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)