1)I have found this problem in a book. Say

. We make the following observations. Since

it must mean that

is even and

. Since

it means that

is a polynomial of even degree and the leading coefficient is

.

Therefore,

. This tells us that

**must** have a minimum value (since it is a continous function). Say

is the point where

is mimimal. Then it means that

. But we know that

because

since the degree of

is

. Thus, by what just stated we have that

. Thus,

since

by hypothesis. So if

is any real number then

for

is the smallest value of the function.

2)A long time ago somebody challenged me to find the shortest checkmate. The following is my solution. The strange think is that it is played by

**black** rather than white!

The are several version by the idea is the same.

WHITE] Play Queens Knight any way.

BLACK] Plays Kings Pawns

WHITE] Plays Sicilian Defense on Kings Side (Move Bishop Pawn)

BLACK] Plays a Checkmate with a Queen.

So the Black player wins in just two moves.