2008 Postal Coaching

Part of Postal Coaching

Subcontests

(6)

3

any set of n points in plane has a free subset of at least \sqrt{n} points

A set of points in the plane is called free if no three points of the set are the vertices of an equilateral triangle. Prove that any set of

n

points in the plane has a free subset of at least

\sqrt{n}

n lines in plane, 7 point exists such any lines pass at least from 3 points

n

straight lines are in the plane so that there exist seven points such that any of these line passes through at least three of these points. Find the largest possible value of

n

determine which of classes X_r, r \in {0, 1, 2\} is the largest, mod 3

Consider the set

A = \{1, 2, 3, ..., 2008\}

. We say that a set is of type

r, r \in \{0, 1, 2\}

, if that set is a nonempty subset of

A

and the sum of its elements gives the remainder

r

when divided by

3

X_r, r \in \{0, 1, 2\}

the class of sets of type

r

. Determine which of the classes

X_r, r \in \{0, 1, 2\}

, is the largest.

6

6

6

6

6