Homework
Easy
1. Suppose
and
are real numbers such that
and
. What is
?
2. The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle
number?
3. If and are real numbers such that
and
, what is
?
4. In a rectangular solid, the areas of the faces are 135, 30, and 50. Find the volume of the
solid.
Medium
5. Consider the following system of 7 linear equations with 7 unknowns:
What is g?
6. Let
be a square, and let
respectively. The line through
divide
parallel to
(it’s not just geometry!)
, , and be positive real numbers such that
.
Hard
and
and the line through
of the area of square
and
Find
be points on
parallel to
into two squares and two nonsquare rectangles. The sum of
the areas of the two squares is
7. Let
and
.
Find
8. Let
, , and
be non-zero real numbers such that
. There are relatively prime positive integers
,
and
, and
so that
. Find
.
9. A rectangular box
is inscribed in a sphere of radius . The surface area of
384, and the sum of the lengths of its 12 edges is 112. What is ?
10. Let
If
, and
be positive real numbers such that
, compute the greatest positive integer less than S.
is