Olympiad problems

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) $\rho v^2$

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)?

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.

2001 Regional Competition For Advanced Students, 1

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

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?

1987 Bundeswettbewerb Mathematik, 1

Tags: number theory , prime , power , digit

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.

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 $abc|(a+b+c)^{7}$

1981 Bundeswettbewerb Mathematik, 1

Tags: digit , number theory , 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 $1$.

2018 Polish Junior MO Finals, 1

Tags: number theory , perfect square , power

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

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.

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.

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 $k=1.$

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 ($k \ge2$)?

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.

1985 AMC 8, 3

Tags: powers of 10 , arithmetic , power

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

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 $10$.

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.

2013 IMAR Test, 2

Tags: digit sum , sequence , number theory , 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 ?

2021 Brazil National Olympiad, 3

Tags: floor function , irrational number , perfect square , power , number theory

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 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.