IMO 2007 - Day 2
Đề ngày 2 đã được post lên Mathlinks.roĐề thi ngày 2 - IMO 2007
Question 4:
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
Question 5:
Let$ a$ and $b$ be positive integers. Show that if $4ab-1$ divides$ (4a^{2}-1)^{2}$, then $a=b$
Question 6:
Let $n$ be a positive integer. Consider
$S = \left\{ (x,y,z) \mid x,y,z \in \{ 0, 1, \ldots, n\}, x+y+z > 0 \right \}$
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 $(0,0,0)$
