VERMONT STATE MATHEMATICS COALITION TALENT SEARCH October 6, 2003
Test 1 of
the 2003 - 2004 school year (Test 2 arrives at
schools November 17, 2003)
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 November 3, 2003 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)
1. Use each of the ten digits 0 through 9
exactly once to form two numbers whose product is as large as possible. If a and b are the two numbers,
find 
.
Answer:
_________________
2. Find all ordered pairs (x, y) which satisfy
the equations
and 
Answer:
_________________
3. In the right triangle ABC, AE intersects BD
so that CD = CE = 8, AD = 16, and BE = 24.
Find
the area of triangle ABF.
Answer:
_________________
4. The roots of the equation 
are
a, b, and c. Find
the value of 
.
Answer:
_________________
5. A triangle has area 12. The points A(6, 4) and B(3, -1) are two of the vertices of a triangle. The third vertex is an x-intercept or a y-intercept. Find the sum of all of these possible x-intercepts and y-intercepts.
Answer: _________________
6. In a special four by four magic square, four
of the entries are
N(1, 3) = 14, N(2, 1) = 12, N(3, 2) = 10, and N(4, 4) = 16, where N(a, b) = m means that the number m is in row a and column b.
The other entries in the magic square are 2, 4, 6, 8, and –1, -3, -5, -7, -9, -11, -13, and –15.
As
in any magic square, when you add the four numbers along any row, along any
column, and along either main diagonal (that goes corner to corner), then the
sum of those four numbers is the same number, say t. This is a special magic square because each
of the following equations is also true:
N(1,
1) + N(1, 4) + N(4, 1) + N(4, 4) = t
N(2,
2) + N(2, 3) + N(3, 2) + N(3, 3) = t
N(1,
2) + N(1, 3) + N(4, 2) + N(4, 3) = t
N(2,
1) + N(2, 4) + N(3, 1) + N(3, 4) = t
N(1,
1) + N(1, 2) + N(2, 1) + N(2, 2) = t
N(1,
3) + N(1, 4) + N(2, 3) + N(2, 4) = t
N(3,
1) + N(3, 2) + N(4, 1) + N(4, 2) = t
N(3,
3) + N(3, 4) + N(4, 3) + N(4, 4) = t.
a) Find t.
b) How many different special magic squares
satisfy the conditions of the problem?
List these special magic squares.
Answer: a) ________, b) _________
7. A circle was drawn and a hexagon was inscribed within the circle. What is the area of the circle if the sides of the hexagon, in order, are 37, 37, 37, 52, 52, and 52, with all measurements in cm.
Answer: _________________
8. The sides of a right triangle have lengths a,
b, and c with 
. Two circles of the
same radius r are inscribed in the triangle as pictured with one circle tangent
to AC and AB, the other circle tangent to AC and BC, and the circles tangent to
each other. Express the radius r in
terms of a, b, and c without using radicals.
Answer:
_________________