WORCESTER POLYTECHNIC INSTITUTE

TWENTIETH ANNUAL INVITATIONAL MATH MEET

OCTOBER 17, 2007

TEAM EXAM QUESTION SHEET WITH ANSWERS

 

  1. A ball was floating in a lake when the lake froze. The ball was removed (without breaking the ice), leaving a hole 24 cm across the top and 8 cm deep. What was the radius of the ball in centimeters?

 

Ans: 13 cm

 

  1. Point of tangency of two spheres described by

x2 + y2 + z2 = 121

(x-4)2 + (y-12)2 + (z-18)2 = 121

Ans: (2,6,9)

 

  1. Consider the graphs of y = Ax2 and y2 + 3 = x2 + 4y, where A is a positive constant and x and y are real variables. In how many points do the two graphs intersect?

 

Ans: 2

 

  1. A certain 5 digit number has the property that if a 1 is placed after it, it is 3 times as large as with a 1 placed before it. What is that number?

 

Ans: 42857

 

  1. We have two concentric circles and wish to find the area of the annulus between them.

If we draw a chord through the outer circle tangent to the inner circle, its length is 20 inches. What is that area?

 

Ans: 100Pi

 

  1. Factor the following polynomial over the reals as completely as possible:

 

Ans: (x-3/2)(x+2)(x2 + 9)(x-1)3

 

7. If z = 2 +√2 i, what is z20 ?? (where i = )

Ans: z = -1048576

 

8. Simplify (a + b)15 mod 15

 

ans:

 

  1. Simplify 3218 mod 7.

Ans: 2

 

  1. Determine the file size in Megabytes (Mb) for a digital recording made with samples of

size 2 bytes taken 44,100 times per second, in stereo, for 40 minutes. Your answer

should be rounded to the nearest tenth of a Mb.

 

Ans: 423.4 Mb

 

 

11. Find the sum 17 + 22 + 27+. . . +182

Ans: 3383

12. If it is given that

logx w=24 logy w =40 logxyz w = 12

then what is logzw ? (x, y and z are all positive numbers)

Ans: 60

 

13. Simplify the following to a single fraction

-5/4 + 5/8 5/16 + 5/32 . . . - 5/1024

Ans: -855/1024

 

14. Determine where p is prime and positive.

Ans: same matrix

(true for odd powers therefore prime)