad, 379
Tags: diophantine , diophantine equation , number theory , algebra , exponential
Find integers $m$ and $n$ such that $(5 + 3 \sqrt2)^m = (3 + 5 \sqrt2)^n$.

2017 India PRMO, 5
Tags: geometric sequence , geometric progression , algebra , exponential
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 $n$.

2006 IMO Shortlist, 7
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]

2016 Dutch IMO TST, 2
Tags: number theory , divisible , divisor , exponential
Determine all pairs $(a, b)$ of integers having the following property: there is an integer $d \ge 2$ such that $a^n + b^n + 1$ is divisible by $d$ for all positive integers $n$.