International Mathematics Olympiad Problems - 2002 | International Mathematics Olympiad (IMO) for Class 10 PDF Download

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Problem 1. S is the set of all (h,k) with h,k non-negative integers such
that h+k < n. Each element of S is colored red or blue, so that if (h,k)
is red and h
0
=h,k
0
=k, then (h
0
,k
0
) is also red. A type 1 subset of S has
n blue elements with di?erent ?rst member and a type 2 subset of S has n
blue elements with di?erent second member. Show that there are the same
number of type 1 and type 2 subsets.
Problem 2. BC is a diameter of a circle center O. A is any point on
the circle with
6 AOC > 60
o
. EF is the chord which is the perpendicular
bisector of AO. D is the midpoint of the minor arc AB. The line through
O parallel to AD meets AC at J. Show that J is the incenter of triangle
CEF.
Problem 3. Find all pairs of integers m > 2,n > 2 such that there are
in?nitely many positive integers k for which k
n
+k
2
-1 divides k
m
+k-1.
Problem 4. The positive divisors of the integer n> 1 are d
1
<d
2
<...<
d
k
, so that d
1
= 1,d
k
= n. Let d = d
1
d
2
+d
2
d
3
+···+d
k-1
d
k
. Show that
d<n
2
and ?nd all n for which d divides n
2
.
Problem 5. Find all real-valued functions on the reals such that (f(x)+
f(y))((f(u)+f(v)) =f(xu-yv)+f(xv+yu) for all x,y,u,v.
Problem 6. n> 2 circles of radius 1 are drawn in the plane so that no line
meets more than two of the circles. Their centers are O
1
,O
2
,···,O
n
. Show
that
P
i<j
1/O
i
O
j
= (n-1)p/4.
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