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

2018 Malaysia National Olympiad, A6
Tags: power , number theory , prime number
Determine the smallest prime $p$ such that $2018!$ is divisible by $p^{3}$ , but not divisible by $p^{4}$.

1945 Moscow Mathematical Olympiad, 096
Tags: power , number theory , 3-digit
Find three-digit numbers such that any its positive integer power ends with the same three digits and in the same order.

1990 Nordic, 2
Tags: power , inequalities
Let $a_1, a_2, . . . , a_n$ be real numbers. Prove $\sqrt[3]{a_1^3+ a_2^3+ . . . + a_n^3} \le \sqrt{a_1^2+ a_2^2+ . . . + a_n^2} $ (1) When does equality hold in (1)?

2002 Estonia National Olympiad, 2
Tags: power , digit , number theory
Does there exist an integer containing only digits $2$ and $0$ which is a $k$-th power of a positive integer ($k \ge2$)?

2009 Thailand Mathematical Olympiad, 7
Tags: sum of powers , power , number theory
Let $a, b, c$ be real numbers, and define $S_n = a^n + b^n + c^n$ for positive integers $n$. Suppose that $S_1, S_2, S_3$ are integers satisfying $6 | 5S_1 - 3S_2 - 2S_3$. Show that $S_n$ is an integer for all positive integers $n$.

2018 NZMOC Camp Selection Problems, 9
Tags: diophantine , diophantine equation , number theory , power
Let $x, y, p, n, k$ be positive integers such that $$x^n + y^n = p^k.$$ Prove that if $n > 1$ is odd, and $p$ is an odd prime, then $n$ is a power of $p$.

2012 India Regional Mathematical Olympiad, 2
Tags: number theory , divide , divisible , power
Let $a,b,c$ be positive integers such that $a|b^3, b|c^3$ and $c|a^3$. Prove that $abc|(a+b+c)^{13}$

2001 Estonia Team Selection Test, 5
Tags: product , power , number theory , digit , prime
Find the exponent of $37$ in the representation of the number $111...... 11$ with $3\cdot 37^{2000}$ digits equals to $1$, as product of prime powers

1996 Estonia National Olympiad, 4
Tags: decimal representation , power , prime
Prove that for each prime number $p > 5$ there exists a positive integer n such that $p^n$ ends in $001$ in decimal representation.

1960 Putnam, B4
Tags: arithmetic sequence , power
Consider the arithmetic progression $a, a+d, a+2d,\ldots$ where $a$ and $d$ are positive integers. For any positive integer $k$, prove that the progression has either no $k$-th powers or infinitely many.

2017 F = ma, 8
Tags: power
8) A train, originally of mass $M$, is traveling on a frictionless straight horizontal track with constant speed $v$. Snow starts to fall vertically and sticks to the train at a rate of $\rho$, where $\rho$ has units of kilograms per second. The train's engine keeps the train moving at constant speed $v$ as snow accumulates on the train. What is the minimum power required from the engine to keep the train traveling at a constant speed v? A) $0$ B) $Mgv$ C) $\frac{1}{2}Mv^2$ D) $\frac{1}{2}pv^2$ E) $\rho v^2$

2012 India Regional Mathematical Olympiad, 2
Tags: power , number theory , divisibility , divisible
Let $a,b,c$ be positive integers such that $a|b^2, b|c^2$ and $c|a^2$. Prove that $abc|(a+b+c)^{7}$

2025 Kosovo National Mathematical Olympiad`, P3
Tags: number theory , decimal representation , power
Find all pairs of natural numbers $(m,n)$ such that the number $5^m+6^n$ has all same digits when written in decimal representation.

2017 Germany, Landesrunde - Grade 11/12, 3
Tags: number theory , prime number , exponent , power , catalan
Find the smallest prime number that can not be written in the form $\left| 2^a-3^b \right|$ with non-negative integers $a,b$.

2017 JBMO Shortlist, NT5
Tags: number theory , power , positive integer , prime
Find all positive integers $n$ such that there exists a prime number $p$, such that $p^n-(p-1)^n$ is a power of $3$. Note. A power of $3$ is a number of the form $3^a$ where $a$ is a positive integer.

1964 All Russian Mathematical Olympiad, 042
Tags: number theory , power
Prove that for no natural $m$ a number $m(m+1)$ is a power of an integer.

2013 Estonia Team Selection Test, 5
Tags: number theory , power
Call a tuple $(b_m, b_{m+1},..., b_n)$ of integers perfect if both following conditions are fulfilled: 1. There exists an integer $a > 1$ such that $b_k = a^k + 1$ for all $k = m, m + 1,..., n$ 2. For all $k = m, m + 1,..., n,$ there exists a prime number $q$ and a non-negative integer $t$ such that $b_k = q^t$. Prove that if $n - m$ is large enough then there is no perfect tuples, and find all perfect tuples with the maximal number of components.

Istek Lyceum Math Olympiad 2016, 3
Tags: power , number theory
Let $n$, $m$ and $k$ be positive integers satisfying $(n-1)n(n+1)=m^k.$ Prove that $k=1.$

2014 Regional Competition For Advanced Students, 3
Tags: power , number theory , recurrence relation , number theory with sequences , sequence , algebra
The sequence $(a_n)$ is defined with the recursion $a_{n + 1} = 5a^6_n + 3a^3_{n-1} + a^2_{n-2}$ for $n\ge 2$ and the set of initial values $\{a_0, a_1, a_2\} = \{2013, 2014, 2015\}$. (That is, the initial values are these three numbers in any order.) Show that the sequence contains no sixth power of a natural number.

2018 Polish Junior MO Finals, 1
Tags: power , number theory , perfect square
Positive odd integers $a, b$ are such that $a^bb^a$ is a perfect square. Show that $ab$ is a perfect square.

2012 India Regional Mathematical Olympiad, 2
Tags: power , number theory , divide , divisible
Let $a,b,c$ be positive integers such that $a|b^4, b|c^4$ and $c|a^4$. Prove that $abc|(a+b+c)^{21}$

2016 IFYM, Sozopol, 8
Tags: power , algebra , sum
Let $a_i$, $i=1,2,…2016$, be fixed natural numbers. Prove that there exist infinitely many 2016-tuples $x_1,x_2…x_{2016}$ of natural numbers, for which the sum $\sum_{i=1}^{2016}{a_i x_i^i}$ is a 2017-th power of a natural number.

2016 Flanders Math Olympiad, 2
Tags: power , number theory , divide , divisor
Determine the smallest natural number $n$ such that $n^n$ is not a divisor of the product $1\cdot 2\cdot 3\cdot ... \cdot 2015\cdot 2016$.

2003 Junior Balkan Team Selection Tests - Romania, 2
Tags: odd , number theory , decimal representation , power
Let $a$ be a positive integer such that the number $a^n$ has an odd number of digits in the decimal representation for all $n > 0$. Prove that the number $a$ is an even power of $10$.

1985 AMC 8, 3
Tags: power , powers of 10 , arithmetic
$ \frac{10^7}{5 \times 10^4}\equal{}$ \[ \textbf{(A)}\ .002 \qquad \textbf{(B)}\ .2 \qquad \textbf{(C)}\ 20 \qquad \textbf{(D)}\ 200 \qquad \textbf{(E)}\ 2000 \]