Problem 4/10. A bag contains
1993
red balls and
1993
black balls. We
remove two balls at a time repeatedly and
(i) discard them if they are of the same color,
(ii) discard the black ball and return to the bag the red ball if they are different
colors.
What is the probability that this process will terminate with one red ball in
the bag?
Problem 5/10. Let
P
be a point on the circumcircle of
4
ABC
, distinct from
A; B
, and
C
. Suppose
BP
meets
AC
at
X
, and
CP
meets
AB
at
Y
. Let
Q
be the point of intersection of the circumcircles of
4
ABC
and
4
AXY
,
with
Q
6
= A
. Prove that
PQ
bisects the segment
XY
. (The various points
of intersection may occur on the extensions of the segments.)