1. Given triangle ABC the point J is the centre of the excircle opposite the
vertex A. This excircle is tangent to the side BC at M, and to the lines AB
and AC at K and L, respectively. The lines LM and BJ meet at F, and the lines
KM and CJ meet at G. Let S be the point of intersection of the lines AF and
BC , and let T be the point of intersection of the lines AG and BC. Prove
that M is the midpoint of ST.
(The excircle of ABC opposite the vertex A is the circle that is tangent to
the line segment BC, to the ray AB beyond B, and to the ray AC beyond C.)
2. If positive reals a_2, a_3,...,a_n satisfy a_2 * a_3 * ... * a_n = 1
and n > 2 prove that
(a_2 + 1)^2 * (a_3 + 1)^3 * ... * (a_n + 1)^n > n^n
3. The "liar's" guessing game is a game played between two players A and B.
The rules of the game depend on two positive integers k and n which are known
to both players.
At the start of the game A chooses integers x and N with 1≦x≦N. Player keeps
x secret, and truthfully tells N to player B. Player B now tries to obtain
information about x by asking player A questions as follows: each question
consists of B specifying an arbitrary set S of positive integers (possibly one
specified in some previous question), and asking A whether x belongs to S.
Player B may ask as many questions as he wishes. After each question, player A
must immediately answer it with yes or no, but is allowed to lie as many times
as she wants; the only restriction is that, among any k+1 consecutive answers,
at least one answer must be truthful.
After B has asked as many questions as he wants, he must specify a set X of at
most n positive integers. If x belongs to X, then wins; otherwise, he loses.
Prove that:
1. If n≧2^k, then B can guarantee a win.
2. For all sufficiently large k , there exists an integer n ≧(1.99)^k such
that B cannot guarantee a win.