International Olympiad Problems - 2004 - International Olympiad (IMO)

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Problem 1. Let ABC be an acute-angled triangle with AB 6= AC. The
circle with diameter BC intersects the sides AB and AC at M and N
respectively. Denote by O the midpoint of the side BC. The bisectors of
the angles
6 BAC and
6 MON intersect at R. Prove that the circumcircles
of the triangles BMR and CNR have a common point lying on the side
BC.
Problem 2. Find all polynomials f with real coe?cients such that for all
reals a,b,c such that ab+bc+ca = 0 we have the following relations
f(a-b)+f(b-c)+f(c-a) = 2f(a+b+c).
Problem 3. De?ne a ”hook” to be a ?gure made up of six unit squares
as shown below in the picture, or any of the ?gures obtained by applying
rotations and re?ections to this ?gure.
Determineallm×nrectanglesthatcanbecoveredwithoutgapsandwithout
overlaps with hooks such that
• the rectangle is covered without gaps and without overlaps
• no part of a hook covers area outside the rectagle.
Problem 4. Let n = 3 be an integer. Let t
1
,t
2
,...,t
n
be positive real
numbers such that
n
2
+1> (t
1
+t
2
+...+t
n
)

1
t
1
+
1
t
2
+...+
1
t
n

.
Show that t
i
,t
j
,t
k
are side lengths of a triangle for all i, j, k with
1=i<j <k=n.
Problem 5. In a convex quadrilateral ABCD the diagonal BD does not
bisect the angles ABC and CDA. The point P lies inside ABCD and
satis?es
6 PBC =
6 DBA and
6 PDC =
6 BDA.
Prove that ABCD is a cyclic quadrilateral if and only if AP =CP.
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Page 2


Problem 1. Let ABC be an acute-angled triangle with AB 6= AC. The
circle with diameter BC intersects the sides AB and AC at M and N
respectively. Denote by O the midpoint of the side BC. The bisectors of
the angles
6 BAC and
6 MON intersect at R. Prove that the circumcircles
of the triangles BMR and CNR have a common point lying on the side
BC.
Problem 2. Find all polynomials f with real coe?cients such that for all
reals a,b,c such that ab+bc+ca = 0 we have the following relations
f(a-b)+f(b-c)+f(c-a) = 2f(a+b+c).
Problem 3. De?ne a ”hook” to be a ?gure made up of six unit squares
as shown below in the picture, or any of the ?gures obtained by applying
rotations and re?ections to this ?gure.
Determineallm×nrectanglesthatcanbecoveredwithoutgapsandwithout
overlaps with hooks such that
• the rectangle is covered without gaps and without overlaps
• no part of a hook covers area outside the rectagle.
Problem 4. Let n = 3 be an integer. Let t
1
,t
2
,...,t
n
be positive real
numbers such that
n
2
+1> (t
1
+t
2
+...+t
n
)

1
t
1
+
1
t
2
+...+
1
t
n

.
Show that t
i
,t
j
,t
k
are side lengths of a triangle for all i, j, k with
1=i<j <k=n.
Problem 5. In a convex quadrilateral ABCD the diagonal BD does not
bisect the angles ABC and CDA. The point P lies inside ABCD and
satis?es
6 PBC =
6 DBA and
6 PDC =
6 BDA.
Prove that ABCD is a cyclic quadrilateral if and only if AP =CP.
1
Problem 6. We call a positive integer alternating if every two consecutive
digits in its decimal representation are of di?erent parity.
Find all positive integers n such that n has a multiple which is alternating.
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