**WORCESTER POLYTECHNIC INSTITUTE**

**TWENTIETH ANNUAL INVITATIONAL MATH MEET**

**OCTOBER 17, 2007**

**TEAM EXAM QUESTION SHEET **

** **

- 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?

- Point of tangency of two spheres described by

** x ^{2} + y^{2} + z^{2} = 121**

** (x-4) ^{2}
+ (y-12)^{2} + (z-18)^{2} =
121**

- Consider
the graphs of
**y = Ax**^{2}and**y**, where A is a^{2}+ 3 = x^{2}+ 4y*positive*constant and x and y are real variables. In how many points do the two graphs intersect?

- 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?

- 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?

- Factor the following polynomial over the reals as completely as possible:

_{}

_{ }

** **7.** **If **z**** = √2 +√2 i** ,
what is **z ^{20}** ? (where i =

** **

8. Simplify **(****a + b) ^{15} mod 15.**

- Simplify
**3**^{218}**mod 7.**

- 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.

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

12. If it is given that

** log _{x}_{
}w=24 log_{y }w_{ }=40 log_{xyz }w = 12 **

then what is** log _{z}w ? (x,y **and

** **

** **

13. Simplify
the following to a single fraction** **

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

14.** **Determine** **_{} where** p
**is prime and positive