Đề thi học sinh giỏi Toán Quốc tế năm 2007

IMO 2007
Ha Noi, Vietnam
Day 1 - 25 July 2007
1 Real numbers a
1
, a
2
, : : :, a
n
are given. For each i, (1  i  n), dene
d
i
= maxfa
j
j 1  j  ig minfa
j
j i  j  ng
and let d = maxfd
i
j 1  i  ng.
(a) Prove that, for any real numbers x
1
 x
2
 
 
 
  x
n
,
maxfjx
i
 a
i
j j 1  i  ng 
d
2
: ()
(b) Show that there are real numbers x
1
 x
2
 
 
 
  x
n
such that the equality holds in (*).
2 Consider ve points A, B, C, D and E such that ABCD is a parallelogram and BCED is a
cyclic quadrilateral. Let ` be a line passing through A. Suppose that ` intersects the interior
of the segment DC at F and intersects line BC at G. Suppose also that EF = EG = EC.
Prove that ` is the bisector of angle DAB.
3 In a mathematical competition some competitors are friends. Friendship is always mutual.
Call a group of competitors a clique if each two of them are friends. (In particular, any group
of fewer than two competitiors is a clique.) The number of members of a clique is called its
size.
Given that, in this competition, the largest size of a clique is even, prove that the competitors
can be arranged into two rooms such that the largest size of a clique contained in one room
is the same as the largest size of a clique contained in the other room.
This le was downloaded from the AoPS  MathLinks Math Olympiad Resources Page
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IMO 2007
Ha Noi, Vietnam
Day 2 - 26 July 2007
4 In triangle ABC the bisector of angle BCA intersects the circumcircle again at R, the per-
pendicular bisector of BC at P , and the perpendicular bisector of AC at Q. The midpoint
of BC is K and the midpoint of AC is L. Prove that the triangles RPK and RQL have the
same area.
5 Let a and b be positive integers. Show that if 4ab 1 divides (4a
2
 1)
2
, then a = b.
6 Let n be a positive integer. Consider
S = f(x; y; z) j x; y; z 2 f0; 1; : : : ; ng; x+ y + z > 0g
as a set of (n+1)
3
1 points in the three-dimensional space. Determine the smallest possible
number of planes, the union of which contains S but does not include (0; 0; 0).
This le was downloaded from the AoPS  MathLinks Math Olympiad Resources Page
Page 2