# AMTNJ 36 Annual Math Contest

```AMTNJ
36th Annual
Math Contest
December 5, 2012
Directions:
&Oslash; 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.
&Oslash; You may only use calculators which are permitted on the
SAT I.
&Oslash; You will have exactly 45 minutes to complete this contest.
Work quickly, work accurately, and good luck.
&Oslash; You may write on this test paper or on any scrap paper
on the Student Response Sheet, to be official.
1)
Find a negative number which when added to its square gives a sum of 506.
2)
The arithmetic mean of a and b is 2. The arithmetic mean of b and c is 9,
and the arithmetic mean of a and c is 4. Find the arithmetic mean of a, b,
and c.
3)
Find the sum of all the distinct prime factors of 2012.
4)
Given the function, f ( x) = x2 + bx + c and f (2) + f (−2) = 20 ,
find f (4) + f (−4) = .
5)
The dimensions of Home Plate, in baseball,
17 inches
8.5 inches
8.5 inches
are given. Find the sum of its area and
perimeter to the nearest tenth of an inch.
6)
If a and b are non-zero real numbers such that :
Then to the nearest integer a − b =
7)
The sum of the squares of three consecutive positive odd integers is 3275.
What is the sum of the three integers?
8)
Find the coordinates of each point P which is 10 units from both the points
(-3, 4) and (9, 4).
9)
Circle C is centered at the origin with a
a + b = 3 and a b + b3 = 0
A
F
B
C
(0, 0)
radius of 13. If AB + DE = 9 , find the
area of rectangle BCDF.
D
E
10)
Four distinct points, M, A, T, and H, are to be randomly selected from 2012
points evenly spaced around a circle. All quadruples are equally likely to be
chosen. What is the probability that the chords MA and TH intersect?
11)
Given that point B is the vertex angle of isosceles triangle V ABC and
1
cos( A) = , find the exact value of cos( B) = .
3
12)
20
∑ (x
2
+ 3 x) + 1 =
x =1
13)
How many different 11 letter “words”(arrangements of letters) can be made
using all the letters in the word “mathematics”?
14)
A circle of radius 1” is always externally tangent to a square whose sides are
12”. As the circle rolls around the perimeter of the square, what is the exact
distance traveled by the center of the circle?
15)
Find the degree measure of the acute angle formed by the asymptotes of the
given hyperbola. Give your answer to the nearest tenth of a degree.
16 x2 − 9 y 2 − 96 x − 36 y − 117 = 0.
```