Olympiad problems

VMEO III 2006 Shortlist, N11

Prove that the composition of the sets of one of the following two forms is finite: (a) $2^{2^n}+1$ (b) $6^{2^n}+1$

2021 Brazil National Olympiad, 3

Tags: number theory , floor function , powers , irrational number , Perfect Squares , Brazilian Math Olympiad , Brazil

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 $n>N$.

2016 IFYM, Sozopol, 8

Tags: algebra , Sum , powers

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 , powers , arithmetic , Why the bump

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

1990 Nordic, 2

Tags: inequalities , powers

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)?

2018 Polish Junior MO Finals, 1

Tags: number theory , Perfect Square , powers , Poland

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

2018 VJIMC, 1

Tags: equations , real numbers , powers

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

2021 Auckland Mathematical Olympiad, 4

Tags: number theory , powers , divides , divisible

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

2018 Malaysia National Olympiad, A6

Tags: number theory , powers , prime numbers

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

2012 India Regional Mathematical Olympiad, 2

Tags: number theory , divides , divisible , powers

Let $a,b,c$ be positive integers such that $a|b^4, b|c^4$ and $c|a^4$. Prove that $abc|(a+b+c)^{21}$

2001 Regional Competition For Advanced Students, 1

Tags: number theory , Digit , Sum of powers , Sum , powers

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?

2012 India Regional Mathematical Olympiad, 2

Tags: number theory , Divisibility , divisible , powers

Let $a,b,c$ be positive integers such that $a|b^2, b|c^2$ and $c|a^2$. Prove that $abc|(a+b+c)^{7}$

1960 Putnam, B4

Tags: Putnam , arithmetic sequence , powers

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.

2006 Junior Tuymaada Olympiad, 2

Tags: number theory , power of number , powers , primes

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 ?

1981 Bundeswettbewerb Mathematik, 1

Tags: Digits , number theory , powers

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 $1$.

2020 Kosovo National Mathematical Olympiad, 1

Tags: national olympiad , Olympiad , algebra , number theory , Compare , powers , POW

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

Istek Lyceum Math Olympiad 2016, 3

Tags: number theory , powers , Istek Lyceum MO

Let $n$, $m$ and $k$ be positive integers satisfying $(n-1)n(n+1)=m^k.$ Prove that $k=1.$

2025 Kosovo National Mathematical Olympiad`, P3

Tags: number theory , decimal representation , powers , Kosovo

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.

MIPT student olimpiad autumn 2022, 3

Tags: powers , MIPT , college contests , 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: {$\phi^{-n} | n \in Z^+$} $\phi=\frac{1+\sqrt5}{2}$

2013 IMAR Test, 2

Tags: powers , digit sum , Sequence , number theory

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 ?

2012 India Regional Mathematical Olympiad, 2

Tags: number theory , divides , divisible , powers

Let $a,b,c$ be positive integers such that $a|b^3, b|c^3$ and $c|a^3$. Prove that $abc|(a+b+c)^{13}$

2021 Brazil Undergrad MO, Problem 3

Tags: Brazilian Undergrad MO 2021 , real analysis , powers , irrational number , approximation , Perfect Square

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 $n>N$.

2005 Austrian-Polish Competition, 10

Tags: inequalities , powers

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

1998 German National Olympiad, 6a

Tags: algebra , system of equations , powers , symmetry , algebra unsolved , Germany , 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}