10

Part of 2000 Harvard-MIT Mathematics Tournament

Problems(6)

2000 Guts #10: Surface area

Source:

What is the total surface area of an ice cream cone, radius

R

H

, with a spherical scoop of ice cream of radius

r

R<r

geometry3D geometry

2000 Advanced Topics #10: Figuring Each Other's Numbers

Source:

I call two people

A

B

and think of a natural number

n

. Then I give the number

n

A

n+1

B

. I tell them that they have both been given natural numbers, and further that they are consecutive natural numbers. However, I don't tell

A

B

's number is and vice versa. I start by asing

A

B

's number. He says "no", Then I ask

B

A

's number, and he says "no" too. I go back to

A

and ask, and so on.

A

B

can both hear each other's responses. Do I ever get a "yes" in response? If so, who responds first with "yes" and how many times does he say "no" before this? Assume that both

A

B

are very intelligent and logical. You may need to consider multiple cases.

2000 Oral #10: Distributing water

Source:

23

frat brothers are sitting in a circle. One, call him Alex, starts with a gallon of water. On the first turn, Alex gives each person in the circle some rational fraction of his water. On each subsequent turn, every person with water uses the same scheme as Alex did to distribute his water, but in relation to themselves. For instance, suppose Alex gave

\frac{1}{2}

\frac{1}{6}

of his water to his left and right neighbors respectively on the first turn and kept

\frac{1}{3}

for himself. On each subsequent turn everyone gives

\frac{1}{2}

\frac{1}{6}

of the water they started the turn with to their left and right neighbors, respectively, and keep the final third for themselves. After

23

turns, Alex again has a gallon of water. What possibilities are there for the scheme he used in the first turn? (Note: you may find it useful to know that

1+x+x^2+\cdot +x^{23}

has no polynomial factors with rational coefficients)

algebrapolynomial

2000 Algebra #10: Smallest Value so Expression is not Prime

Source:

Find the smallest positive integer

a

x^4+a^2

is not prime for any integer

x

2000 HMMT RMT Geometry #10

Source:

C_1

C_2

be two concentric reflective hollow metal spheres of radius

R

R\sqrt3

respectively. From a point

P

on the surface of

C_2

, a ray of light is emitted inward at

30^o

from the radial direction. The ray eventually returns to

P

. How many total reflections off of

C_1

C_2

2000 HMMT RMT Team #10

Source:

How many times per day do at least two of the three hands on a clock coincide?

geometryalgebra