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: _________________