1999 North Macedonia National Olympiad

1

In a set of 21 real numbers, the sum of any 10 numbers is less than the sum of the remaining 11 numbers. Prove that all the numbers are positive.

2

We are given 13 apparently equal balls, all but one having the same weight (the remaining one has a different weight). Is it posible to determine the ball with the different weight in 3 weighings?

3

Let the two tangents from a point A outside a circle k touch k at M and N. A line p through A intersects k at B and C, and D is the midpoint of MN. Prove that MN bisects the angle BDC

4

Do there exist 100 straight lines on a plane such that they intersect each other in exactly 1999 points?

5

If a,b,c are positive numbers with a2+b2+c2=1, prove that a+b+c+1abc43