**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)
= xy ^{4} – y^{4} + x^{3} y^{2} – x^{2}
y^{2} – x^{3} + x^{2} – 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:

** x ^{5}
+ x^{4} + 4 x^{3} – x^{2} – 19 x + 14**

**Ans****:
(x-1) ^{2}(x+2)(x^{2} + 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 **2****Π****/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 **8x ^{3}
+4x^{2} - 42x - 45** is
divisible by

**Ans****:
3/2**

14. How many real solutions are there to the
equation: **x ^{256} – 256^{32} = 0 ?**

**Ans****: 2**