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.

AND:
OR:
NO:

Found problems: 86

2016 Portugal MO, 5
Tags: power of 2 , number theory
Determine all natural numbers $x, y$ and $z$ such that the number $2^x +4^y +8^z +16^2$ is a power of $2$.

1968 Spain Mathematical Olympiad, 7
Tags: number theory , digit , power of 2
In the sequence of powers of $2$ (written in the decimal system, beginning with $2^1 = 2$) there are three terms of one digit, another three of two digits, another three of $3$, four out of $4$, three out of $5$, etc. Clearly reason the answers to the following questions: a) Can there be only two terms with a certain number of digits? b) Can there be five consecutive terms with the same number of digits? c) Can there be four terms of n digits, followed by four with $n + 1$ digits? d) What is the maximum number of consecutive powers of $2$ that can be found without there being four among them with the same number of digits?

2007 Regional Olympiad of Mexico Center Zone, 4
Tags: number theory , power of 2
Is there a power of $2$ that when written in the decimal system has all its digits different from zero and it is possible to reorder them to form another power of $2$?

2005 Germany Team Selection Test, 2
Tags: polynomial , function , power of 2 , algebra
For any positive integer $ n$, prove that there exists a polynomial $ P$ of degree $ n$ such that all coeffients of this polynomial $ P$ are integers, and such that the numbers $ P\left(0\right)$, $ P\left(1\right)$, $ P\left(2\right)$, ..., $ P\left(n\right)$ are pairwisely distinct powers of $ 2$.

2019 Ecuador NMO (OMEC), 3
Tags: divisible , divide , power of 2 , number theory
For every positive integer $n$, find the maximum power of $2$ that divides the number $$1 + 2019 + 2019^2 + 2019^3 +.. + 2019^{n-1}.$$

1984 IMO, 3
Tags: number theory , equation , divisibility , power of 2
Let $a,b,c,d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^k$ and $b+c=2^m$ for some integers $k$ and $m$, then $a=1$.

1994 All-Russian Olympiad, 5
Tags: recursive , sequence , power of 2 , number theory
Let $a_1$ be a natural number not divisible by $5$. The sequence $a_1,a_2,a_3, . . .$ is defined by $a_{n+1} =a_n+b_n$, where $b_n$ is the last digit of $a_n$. Prove that the sequence contains infinitely many powers of two. (N. Agakhanov)

2016 Lusophon Mathematical Olympiad, 6
Tags: diophantine equation , power of 2 , number theory
Source: Lusophon MO 2016 Prove that any positive power of $2$ can be written as: $$5xy-x^2-2y^2$$ where $x$ and $y$ are odd numbers.

2015 Balkan MO Shortlist, N5
Tags: number theory , maximum , power of 2 , divide , product
For a positive integer $s$, denote with $v_2(s)$ the maximum power of $2$ that divides $s$. Prove that for any positive integer $m$ that: $$v_2\left(\prod_{n=1}^{2^m}\binom{2n}{n}\right)=m2^{m-1}+1.$$ (FYROM)

2018 Hanoi Open Mathematics Competitions, 7
Tags: power of 2 , sum , max , number theory
Some distinct positive integers were written on a blackboard such that the sum of any two integers is a power of $2$. What is the maximal possible number written on the blackboard?

1998 Rioplatense Mathematical Olympiad, Level 3, 4
Tags: number theory , power of 2
Let $M$ be a subset of $\{1,2,..., 1998\}$ with $1000$ elements. Prove that it is always possible to find two elements $a$ and $b$ in $M$, not necessarily distinct, such that $a + b$ is a power of $2$.