1) The probability that Larry gets a six on his first throw is . The probability that it is his second throw is . The probability that he gets a six on the throw is
This is a geometric series with first term , with common ratio
.
. Finding the sum using
gives .
2) 7 Let n be the largest of the numbers. If n is removed the average must be greater than
. If 1 is removed the average must be less than . Therefore
and , hence n = 69 or 70. Since numbers, n must equal 69. If n = 69 then the missing number must be 7, since
is the average of n-1
gives n = 7.
3) therefore and is the smallest angle.
gives ,
4) 3 . This is real, only if . Solving for n , gives n = 0, 1, or -1. Therefore, there are 3 possible values for n .
5) no solutions. Adding all the given equations and rearranging the terms results in
or squares that add to 175; therefore, there are no solutions.
. There are no perfect

6) and
Let x be the angle opposite the side of length c .
or . By the Law of Cosines:
. Therefore,
7) 5
Therefore,
. Since  ABO
60
, then and
8) 0 log(4)+log(f)-log(5)-log(g)-log(3)-log(d)+log(g)-log(6)-log(f)+log(9)+ log(h)+ log(5)+log(d)-log(2)-log(h)= [log(2)+log(2)]-log(3)-[log(2)+log(3)]-[log(3)+log(3)]-log(2)=0
9) If the hexagon is ABCDEF and Willie starts at vertex A, he will pass through B and C and end up halfway to point D, called that point P. Since AB = BC = 2 and then AC = . is a right triangle with right angle point C. By the Pythagorean theorem
10) Since , solving for q gives
.
11) 69 Adding the two equations gives
.
Therefore,
12) remainder of
Dividing gives a quotient of
. Since the remainder must be 0, and
and a
13) 16
; therefore, a =1 and b = -2.
Each of the four playoff games has 2 possible outcomes. For each sequence of 4 outcomes, the prizes are awarded different ways. Thus, there are possible outcomes.

14) 4 If opposite side a is acute, then by the Law of Cosines
and Therefore, solving the inequalities
and gives integer answers of 22, 23, 24, and 25.
15) Jan. 27 th . For geometric series, . Therefore,
and or n = 27.
,