Found problems: 86
Determine all pairs $(a,b)$ of positive integers such that $(a+b)^3-2a^3-2b^3$ is a power of two.
Prove that there is a unique positive integer formed only by the digits $2$ and $5$, which has $ 2007$ digits and is divisible by $2^{2007}$.
Find all numbers that can be expressed in exactly $2010$ different ways as the sum of powers of two with non-negative exponents, each power appearing as a summand at most three times. A sum can also be made from just one summand.
Prove that every $1000$-element subset $M$ of the set $\{0,1,...,2001\}$ contains either a power of two or two distinct numbers whose sum is a power of two.
Show that if $n \ge 4, n \in N$ and $\big [ \frac{2^n}{n} ]$ is a power of $2$, then $n$ is a 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?
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.
Determine all pairs $(a,b)$ of positive integers such that $(a+b)^3-2a^3-2b^3$ is a power of two.
Find all pairs of natural numbers $a,b$ , with $a\ne b$ , such that $a+b$ and $ab+1$ are powers of $2$.
In the first $1999$ cells of the computer are written numbers in the specified order:: $1$, $2$, $4$,$... $, $2^{1998}$. Two programmers take turns reducing in one move per unit number in five different cells. If a negative number appears in one of the cells, then the computer breaks down and the broken repairs are paid for. Which programmer can protect himself from financial losses, regardless of his partner’s moves, and how should he do this act?
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 $a=1$.