International Mathematics Olympiad Problems - 2000 | International Mathematics Olympiad (IMO) for Class 10 PDF Download

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Problem 1. AB is tangent to the circles CAMN and NMBD. M lies
between C and D on the line CD, and CD is parallel to AB. The chords
NA andCM meet atP; the chordsNB andMD meet atQ. The raysCA
andDB meet atE. Prove thatPE =QE.
Problem 2. A,B,C are positive reals with product 1. Prove that (A-1+
1
B
)(B-1+
1
C
)(C-1+
1
A
)= 1.
Problem 3. k is a positive real. N is an integer greater than 1. N points
are placed on a line, not all coincident. A move is carried out as follows.
Pick any two pointsA andB which are not coincident. Suppose thatA lies
to the right ofB. ReplaceB by another pointB
0
to the right ofA such that
AB
0
=kBA. For what values ofk can we move the points arbitrarily far to
the right by repeated moves?
Problem 4. 100 cards are numbered 1 to 100 (each card di?erent) and
placed in 3 boxes (at least one card in each box). How many ways can this
be done so that if two boxes are selected and a card is taken from each, then
the knowledge of their sum alone is always su?cient to identify the third
box?
Problem 5. Can we ?ndN divisible by just 2000 di?erent primes, so that
N divides 2
N
+1? [N may be divisible by a prime power.]
Problem 6. A
1
A
2
A
3
is an acute-angled triangle. The foot of the altitude
fromA
i
isK
i
and the incircle touches the side opposite A
i
atL
i
. The line
K
1
K
2
is re?ected in the line L
1
L
2
. Similarly, the line K
2
K
3
is re?ected in
L
2
L
3
andK
3
K
1
is re?ected inL
3
L
1
. Show that the three new lines form a
triangle with vertices on the incircle.
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