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2008 LSU Math Contest

14
Suppose the equation x 2+bx+4 = 0 has exactly one solution.
The best statement about b is:

A b must be equal to 4

B b must be equal to −4

C b must be equal to 2

D b must be equal to −2

E none of the above
15
Of the following statements, the one that is incorrect is:

A Doubling the base of a rectangle doubles the area.

B Doubling the altitude of a triangle doubles the area.

C Doubling the radius of a circle doubles the area.

D Doubling the denominator of a positive fraction and dividing
its numerator by 2 changes the fraction .

E Doubling a given number may make it less than it originally
was.
16
Suppose the triangle
ΔABC shown in the
diagram is an equilateral
triangle. If the line BC is
described by an equation
of the form y = mx + b
for some m and b, then
find m.

17
During the final game of a basketball tournament, only 7 players
from the tournament winning team played. The scoring
average of all 7 players was 13. The scoring average of everyone
but the point guard was 12. How many points did the
point guard score?

18
A small circle just fits inside a semicircle. What is the ratio
of the area of the small circle to the area of the semicircle
containing it?

Questions 19 - 28 Exact Answers

These next ten questions require exact numerical or algebraic
answers. Hand-written exact answers must be written
on the answer sheet with fractions reduced , radicals simplified ,
and denominators rationalized (Improper fractions can
be left alone or changed to mixed fractions). Do not make
an approximation for π or other irrational numbers. Answers
must be exact. Large numbers should not be multiplied out ,
i.e., do not try to multiply out 20! or 640.
19 An airport has a moving sidewalk. Standing on the
moving sidewalk, Joe takes 3 minutes to reach the end.
Walking next to the moving sidewalk, Joe takes 4 minutes
to reach the end. How long does it take Joe to reach
the end if he walks on the moving sidewalk?
20 John earned some money. When he was paid, his employer
had already deducted 20% federal and 2% state
income taxes. He was paid $792. How much, to the
nearest dollar, did he earn?
21 A six place number is formed by repeating a three place
number: For example, 256,256, or 678,678, etc. Call
these numbers copycats. Find greatest common divisor
of all the copycats.
22 The squares ABCD and
EFGH have equal areas.
The area of the
shaded square is 1/9 of
the area of the square
ABCD. Find the area of
the square ARGS if the
area of the shaded square
is 49in2.
23 For how many different positive integers does differ
from by less than 1?
24 What is the last digit of 22008?
25 Let

Compute the integer x.

26 A line is called a supporting line of a figure if the line
meets the figure but the whole figure is on one side of
the line. For example, the line that contains the side of
a square is a supporting line of the square. Consider the
figure below drawn on the coordinate plane - it is half
of the unit circle.

If a supporting line passes through A, then how large of
slope can the line have and how small of slope can the
line have?

27 Suppose A and B are points on the circle with center
C. The angle ∠ACB is 30°. If a point D is randomly
chosen on the circle, then what is the probability that
the triangle ΔABD is obtuse?
28 A room is 30 feet square and 12 feet high. A spider
is located in one of the corners on the floor. An unsuspecting
fly rests at the diagonally opposite corner on
the ceiling. If the fly does not move, what is the shortest
distance the spider must crawl, always touching a
surface of the room, to catch the fly?
Tie Breaker

Please give a detailed explanation on the answer sheet to your
solution to Question 28 above.

This tie breaker question is graded as an essay question
i.e., it is graded for the clarity of explanation and argument as
well as correctness. It is the only question graded for partial
credit.

It is graded only to separate first, second, and third place ties.
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