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

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

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.

2018 VJIMC, 1
Tags: equation , power , real number
Find all real solutions of the equation \[17^x+2^x=11^x+2^{3x}.\]

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

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.

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

2021 Auckland Mathematical Olympiad, 4
Tags: number theory , divide , divisible , power
Prove that there exist two powers of $7$ whose difference is divisible by $2021$.

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

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

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 $...201320132013$.

2021 Brazil Undergrad MO, Problem 3
Tags: real analysis , irrational number , approximation , perfect square , 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 of the form $m^2-k$ com $m \in \mathbb{Z}$ for every integer $n>N$.

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 $a,b$.

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.

1998 German National Olympiad, 6a
Tags: system of equations , symmetry , algebra , power , real
Find all real pairs $(x,y)$ that solve the system of equations \begin{align} x^5 &= 21x^3+y^3 \\ y^5 &= x^3+21y^3. \end{align}

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 $n$.

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.

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.

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 $a = b = 0$.

2006 Junior Tuymaada Olympiad, 2
Tags: number theory , power of number , prime , power
Ten different odd primes are given. Is it possible that for any two of them, the difference of their sixteenth powers to be divisible by all the remaining ones ?

MIPT student olimpiad autumn 2022, 3
Tags: ratio , power
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: {$\phi^{-n} | n \in Z^+$} $\phi=\frac{1+\sqrt5}{2}$

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

2015 Estonia Team Selection Test, 7
Tags: digit , consecutive , power , number theory
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.

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.

2013 Saudi Arabia Pre-TST, 2.3
Tags: number theory , power , last digit
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 $\underbrace{0...0}_\text{n-1}1$.