International Mathematics Olympiad Problems - 2009 | International Mathematics Olympiad (IMO) for Class 10 PDF Download

 Page 1


Wednesday, July 15, 2009
Problem 1. Let n be a positive integer and let a
1
,...,a
k
(k= 2) be distinct integers in the set
{1,...,n}suchthatndividesa
i
(a
i+1
-1)fori = 1,...,k-1. Provethatndoesnotdividea
k
(a
1
-1).
Problem 2. Let ABC be a triangle with circumcentre O. The points P and Q are interior points
of the sides CA and AB, respectively. Let K, L and M be the midpoints of the segments BP, CQ
and PQ, respectively, and let G be the circle passing through K, L and M. Suppose that the line
PQ is tangent to the circle G. Prove that OP =OQ.
Problem 3. Suppose that s
1
,s
2
,s
3
,... is a strictly increasing sequence of positive integers such
that the subsequences
s
s1
,s
s2
,s
s3
,... and s
s1+1
,s
s2+1
,s
s3+1
,...
are both arithmetic progressions. Prove that the sequence s
1
,s
2
,s
3
,... is itself an arithmetic pro-
gression.
Page 2


Wednesday, July 15, 2009
Problem 1. Let n be a positive integer and let a
1
,...,a
k
(k= 2) be distinct integers in the set
{1,...,n}suchthatndividesa
i
(a
i+1
-1)fori = 1,...,k-1. Provethatndoesnotdividea
k
(a
1
-1).
Problem 2. Let ABC be a triangle with circumcentre O. The points P and Q are interior points
of the sides CA and AB, respectively. Let K, L and M be the midpoints of the segments BP, CQ
and PQ, respectively, and let G be the circle passing through K, L and M. Suppose that the line
PQ is tangent to the circle G. Prove that OP =OQ.
Problem 3. Suppose that s
1
,s
2
,s
3
,... is a strictly increasing sequence of positive integers such
that the subsequences
s
s1
,s
s2
,s
s3
,... and s
s1+1
,s
s2+1
,s
s3+1
,...
are both arithmetic progressions. Prove that the sequence s
1
,s
2
,s
3
,... is itself an arithmetic pro-
gression.
Thursday, July 16, 2009
Problem 4. Let ABC be a triangle with AB = AC. The angle bisectors of
6 CAB and
6 ABC
meet the sides BC and CA at D and E, respectively. Let K be the incentre of triangle ADC.
Suppose that
6 BEK = 45
?
. Find all possible values of
6 CAB.
Problem5. Determineallfunctionsf fromthesetofpositiveintegerstothesetofpositiveintegers
suchthat, forallpositiveintegersaandb, thereexistsanon-degeneratetrianglewithsidesoflengths
a, f(b) and f(b+f(a)-1).
(A triangle is non-degenerate if its vertices are not collinear.)
Problem 6. Let a
1
,a
2
,...,a
n
be distinct positive integers and let M be a set of n - 1 positive
integers not containing s =a
1
+a
2
+···+a
n
. A grasshopper is to jump along the real axis, starting
at the point 0 and making n jumps to the right with lengths a
1
,a
2
,...,a
n
in some order. Prove that
the order can be chosen in such a way that the grasshopper never lands on any point in M.