AMTNJ
35th Annual
Math Contest
December 7, 2011
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.
 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)
Given the set of integers from -50 to 50 inclusive, how many of these
integers have the property that the square of the integer has a unit’s
digit of 1?
2) Point D is on the hypotenuse, AB , of right ABC such that CD=BD and
m B  40 . Find m ADC .
3) How many different 11 letter “words” can be made using all the letters in
the word Mississippi ?
4) Given: 2011x2  2010 x  1  0 , find the sum of the roots.
5) Find all real numbers x, such that: 6  2 x  x 2  3x .
6) Points P and Q lie on line segment AB , with AP:PB = 1:5, and PQ:QB =
2:3. if PQ = 8, find AB =
7) Layla, Julian and Pressley always run at constant rates. In a two person
100 yard dash, Layla gave Julian a 10-yard start, and they tied. Julian
could have done the same with Pressley. How many yards head start
should Layla give Pressley if Layla wants to tie Pressley in a 100-yard
dash?
8) The six edges of tetrahedron ABCD measure 7, 13, 18, 27, 36, and 41
units. If the length of edge AB is 41, find the length of edge CD.
9) A lattice point is a point in a plane with integer coordinates. How many
lattice points are on the line segment whose endpoints are (3, 17) and (48,
281)? Include the endpoints.
10)
Four whole numbers when added three at a time, give the sums 180,
197, 208, and 222. What is the largest of the four numbers?
11)
Given ABC , with m ABC  120 , AB =3, and BC = 4. If
perpendiculars are constructed to AB at A and to BC at C, they meet at
point D. Find the length of CD to the nearest 100th.
12)
An “unfair” coin has a 2/3 probability of turning up heads. If this
coin is tossed 50 times, what is the exact probability that the total number
of heads is even?
13)
Bob and Bill start their new jobs on the same day. Bob’s schedule is
3 work-days followed by 1 rest-day. Bill’s schedule is 7 work-days
followed by 3 rest-days. On how many of their first 1000 days do both
have rest days on the same day?
14)
If cos( A)  cos2 (60)  sin 2 (60) , find the exact value of sin( A) .
15)
Let A = 10, B = 20, and C = 40. Write 3log x A  4log x B  7log x C , in
the form M log x N if M and N are integers and N is as small as possible.