International Mathematics Olympiad Problems - 2007

``` Page 1

July 25, 2007
Problem 1. Real numbers a
1
,a
2
,...,a
n
are given. For each i (1 =i =n) de?ne
d
i
= max{a
j
: 1 =j =i}-min{a
j
:i =j =n}
and let
d = max{d
i
: 1 =i =n}.
(a) Prove that, for any real numbers x
1
=x
2
= ··· =x
n
,
max{|x
i
-a
i
| : 1 =i =n} =
d
2
. (*)
(b) Show that there are real numbers x
1
= x
2
= ··· = x
n
such that equality holds
in (*).
Problem 2. Consider ?ve pointsA,B,C,D andE such thatABCD 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.
Problem 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 competitors 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 in 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.
Page 2

July 25, 2007
Problem 1. Real numbers a
1
,a
2
,...,a
n
are given. For each i (1 =i =n) de?ne
d
i
= max{a
j
: 1 =j =i}-min{a
j
:i =j =n}
and let
d = max{d
i
: 1 =i =n}.
(a) Prove that, for any real numbers x
1
=x
2
= ··· =x
n
,
max{|x
i
-a
i
| : 1 =i =n} =
d
2
. (*)
(b) Show that there are real numbers x
1
= x
2
= ··· = x
n
such that equality holds
in (*).
Problem 2. Consider ?ve pointsA,B,C,D andE such thatABCD 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.
Problem 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 competitors 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 in 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.
July 26, 2007
Problem 4. In triangle ABC the bisector of angle BCA intersects the circumcircle
again atR, the perpendicular bisector ofBC atP, and the perpendicular bisector ofAC
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.
Problem 5. Let a and b be positive integers. Show that if 4ab-1 divides (4a
2
-1)
2
,
then a =b.
Problem 6. Let n be a positive integer. Consider
S ={(x,y,z) : x,y,z ?{0,1,...,n}, x+y +z > 0}
as a set of (n+1)
3
-1 points in three-dimensional space. Determine the smallest possible
number of planes, the union of which contains S but does not include (0,0,0).
```

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## Olympiad Preparation for Class 10

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