1

Part of 2006 Stanford Mathematics Tournament

Problems(4)

Difference in Rate between Riding and Walking

Source:

3/15/2010

After a cyclist has gone

\frac{2}{3}

of his route, he gets a flat tire. Finishing on foot, he spends twice as long walking as he did riding. How many times as fast does he ride as walk?

ratio

Cones from Rotating a Triangle about an Axis

Source:

3/16/2010

Given

\triangle{ABC}

, where

A

is at

(0,0)

B

is at

(20,0)

, and

C

is on the positive

y

-axis. Cone

M

is formed when

\triangle{ABC}

is rotated about the

x

-axis, and cone

N

is formed when

\triangle{ABC}

is rotated about the

y

-axis. If the volume of cone

M

minus the volume of cone

N

140\pi

, find the length of

\overline{BC}

geometry3D geometryrotation

Sequence and Divisibility

Source:

3/15/2010

A finite sequence of positive integers

m_i

for i\equal{}1,2,...,2006 are defined so that m_1\equal{}1 and m_i\equal{}10m_{i\minus{}1} \plus{}1 for

i>1

. How many of these integers are divisible by

37

Romance Crisis

Source:

3/20/2010

A college student is about to break up with her boyfriend, a mathematics major who is apparently more interested in math than her. Frustrated, she cries, ”You mathematicians have no soul! It’s all numbers and equations! What is the root of your incompetence?!” Her boyfriend assumes she means the square root of himself, or the square root of i. What two answers should he give?

trigonometry