5

Part of 2006 Harvard-MIT Mathematics Tournament

Problems(4)

2006 Algebra #5: Continuously Moving Hands

Source:

Tim has a working analog 12-hour clock with two hands that run continuously (instead of, say, jumping on the minute). He also has a clock that runs really slow—at half the correct rate, to be exact. At noon one day, both clocks happen to show the exact time. At any given instant, the hands on each clock form an angle between

0^\circ

180^\circ

inclusive. At how many times during that day are the angles on the two clocks equal?

2006 Calculus #5: Radical Integral

Source:

\displaystyle\int_0^1\dfrac{dx}{\sqrt{x}+\sqrt[3]{x}}

calculusintegrationlogarithms

2006 Combinatorics #5: Freshmen and Handouts

Source:

Fifteen freshmen are sitting in a circle around a table, but the course assistant (who remains standing) has made only six copies of today’s handout. No freshman should get more than one handout, and any freshman who does not get one should be able to read a neighbor’s. If the freshmen are distinguishable but the handouts are not, how many ways are there to distribute the six handouts subject to the above conditions?

countingdistinguishability

2006 Geometry #5: Area of Quadrilateral

Source:

ABC

has side lengths

AB=2\sqrt{5}

BC=1

CA=5

D

AC

CD=1

F

is a point such that

BF=2

CF=3

E

be the intersection of lines

AB

DF

. Find the area of

CDEB