Olympiad problems

1987 Bundeswettbewerb Mathematik, 1

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.

2018 Polish Junior MO Finals, 1

Tags: perfect square , power , number theory

Positive odd integers $a, b$ are such that $a^bb^a$ is a perfect square. Show that $ab$ is a perfect square.

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 ($k \ge2$)?

2013 Estonia Team Selection Test, 5

Tags: power , number theory

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.

2020 Kosovo National Mathematical Olympiad, 1

Tags: power , compare , pow , algebra , number theory

Compare the following two numbers: $2^{2^{2^{2^{2}}}}$ and $3^{3^{3^{3}}}$.

2018 VJIMC, 1

Tags: equation , power , real number

Find all real solutions of the equation \[17^x+2^x=11^x+2^{3x}.\]

2001 Regional Competition For Advanced Students, 1

Tags: digit , power , sum of powers , sum , number theory

Let $n$ be an integer. We consider $s (n)$, the sum of the $2001$ powers of $n$ with the exponents $0$ to $2000$. So $s (n) = \sum_{k=0}^{2000}n ^k$ . What is the unit digit of $s (n)$ in the decimal system?

2025 Kosovo National Mathematical Olympiad`, P3

Tags: power , decimal representation , number theory

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.

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 $n$.

2021 Brazil Undergrad MO, Problem 3

Tags: power , approximation , irrational number , perfect square , real analysis

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 $n>N$.

1998 German National Olympiad, 6a

Tags: power , real , algebra , symmetry , system of equations

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}

2015 Estonia Team Selection Test, 7

Tags: number theory , digit , power , consecutive

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.

2018 Malaysia National Olympiad, A6

Tags: prime number , power , number theory

Determine the smallest prime $p$ such that $2018!$ is divisible by $p^{3}$ , but not divisible by $p^{4}$.

2013 Estonia Team Selection Test, 5

Tags: power , number theory

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.

2003 Junior Balkan Team Selection Tests - Romania, 2

Tags: odd , decimal representation , power , number theory

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 $10$.

2013 Saudi Arabia Pre-TST, 2.3

Tags: power , last digit , number theory

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 $\underbrace{0...0}_\text{n-1}1$.

1945 Moscow Mathematical Olympiad, 096

Tags: number theory , power , 3-digit

Find three-digit numbers such that any its positive integer power ends with the same three digits and in the same order.

2021 Auckland Mathematical Olympiad, 4

Tags: number theory , power , divide , divisible

Prove that there exist two powers of $7$ whose difference is divisible by $2021$.

2001 Estonia Team Selection Test, 5

Tags: prime , product , power , digit , 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

1985 AMC 8, 3

Tags: power , powers of 10 , arithmetic

$ \frac{10^7}{5 \times 10^4}\equal{}$ \[ \textbf{(A)}\ .002 \qquad \textbf{(B)}\ .2 \qquad \textbf{(C)}\ 20 \qquad \textbf{(D)}\ 200 \qquad \textbf{(E)}\ 2000 \]

2018 NZMOC Camp Selection Problems, 9

Tags: diophantine equation , power , diophantine , number theory

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 $p$.

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.

2006 Junior Tuymaada Olympiad, 2

Tags: number theory , prime , power , power of number

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 ?

2014 Contests, 3

Tags: algebra , power , number theory with sequences , sequence , number theory , recurrence relation

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.

2016 Flanders Math Olympiad, 2

Tags: power , divide , divisor , number theory

Determine the smallest natural number $n$ such that $n^n$ is not a divisor of the product $1\cdot 2\cdot 3\cdot ... \cdot 2015\cdot 2016$.