VERMONT STATE MATHEMATICS COALITION TALENT SEARCH
Test 1 of the 2002-2003 school year, October 7, 2002
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 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)
- 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.
Answer: ___________,___________,___________
-
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.
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- 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.
Answer: ___________,___________,___________
- 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) 
= ________,
=________
- 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.
Answer: a)______________, b)______________
- 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).
Answer: a)______________, b)______________
- 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.
Answer: _________________
-
| In the early 1500s, the mathematician Tartaglia solved
the general cubic equation. That success began a search among
mathematicians for a solution to the general quartic (fourth degree)
equation. The following system of equations in the box leads to a
quartic equation in b.
a) Write the quartic equation in b.
b) Find solutions of this system of equations for real numbers (a,
b, c) correct to the nearest hundredth.
Answer: a)______________, b)______________
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