International Mathematics Olympiad Problems - 2005 | International Mathematics Olympiad (IMO) for Class 10 PDF Download

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Problem 1. Six points are chosen on the sides of an equilateral triangle
ABC: A
1
, A
2
onBC, B
1
, B
2
onCA andC
1
, C
2
onAB, such that they are
the vertices of a convex hexagon A
1
A
2
B
1
B
2
C
1
C
2
with equal side lengths.
Prove that the lines A
1
B
2
,B
1
C
2
and C
1
A
2
are concurrent.
Problem 2. Let a
1
,a
2
,... be a sequence of integers with in?nitely many
positive and negative terms. Suppose that for every positive integer n the
numbers a
1
,a
2
,...,a
n
leave n di?erent remainders upon division by n.
Prove that every integer occurs exactly once in the sequence a
1
,a
2
,....
Problem 3. Let x,y,z be three positive reals such that xyz = 1. Prove
that
x
5
-x
2
x
5
+y
2
+z
2
+
y
5
-y
2
x
2
+y
5
+z
2
+
z
5
-z
2
x
2
+y
2
+z
5
= 0.
Problem4. Determine all positive integers relativelyprime to allthe terms
of the in?nite sequence
a
n
= 2
n
+3
n
+6
n
-1, n= 1.
Problem 5. Let ABCD be a ?xed convex quadrilateral with BC = DA
and BC not parallel with DA. Let two variable points E and F lie of the
sides BC and DA, respectively and satisfy BE = DF. The lines AC and
BD meet atP, the linesBD andEF meet atQ, the linesEF andAC meet
at R.
Prove that the circumcircles of the triangles PQR, as E and F vary, have a
common point other than P.
Problem 6. In a mathematical competition, in which 6 problems were
posed to the participants, every two of these problems were solved by more
than
2
5
ofthecontestants. Moreover, nocontestantsolvedallthe6problems.
Show that there are at least 2 contestants who solved exactly 5 problems
each.
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