KANSAS STATE UNIVERSITY
Department of Mathematics
S. Thomas Parker Mathematical
Competition
April 17, 1999
First Place: Ali
Mohammad, $300 Award
Seccond Place: Trevor
Fast, $200 Award
Third Place: Peter
Pauzauskie, $100 Award
Honorable: Jesus
V. Hernandez, $20 Award
Problems
Instructions: Put your name on
all papers you use and turn them all in. Try to solve as many problems
as you can, in any order. For any problem you try, give as complete an
answer as you can. Include a clearly written explanation of how you found
your answer and why it is true. You may use drawings or calculations to
help you for your justification, but your explanation should be convincing.
1. A circular conical tank, with vertical axis, tip (vertex)
at the bottom, 4 ft in height, and 1 ft in radius, is filled
with water. A plug in the bottom is pulled and water flows out at
a rate (ft3/sec.) proportional to the water pressure at the
bottom of the tank. If the water level drops at rate of 1 in/sec.
at the beginning when the plug is pulled, how long does it take for
the tank to become empty? (Hint: the water pressure at the bottom
is proportional to the depth of water.)
2. A compass is used to draw a circle on a
plane. Then, without changing the compass opening, the compass is used
to draw a circle on a sphere of radius larger than the compass
opening. This circle partitions the sphere into two regions. Which is greater,
the area of the disk enclosed by the circle in the plane or the surface
area of the smaller region on the sphere?
3. Suppose that a real valued function f(x),
and the derivatives f'(x) and
f''(x) are defined for every real number
x. If f(0)=0, f'(0)=f'(1)=0,
and |f''(x)|<a
for 0=< x =<1, show that f(1)<a/4.
4. Prove that the average of the numbers n
sin(no) for n=2,
4, 6, . . . , 180 is cot(1o).
Solution to these Problems are at
Hale Library Reserve Desk.
Ask for Math 000 or S. Thomas Parker Mathematical Competition