(1) Let f ( n ) be the sum of a positive integer n and its digits. Find all numbers such that f ( n ) = 2010.
(2) For which k does the system


 ( x

2 
)
2 y

2 y
2


0
2
has exactly two real solutions?
(3) How many pairs of positive integers have greatest common divisor 3! and least common multiple 18!?
(4) Suppose you have a piece of paper which you cut into either four or sixteen pieces. After that you cut again some of the pieces into either four or sixteen smaller pieces. Suppose you have nothing else to do, so you keep repeating this procedure cutting some of the pieces into either four or sixteen smaller pieces. Can you end up with 2010 pieces at some stage of your cutting process?
(5) A group of 150 high school students visited Missouri State University for Computer
Science, Engineering, and Math Day. The students could participate in at most two of the three Science, Engineering, and Math workshops but must attend at least one. Suppose
100 students participated in Computer Science, 90 participated in Engineering, and 80 participated in Math. If 87 participated in both Computer Science and Engineering or in both Computer Science and Math. How many students participated in both Engineering and Math?
(6) Find the smallest positive integer N such that
1
3
N seventh power.
is a perfect cube and
1
7
N is a perfect
(7) How many ways are there to assign the numbers 1, 3, 5, 7, and 9 to c , o , k , i , and e so that the numerical equivalent to ‘ cookie ’ is divisible by 3?
(8) If x + y + z = 5, xy + xz + yz = 6, and xyz = 4, find x
3
+ y
3
+ z
3
.
(9) Suppose x
3
– 6 x + k and x
3
– 10 x + 2 k have a common root, What must k be?
(10) A polynomial P ( x ) of degree three is known to have P (1) = 1, P (2) = 1, P (3) = 2, and
P (4) = 2. Compute P (5)