VERMONT STATE MATHEMATICS COALITION TALENT SEARCH CONTEST
Test 4 of the 20002001 school year Feb. 14 , 2001
Student Name ________________________ School ____________________________
Grade ________ Math Department Head _______________________________
Directions: Solve as many as you can of the problems and
list your solutions on this sheet of paper. On separate sheets, in an organized
way, show how you solved the problems. You will be awarded full credit
for a complete correct answer which is adequately supported by mathematical
reasoning. You can receive half credit for correct answers which are the
result of guesses, conjectures or incomplete solutions. Included as incomplete
solutions are solutions that list some, but not all, solutions when the
problem asks for solutions of equations. The decisions of the graders are
final. You may earn bonus points for "commendable solutions" solutions
that display creativity, ingenuity and clarity. Your answers and solutions
must be postmarked by March 14, 2001 and submitted to Tony Trono, Vermont
State Mathematics Coalition, 419 Colchester Avenue, Burlington, VT 05401.
(For Coalition information and a copy of the test: http://www.vermontinstitutes.org/vsmc)

There are five parts to this problem. The answer to part a) will be used
to solve part b). The answer to part b) will be used to solve part c),
etc. "YPA" will mean "your previous answer". Your final answer is the answer
to part e).

If = 39, =
47, = 138,
and if =
5, = 43,
and = 201,
then find n> 0 for which = .

Let k = YPA + 1. The equation has
solutions which are the squares of those of .
Find the sum p + q.

Two circles are externally tangent. The length of the line segment connecting
their centers is YPA and the length of the common external tangent is 24.
Find the radius of the larger circle.

Let k = YPA. Find the area of the ellipse with equation

The circumference of a circle O is YPA. A chord is drawn perpendicular
to the diameter AB at the midpoint of OB. Find the area of the smaller
segment of the circle thus formed.
Answer: a) _________ Answer: b)__________ Answer: c)__________
Answer: d) _________________ Answer: e)_________________

Vermont license plates (not vanity plates) may contain letters, but the
last two places do not contain letters. Suppose that eight cars pass you
and all have digits in the last two places. What is the probability that
at least one of the cars has a plate that ends in a double digit (00, 11,
22, …, 99)?
Answer: _________________

Let (m, n, p, q) be a solution of 2m + 3n + 7p + 12q = 103, where m, n
p, q are nonnegative integers. Find (m, n, p, q) so that is
as large as possible.
Answer: _________________

The sequence (1, 3, 4, 9, 10, 12, 13, … ) consists of all positive integers
which are powers of 3 or sums of distinct powers of 3. For example, ,
and . What
is the 100^{th} term of the sequence?
Answer: _________________

There are two buildings, a school and a gymnasium, opposite each other
along a road. A ladder set in the road leaning against the school has its
top h feet above the road. When the ladder is shifted, without moving the
base, to lean against the gymnasium, then its top is k feet above the road.
When leaning against the school, the angle made by the ladder and the road
is 75º. When leaning against the gymnasium, the angle made by the
ladder and the road is 45º. Express the width of the road in terms
of h and k and in simplest form.
Answer: _________________

Find the rational value of x for which is
four times the value of .
Answer: _________________

The number of terms of an arithmetic progression is even. The sum of the
odd numbered terms is 48, and the sum of the even numbered terms is 78.
The last term of the AP exceeds the first term by 57.5. Find the last term
of the AP.
Answer: _________________

A pentagon inscribed in a circle has sides of lengths 16, 16, 16, 16 and
6. Find the area of the pentagon.
Answer: _________________