MathDB

Fermi Questions
p1. What is

\sin (1000)

? (note: that's

1000

radians, not degrees)
p2. In liters, what is the volume of

10

million US dollars' worth of gold?
p3. How many trees are there on Earth?
p4. How many prime numbers are there between

10^8

and

10^9

?
p5. What is the total amount of time spent by humans in spaceflight?
p6. What is the global domestic product (total monetary value of all goods and services produced in a country's borders in a year) of Bangladesh in US dollars?
p7. How much time does the average American spend eating during their lifetime, in hours?
p8. How many CHMMC-related emails did the directors receive or send in the last month?
Suspiciously Familiar. . .
p9. Suppose a farmer learns that he will die at the end of the year (day

365

, where today is day

0

) and that he has

100

sheep. He decides to sell all his sheep on one day, and that his utility is given by

ab

where

a

is the money he makes by selling the sheep (which always have a fixed price) and

b

is the number of days he has left to enjoy the profit; i.e.,

365 - k

where

k

is the day number. If every day his sheep breed and multiply their numbers by

(421 + b)/421

(yes, there are small, fractional sheep), on which day should he sell out?
p10. Suppose in your sock drawer of

14

socks there are

5

different colors and

3

different lengths present. One day, you decide you want to wear two socks that have either different colors or different lengths but not both. Given only this information, what is the maximum number of choices you might have?
I'm So Meta Even This Acronym
p11. Let

\frac{s}{t}

be the answer of problem

13

, written in lowest terms. Let

\frac{p}{q}

be the answer of problem

12

, written in lowest terms. If player

1

wins in problem

11

, let

n = q

. Otherwise, let

n = p

. Two players play a game on a connected graph with

n

vertices and

t

edges. On each player's turn, they remove one edge of the graph, and lose if this causes the graph to become disconnected. Which player (first or second) wins?
p12. Let

\frac{s}{t}

be the answer of problem

13

, written in lowest terms. If player

1

wins in problem

11

, let

n = t

. Otherwise, let

n = s

. Find the maximum value of

\frac{x^n}{1 + \frac12 x + \frac14 x^2 + ...+ \frac{1}{2^{2n}} x^{2n}}

for

x > 0

.
p13. Let

\frac{p}{q}

be the answer of problem

12

, written in lowest terms. Let

y

be the largest integer such that

2^y

divides

p

. If player

1

wins in problem

11

, let

z = q

. Otherwise, let

z = p

. Suppose that

a_1 = 1

and

a_{n+1} = a_n -\frac{z}{n + 2}+\frac{2z}{n + 1}-\frac{z}{n}

What is

a_y

?
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement