Saturday, March 29, 2008

Is this linear? Are you sure???

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that whenever $a$, $b \in \mathbb{R}$ are such that $a-b \in \mathbb{Q}$, then also $f(a)-f(b)\in \mathbb{Q}$. Prove that $f$ is of the form $f(x) = qx + r$, and that $q$ is rational.

Update (16/04/2008): Solution posted! (show solution)

Tuesday, March 18, 2008

Boring books

There are strange boring books. In every page of those books there is only one sentence saying: 'the sententence at page $p$ is $X$', where $p$ is a positive integer less or equal of that book lenght -in pages- and $X$ belong to $\{true, false\}$.
  • For which books does exist a function from the pages to $\{true, false\}$ that makes true the sentence written in each page?
  • If such a function exists, how many different functions make the job?

Update (08/04/2008): Solution posted! (show solution)

Saturday, March 8, 2008

Bijective function

Let $f : \mathbb{Z} \rightarrow \mathbb{Z}$ be a bijective function. Prove that you can find two other bijective functions $u$, $v$ from $\mathbb{Z}$ to $\mathbb{Z}$ such that $f(x) = u(x) + v(x)$, $\forall x \in \mathbb{Z}$.

Update (23/03/2008): Solution posted! (show solution)

Sunday, March 2, 2008

Looks like SPAM...

One million (1000000) people receive an e-mail with the following message:
  • If *exactly* one of you will reply to this e-mail, everyone of you will win 100\$. In any other case, everybody will win nothing.
Suppose that the people receiving this e-mail are all very intelligent, they are willing to win the prize, and they cannot comunicate with each other. The only thing that they can do is just replying, or not. Ah, the e-mail is not SPAM, it is serious :)

What is the best strategy that they can adopt to maximize the probability of succeding? What is (more or less) the probability of success?

Update (20/03/2008): Solution posted! (show solution)