AMTNJ 36th Annual Math Contest December 5, 2012 Directions: Ø 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. Ø You may only use calculators which are permitted on the SAT I. Ø You will have exactly 45 minutes to complete this contest. Work quickly, work accurately, and good luck. Ø 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) 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.