Show that \(\text{Nim}(x_1,\dots,x_k)\) is an losing position if and only if \(x_1 \oplus \dots \oplus x_k = 0\). \(x \oplus y\) denotes the nim-sum of \(x\) and \(y\).
You meet an alien, who you learn is thinking of a positive integer \(n\). They ask the following three questions.
“Am I the kind who could ask whether \(n\) is divisible by no primes other than \(2\) or \(3\)?"
“Am I the kind who could ask whether the sum of the divisors of \(n\) (including \(1\) and \(n\) themselves) is at least twice \(n\)?"
“Is \(n\) divisible by 3?"
Is this alien a Crick or a Goop?
There is a secret gathering of a group of \(n\) aliens in a very dark room. You cannot see anyone in the room, but you hear the following questions.
“Is at least one of us a Goop?"
“Is the number of Goops amongst us an even number?"
“Is the number of Goops amongst us a multiple of 3?"
\(\dots\)
“Is the number of Goops amongst us a multiple of \(n\)?"
What are all the possible values of \(n\) such that this gathering can happen? Note that each of the \(n\) aliens have asked exactly one question.
The pigeonhole principle is often called “Dirichlet’s box principle". Dirichlet made good use of this tool to show a fundamental result in Diophantine approximation, now commonly known as the Dirichlet Approximation Theorem. You will now prove it yourself!
Suppose \(\alpha\) is any irrational real number and \(N\geq 1\) is any positive integer. Show that there is an integer \(1\leq q\leq N\) and an integer \(p\) such that \[\left| q \alpha - p \right| < \frac{1}{N}.\]
What’s the sum of the Fibonacci numbers \(F_0+F_1+F_2+...+F_n\)?
What’s the sum \(\frac{F_2}{F_1}+\frac{F_4}{F_2}+\frac{F_6}{F_3}+...+\frac{F_{18}}{F_9}+\frac{F_{20}}{F_{10}}\)?
We have a sequence where the first term (\(x_1\)) is equal to \(2\), and each term is \(1\) minus the reciprocal of the previous term (which we can write as \(x_{n+1}=1-\frac{1}{x_n}\)).
What’s \(x_{57}\)?