WORCESTER POLYTECHNIC INSTITUTE

TWENTY-FIRST ANNUAL INVITATIONAL MATH MEET

OCTOBER 21, 2008

TEAM EXAM QUESTION SHEET WITH ANSWERS

 

 

1.      Suppose a and b are digits (integers from 0 through 9). What number divides abba for all choices of a and b?

Ans: 11

 

 

2.      Find all points (x,y) in the Euclidean plane satisfying F(x,y) = 0 where

 

F(x,y) = xy4 y4 + x3 y2 x2 y2 x3 + x2 x +1

 

Ans: union of the lines: x = 1, y = 1 and y= -1.

 

 

3.      Find x > 0 satisfying the following equation

 

 

Ans: 1/20

 

 

4.      Arithmetic series sum: 200 + 205 + 210 + 215 + . . . 2745

 

Ans: 750975

 

 

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

 

x5 + x4 + 4 x3 x2 19 x + 14

Ans: (x-1)2(x+2)(x2 + x + 7)

 

 

6.      A right circular cone is formed by cutting up a circular piece of paper which is 10 units in diameter. A sector of angular measure 2Π/3 radians is removed and the remaining paper formed into the cone. What is its volume?

 

Ans:

 

 

7.      Hoodsie the cow is tethered to a corner of a barn which is 20′ by 40′. Her rope is 60′ long. How much grazing area does she have? No barn doors are open.

 

Ans: 3200Pi

 

 

8.      How many zeroes are at the end of the expansion of 31!

 

Ans: 7

 

 

9.      A circle inscribed in an equilateral triangle and a square inscribed in the circle. What is the ratio of the area of the triangle to that of the square?

 

Ans: 3:2

 

 

10.  What is the minimum value of if 5x + 12y = 60?

 

Ans: 60/13

 

 

11.  The sum of an infinite geometric series with -1< r < 1 as its common ratio, is 15. The sum of the squares of the terms in this series is 45. What is the first term in this series?

 

Ans: a = 5

 

 

12.  How many permutations of the letters D,O,R,E,M,I do not contain the word DO, RE, MI; that is none of the words DO, RE and MI appears as consecutive letters?

 

Ans: 426

 

 

13.  For some real number r, the polynomial 8x3 +4x2 - 42x - 45 is divisible by (x r)2. What is r ?

 

Ans: 3/2

 

 

14.  How many real solutions are there to the equation: x256 25632 = 0 ?

 

Ans: 2