AMTNJ
34th Annual
Math Contest
December 8, 2010
Directions:
 Answers should only be in the form specified: decimals must be at least
three decimal places rounded or truncated. For example, 2/3 = 0.666 or
0.667. Fractions and irrational quantities must be in simplest form. In
some cases, the desired form of an answer is specified. No other form
will be accepted in those cases.
 You may only use calculators which are permitted on the SAT I’s.
 You will have exactly 45 minutes to complete this contest. Work
quickly, work accurately, and good luck.
 You may write on this test paper or on any scrap paper provided by your
teacher, but your answers must be written on the Student Response Sheet,
to be official.
1)
Larry, Curley, and Moe repeatedly take turns tossing a normal die. Larry begins, Curley
always follows Larry, and Moe always follows Curley, with Larry then starting the rotation
again. Find the probability that Larry is the first person to toss a six. (The probability of
obtaining a six on any toss is
2)
, independent of the outcome of any other toss.)
A set of consecutive positive integers beginning with 1 is written on a blackboard. One
number is erased. The average (arithmetic mean) of the remaining number is
. What
number was erased?
3)
If
this is true?
4)
Given
, what is the smallest positive value of x for which
, for how many integers n is
an integer?
5)
How many ordered triples (x, y, z) of integers satisfy the following system of equations?
6)
In a triangle with sides a, b, and c,
the angle opposite the side of length c ?
7)
In circle O,
What is the measure of
C
AD is a diameter,
is a chord,
, and
.
60°
Find the exact length of
B
A
60°
5
O
D
8)
Find the exact value of the following expression
using log base 10.
9)
A park is in the shape of a regular hexagon 2 km on a side. Starting at a corner, Willie
walks along the perimeter for a distance of 5 km. How many kilometers is Willie from his
starting point?
10)
Suppose x and y are inversely proportional and positive. If x increases by
decreases by what percent (in terms of p)?
11)
If (x, y) is a solution of the system:
?
and
, then y
then
12)
If a and b are integers such that
13)
At the end of a golf tournament, the top 5 golfers have a playoff. First #5 plays #4. The
loser receives the 5th prize and the winner plays #3 in another round of golf. The loser of
this round receives the 4th prize and the winner plays #2. The loser of this round receives
the 3rd prize and the winner plays #1. The winner of this round gets the first prize and the
loser gets the second prize. In how many orders can the golfers #1 through #5 receive the
prizes?
14)
For how many integers x does a triangle with sides 10, 24, and x have all of its angles
acute?
15)
Layla talked her grandparents into giving her some money. On January 1st they gave her 1
penny, on January 2nd they gave her two cents, and continued to double the amount of
money they gave her each day. On what day of the year did the total her grandparents give
her first exceed $1,000,000?
is a factor of
then b = ?