MathDB

p1. Consider the following function

f(x) = \left(\frac12 \right)^x - \left(\frac12 \right)^{x+1}

. Evaluate the infinite sum

f(1) + f(2) + f(3) + f(4) +...

p2. Find the area of the shape bounded by the following relations

y \le |x| -2

y \ge |x| - 4

y \le 0

where |x| denotes the absolute value of

x

.
p3. An equilateral triangle with side length

6

is inscribed inside a circle. What is the diameter of the largest circle that can fit in the circle but outside of the triangle?
p4. Given

\sin x - \tan x = \sin x \tan x

, solve for

x

in the interval

(0, 2\pi)

, exclusive.
p5. How many rectangles are there in a

6

6

square grid?
p6. Find the lateral surface area of a cone with radius

3

and height

4

.
p7. From

9

positive integer scores on a

10

-point quiz, the mean is

8

, the median is

8

, and the mode is

7

. Determine the maximum number of perfect scores possible on this test.
p8. If

i =\sqrt{-1}

, evaluate the following continued fraction:

2i +\frac{1}{2i +\frac{1}{2i+ \frac{1}{2i+...}}}

p9. The cubic polynomial

x^3-px^2+px-6

has roots

p, q

, and

r

. What is

(1-p)(1-q)(1-r)

?
p10. (Variant on a Classic.) Gilnor is a merchant from Cutlass, a town where

10\%

of the merchants are thieves. The police utilize a lie detector that is

90\%

accurate to see if Gilnor is one of the thieves. According to the device, Gilnor is a thief. What is the probability that he really is one?
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

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