## VERMONT STATE MATHEMATICS COALITION TALENT SEARCH

Test 1 of the 2002-2003 school year, October 7, 2002

Student Name ________________________ School ____________________________

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 4, 2002 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. Palindromes are words or sentences in which the letters are the same when read backwards, but numbers can have the same property. Our year 2002 is a palindrome: reading it backward produces the same number. Other numbers that are palindromes are 33, 757, 1001, and 23432. Write the year 2002 after itself to produce the number 20022002. This number is also a palindrome. The problem: Write this number 20022002 as the product of three numbers so that
• each of the three numbers is a palindrome, and
• each number contains at least two digits.

2.
 Complete the pictured five by five array using only the digits 1, 2, 3, or 4 so that: the sum of the digits in any row and any column will be 13, the sum of the digits along one diagonal is 7, the sum of the digits along the other diagonal is greater than 10, and digits are different when they are in squares that are adjacent horizontally or vertically. Six of the digits are included to start the solution.
3. A triangle with sides of lengths 3, 4, and 5 has two circles of the same radius drawn into the triangle so that:
• the circles are tangent to each other, and
• each circle is tangent to a leg and the hypotenuse of the triangle.
Find the radius of the circles.

4. a) A sequence is defined as follows: , and for n = 2, 3, 4, ... .
Find a formula for in terms of n and use the formula to evaluate .

b) Another sequence is defined by the formula and for n = 2, 3, 4, ... .
Find a formula for in terms of n and use the formula to evaluate .

Answer: a) = ________, = ________, b) = ________, =________

5. In a special coordinate system, the unit length on the y-axis is five times the unit length on the x-axis. In this system, the lines with equations are perpendicular.

a) Find m.

b) Find the coordinates of the point of intersection of the two lines.

6. Let f(x) = and let g(x) = f(f(f(x))).

a) Find the area bounded by the graph of g(x) and the x-axis.

b) Find the length of the graph of graph of g(x).

7. The triangle ABC has sides of these lengths: AB = 5, AC = 3, BC = 4. Point Y lies on AB so that AY = AC. Point X lies on AB so that BX = BC. Find the measure of the angle XCY.