5

Part of 2004 Harvard-MIT Mathematics Tournament

Problems(6)

2004 Algebra #5

Source:

There exists a positive real number

x

\cos (\arctan (x)) = x

. Find the value of

x^2

trigonometryquadratics

2004 Calculus #5

Source:

A mouse is sitting in a toy car on a negligibly small turntable. The car cannot turn on its own, but the mouse can control when the car is launched and when the car stops (the car has brakes). When the mouse chooses to launch, the car will immediately leave the turntable on a straight trajectory at

1

meter per second. Suddenly someone turns on the turntable; it spins at

30

rpm. Consider the set

S

of points the mouse can reach in his car within

1

second after the turntable is set in motion. What is the area of

S

, in square meters?

calculusgeometry

2004 Combinatorics #5

Source:

A best-of-9 series is to be played between two teams; that is, the first team to win 5 games is the winner. The Mathletes have a chance of

\tfrac{2}{3}

of winning any given game. What is the probability that exactly 7 games will need to be played to determine a winner?

2004 HMMT Geometry # 5

Source:

Find the area of the region of the

xy

-plane defined by the inequality

|x|+|y|+|x+y| \le 1

2004 General, part 1 #5

Source:

A rectangle has perimeter

10

\sqrt{15}

. What is its area?

2004 General, part 2 #5

Source:

Eight strangers are preparing to play bridge. How many ways can they be grouped into two bridge games - that is, into unordered pairs of unordered pairs of people?