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: 25

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 $p$.

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.

2009 Junior Balkan Team Selection Tests - Romania, 2
Tags: number theory , Power , consecutive
A positive integer is called [i]saturated [/i]i f any prime factor occurs at a power greater than or equal to $2$ in its factorisation. For example, numbers $8 = 2^3$ and $9 = 3^2$ are saturated, moreover, they are consecutive. Prove that there exist infinitely many saturated consecutive numbers.

2016 Flanders Math Olympiad, 2
Tags: Power , number theory , divides , 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 $10$.

1987 Bundeswettbewerb Mathematik, 1
Tags: number theory , Digit , prime , Power
Let $p>3$ be a prime and $n$ a positive integer such that $p^n$ has $20$ digits. Prove that at least one digit appears more than twice in this number.

2013 Saudi Arabia Pre-TST, 2.3
Tags: number theory , Power , last digits
The positive integer $a$ is relatively prime with $10$. Prove that for any positive integer $n$, there exists a power of $a$ whose last $n$ digits are $\underbrace{0...0}_\text{n-1}1$.

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.

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.

2001 Estonia Team Selection Test, 5
Tags: primes , Product , Power , number theory , Digits
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

2015 Estonia Team Selection Test, 7
Tags: number theory , Digits , 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.

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) $\rho v^2$

2001 Estonia Team Selection Test, 5
Tags: primes , Product , Power , number theory , Digits
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

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.

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

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

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

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.

1996 Estonia National Olympiad, 4
Tags: decimal representation , primes , 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.

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

2014 Contests, 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.

2015 Estonia Team Selection Test, 7
Tags: number theory , Digits , 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.

2009 Thailand Mathematical Olympiad, 7
Tags: number theory , Sum of powers , Power
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 $n$.

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.

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.