3

Part of 2004 China National Olympiad

Problems(2)

Minimum value of the number of points in the set M

Source: Chinese MO 2004

M

be a set consisting of

n

points in the plane, satisfying: i) there exist

7

M

which constitute the vertices of a convex heptagon; ii) if for any

5

M

which constitute the vertices of a convex pentagon, then there is a point in

M

which lies in the interior of the pentagon. Find the minimum value of

n

.
Leng Gangsong

combinatorics proposedcombinatorics

Only a finite number of integers cannot be represented

Source: Chinese MO 2004

Prove that every positive integer

n

, except a finite number of them, can be represented as a sum of

2004

positive integers:

n=a_1+a_2+\cdots +a_{2004}

1\le a_1<a_2<\cdots <a_{2004}

a_i \mid a_{i+1}

1\le i\le 2003

inductionnumber theory proposednumber theory