4. A set of positive integers is called "fragrant" if it contains at least two
elements and each of its elements has a prime factor in common with at
least one of the other elements. Let P(n)=n^2+n+1. What is the least
possible positive integer value of b such that there exists a non-negative
integer a for which the set {P(a+1),P(a+2),...,P(a+b)} is "fragrant"
5. The equation
(x-1)(x-2)...(x-2016)=(x-1)(x-2)...(x-2016)
is written on the board, with 2016 linear factors on each side. What is the
least possible value of k which it is possible to erase exactly of these
4032 linear factors so that at least one factor remains on each side and
the resulting equation has no real solutions?
6. There are n≧2 line segments in the plane such that every two segments
cross and no three segments meet at a point. Geoff has to choose an
endpoint of each segment and place a frog on it facing the other endpoint.
Then he will clap his hands n-1 times. Every time he claps, each frog will
immediately jump forward to the next intersection point on its segment.
Frogs never change the direction of their jumps. Geoff wishes to place the
frogs in such a way that no two of them will every occupy the same
intersection point at the same time.
(a) Prove that Geoff can always fulfill his wish if n is odd.
(b) Prove that Geoff can never fulfill his wish if n is even.
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