News: IMO Day 1 (25 July 2007)
The first IMO 2007 paper was sat in Hanoi on 25 July 2007; the
problems and the countries submitting them are shown.
Real numbers a_{1}, a_{2},
…, a_{n} are given. For
each i (1 ≤ i ≤ n)
define
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 (*).
(New Zealand)
Consider five points A, B, C, D and
E such that ABCD is a parallelogram and BCED is a
cyclic quadrilateral. Let ℓ be a line passing through A.
Suppose that ℓ intersects the interior of the segment DC
at F and intersects line BC at G. Suppose also
that EF = EG = EC. Prove that ℓ is the
bisector of angle DAB.
(Luxembourg)
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.
(Russia)
