International Mathematics Olympiad Problems - 2001 | International Mathematics Olympiad (IMO) for Class 10 PDF Download

 Page 1


 
Problem 1
Let ABC be an acute-angled triangle with circumcentre O. Let P on BC be the foot of the altitude from A. 
Suppose that 
BCA
ABC 30


. 
Prove that 
CAB
COP 90


. 
Problem 2
Prove that 
a
 


a
2
 8bc

b
 


b
2
 8ca

c
 


c
2
 8ab
 1
for all positive real numbers a, b and 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. 
Problem 4
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 
Sa 
i1
n
k
i
a
i
.
Prove that there are two permutations b and c, b	 c, such that n is a divisor of Sb
 Sc. 
Page 2


 
Problem 1
Let ABC be an acute-angled triangle with circumcentre O. Let P on BC be the foot of the altitude from A. 
Suppose that 
BCA
ABC 30


. 
Prove that 
CAB
COP 90


. 
Problem 2
Prove that 
a
 


a
2
 8bc

b
 


b
2
 8ca

c
 


c
2
 8ab
 1
for all positive real numbers a, b and 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. 
Problem 4
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 
Sa 
i1
n
k
i
a
i
.
Prove that there are two permutations b and c, b	 c, such that n is a divisor of Sb
 Sc. 
Problem 5
In a triangle ABC, let AP bisect 
BAC , with P on BC, and let BQ bisect 
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. Suppose that 
ac bd b d a
 cb d
 a c.
Prove that ab cd is not prime. 
2