Sunday, April 20, 2008

 

 

Election time

In Italy, there are just three parties: L (the left), R (the right), and N (north league). Elections are done, L and R get each 49% of votes, N just 2%. Since an agreement cannot be found, two tie-break rounds are done, where the two biggest parties are matched, and at the second round the winner is matched against the other party. Prove that there is a strategy for N's electors to make their party win the tie-break rounds, knowing that:
  • L's electors, if they cannot vote for L they will vote for R, because they hate N.
  • R's electors, if they cannot vote for R they will vote for N, because they hate L.

 

 

 

Wednesday, April 16, 2008

 

 

Minimal values

Let f:𝕉𝕉 be a function (any function). Let's call
M:={x𝕉:εx>0:f(x)f(y)whenevery-x<εx}

the set of points where f has a (non-strict) local minimum. Let V=f(M) be the set of the local minimal values. Can V have positive Lebesgue measure? (note that M surely can, just take as f a constant function, and every point is a local minimum).

 

 

 

Friday, April 4, 2008

 

 

An elettric circuit

There exists an electric circuit that has an interruption. The -proper- length of the interruption is L0 . A tract of electric wire runs along the circuit with speed v . The -proper- length of the wire is L0 too. In circuit reference system wire length is L<L0 , so the circuit is never closed. In wire reference system interruption length is L<L0 , so at some instant the circuit is closed. How can you explain the apparent contradiction?
Update(22/04/2008): Solution posted! (show solution)