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