News: IMO Day 2 (26 July 2007)
The second IMO 2007 paper was sat in Hanoi on 26 July 2007;
the problems and the countries submitting them are shown.
Coordination takes place on 27–28 July and the final Jury
meeting to approve scores and determine medal boundaries is scheduled
for 22:00 local time (16:00 BST) on 28 July.
In triangle ABC the bisector of angle BCA intersects
the circumcircle again at R, the perpendicular 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.
(Czech Republic)
Let a and b be positive integers. Show that if
4ab − 1 divides
(4a^{2} − 1)^{2}, then
a = b.
(United Kingdom)
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
threedimensional space. Determine the smallest possible number of
planes, the union of which contains S but does not include
(0, 0, 0).
(Netherlands)
