6

Part of 2008 Postal Coaching

Problems(3)

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

Source: Indian Postal Coaching 2008 set 4 p6

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}

combinatoricscombinatorial geometryEquilateralpoints

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

Source: Indian Postal Coaching 2008 set 5 p6

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

combinatorial geometrylinespoints

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

Source: Indian Postal Coaching 2008 set 6 p6

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.

combinatoricsnumber theoryremainder