Washington, DC, United States of America河南豆
July 8–9, 2001
Problems
Each problem is worth ven points.
Problem 1
Let ABC be an acute-angled triangle with circumcentre O . Let P on BC be the foot of the altitude from A .
Suppo that BCA ABC 30 .
Prove that CAB COP 90 .
Problem 2
Prove that
a a 2 8
b c
b b 2 8
网络教育培训c a c c 2 8 a b 1for all positive real numbers a ,b an
d c .
Problem 3
Twenty-one girls and twenty-one boys took part in a mathematical contest.
• Each contestant solved at most six problems.
• For each girl and each boy, at least one problem was solved by both of them.
Prove that there was a problem that was solved by at least three girls and at least three boys.
金属英文Problem 4
关于中秋的画
Let n be an odd integer greater than 1, and let k 1,k 2,…,k n be given integers. For each of the n permutations
a a 1,a 2,…,a n of 1,2,…,n , let
S a i 1
n
k i a i .
offorProve that there are two permutations b and c , b c , such that n is a divisor of S b S c .
2IMO 2001 Competition Problems陶笛怎么吹
Problem 5
In a triangle ABC, let AP bict BAC, with P on BC, and let BQ bict ABC, with Q on CA.
It is known that BAC 60 and that AB BP AQ QB.
What are the possible angles of triangle ABC?
Problem 6
Let a,b,c,d be integers with a b c d 0. Suppo that
古运河畔a c
b d b d a美女黑
c b
d a c .
Prove that a b c d is not prime.
