**51. If [x] denotes the greatest integer not exceeding x, then Int_0^Inf [x]e^(-x) dx =?**

A) e/(e^2-1)

B) 1/(e-1)

C) (e-1)/e

D) 1

E) +Inf

I'm unsure how to answer this question fully since I dont know how to deal with [x]. I tried approximating it by just replacing it with x and integrating that way and came up with 1. However this is the wrong answer, the correct answer is infact 1/(e-1).

**54. The four shaded circles in Figure 1 above are congruent and each is tangent to the large circle and to two of the other shaded cricles. Figure 2 is the result of replacing each of the shaded circles in Figure 1 by a figure that is geometrically similar to Figure 1. What is the ratio of the area of the shaded portion of Figure 2 to the area of the shaded portion of Figure 1?**

A) 1/2Root(2)

B) 1/(1+Root(2))

C) 4/(1+Root(2))

D) ( Root(2)/(1+Root(2)) )^2

E) ( 2/(1+Root(2)) )^2

So the number we are looking for is Area of Shaded Region in Figure 2 / Area of Shaded Region in Figure 1. Just by inspection alone we can deduce that the ration must be lower then 1 so we can eliminate C as an answer.

I did some of my own calculations (hard to explain without the figures) but I determined the answer to be 4R^2 where R is the radius of the 4 inscribed circles in Figure 1 (Letting the radius of the main circle to be 1). What did everyone else get?

**55. For how many positive integers k does the ordinary decimal respresentation of the integer k! end in exactly 99 zeros.**

A) None

B) One

C) Four

D) Five

E) Twenty-four

I'm unsure how to start this problem on so many levels. At first I though I could count how many orders of 10 were being multiplied until there were 99 but then I realized the other factors could produce extra zeroes as well.

**64. For each positive integer n, let f_n be the function defined on the interval [0,1] by f_n(x) = x^n /(1+x^n) which of the following statements is true?**

I. The sequence converges pointwise.

II. The sequence converges uniformly

III. lim_n^Inf Integral_0^1 f_n(x) dx = Integral_0^1 lim_n^Inf f_n(x) dx

I know the function converges pointwise and not uniformly since convergence depends on the value of x. My problem is that I'm unsure to k now for certain when its ok to pass the limit for it. I know if it was converging uniformly it would be alright since f is continuous on the interval, is there any rule here that I should be made aware of?

**65. Which of the following statements are true about the open interval (0,1) and the closed interval [0,1]?**

I. There is a continuous function from (0,1) onto [0,1]

II. There is a continuous function from [0,1] onto (0,1)

III. There is a continuous bijective function from (0,1) to [0,1]

So, I know there is a bijection between the two sets however its not continuous from what I recall. If III is correct then both I and II are correct therefore D is wrong as a possible answer (I and III). How did you guys go about checking each of these individually though?