**APMO 2000**

I've just joined in the 13^{th} APMO (8/3/2000 in Vietnam). I think
I had a half correct. Here're the 5 problems, must be done in 4 hours:

- sums of 4 numbers in each side are the same
- sums of 4 squares of number in each side are the same

Find all ways to fill 9 number {1,2,3,4,5,6,7,8,9} in 9 circles so that:
Problem 2: |

**VMO 2000**

The 38^{th} VMO 1999-2000 was held in March, 13 and 14. There're 3 hours for 3 problems each day. And this is the 6 problems:

1. Find the locus of midpoints of MN.

2. Prove that the circumcircle of MNQ go through a fixed point.

1. Prove that there're at least 9 numbers a in [1 ; 3

2. How many a in [1 ; 3

1. Show that there's exactly one quadratic trinomial has form f(x) = x

2. There's not a g(x) = x + c so that P

(1) not 3 points in the same line

(2) not 4 points in the same circle

(3) all circumcircles of three points have the same radius.

Give A