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.

AND:
OR:
NO:

Found problems: 49

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) $\rho v^2$

2005 Austrian-Polish Competition, 10
Tags: power , inequalities
Determine all pairs $(k,n)$ of non-negative integers such that the following inequality holds $\forall x,y>0$: \[1+ \frac{y^n}{x^k} \geq \frac{(1+y)^n}{(1+x)^k}.\]

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

1995 North Macedonia National Olympiad, 3
Tags: number theory , consecutive , power
Prove that the product of $8$ consecutive natural numbers can never be a fourth power of natural number.

1955 Moscow Mathematical Olympiad, 303
Tags: number theory , trinomial , integer , power , algebra
The quadratic expression $ax^2+bx+c$ is the $4$-th power (of an integer) for any integer $x$. Prove that $a = b = 0$.

2012 India Regional Mathematical Olympiad, 2
Tags: divide , power , divisible , number theory
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: algebra , sum , power
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.

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

MIPT student olimpiad autumn 2022, 3
Tags: power , ratio
How many ways are there (in terms of power) to represent the number 1 as a finite number or an infinite sum of some subset of the set: {$\phi^{-n} | n \in Z^+$} $\phi=\frac{1+\sqrt5}{2}$

2015 Estonia Team Selection Test, 7
Tags: number theory , digit , consecutive , power
Prove that for every prime number $p$ and positive integer $a$, there exists a natural number $n$ such that $p^n$ contains $a$ consecutive equal digits.

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

2014 Saudi Arabia Pre-TST, 3.4
Tags: number theory , power , last digit
Prove that there exists a positive integer $n$ such that the last digits of $n^3$ are $...201320132013$.

2001 Estonia Team Selection Test, 5
Tags: prime , number theory , product , power , digit
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

1990 Nordic, 2
Tags: inequalities , power
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)?

VMEO III 2006 Shortlist, N11
Tags: number theory , power
Prove that the composition of the sets of one of the following two forms is finite: (a) $2^{2^n}+1$ (b) $6^{2^n}+1$

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 $abc|(a+b+c)^{13}$

1981 Bundeswettbewerb Mathematik, 1
Tags: number theory , digit , power
Let $a$ and $n$ be positive integers and $s = a + a^2 + \cdots + a^n$. Prove that the last digit of $s$ is $1$ if and only if the last digits of $a$ and $n$ are both equal to $1$.

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 $a,b$.

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

2021 Brazil National Olympiad, 3
Tags: number theory , perfect square , floor function , irrational number , power
Find all positive integers \(k\) for which there is an irrational \(\alpha>1\) and a positive integer \(N\) such that \(\left\lfloor\alpha^{n}\right\rfloor\) is a perfect square minus \(k\) for every integer \(n\) with \(n>N\).

2014 Regional Competition For Advanced Students, 3
Tags: number theory , algebra , power , recurrence relation , number theory with sequences , sequence
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.

2013 IMAR Test, 2
Tags: number theory , digit sum , sequence , power
For every non-negative integer $n$ , let $s_n$ be the sum of digits in the decimal expansion of $2^n$. Is the sequence $(s_n)_{n \in \mathbb{N}}$ eventually increasing ?

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.

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.