3

Part of 2009 Saint Petersburg Mathematical Olympiad

Problems(2)

Source: St Petersburg Olympiad 2009, Grade 10, P4

Streets of Moscow are some circles (rings) with common center

O

and some straight lines from center

O

to external ring. Point

A,B

- two crossroads on external ring. Three friends want to move from

A

B

. Dima goes by external ring, Kostya goes from

A

O

B

. Sergey says, that there is another way, that is shortest. Prove, that he is wrong.

Square trinomials

Source: St Petersburg Olympiad 2009, Grade 10, P3

f(x),g(x),h(x)

are square trinomials with discriminant, that equals

2

f(x)+g(x),f(x)+h(x),g(x)+h(x)

are square trinomials with discriminant, that equals

1

f(x)+g(x)+h(x)