MathDB

1999 Advanced Topics #3: Unfair Coin

Source:

6/21/2012

An unfair coin has the property that when flipped four times, it has the same nonzero probability of turning up

2

heads and

2

tails (in any order) as

3

heads and

1

tail (in any order). What is the probability of getting a head in any one flip?

probability

1999 Algebra #3: Floating Platform

Source:

6/21/2012

Alex is stuck on a platform floating over an abyss at

1

ft/s. An evil physicist has arranged for the platform to fall in (taking Alex with it) after traveling

100

ft. One minute after the platform was launched, Edward arrives with a second platform capable of floating all the way across the abyss. He calculates for 5 seconds, then launches the second platform in such a way as to maximize the time that one end of Alex's platform is between the two ends of the new platform, thus giving Alex as much time as possible to switch. If both platforms are

5

ft long and move with constant velocity once launched, what is the speed of the second platform (in ft/s)?

1999 HMMT Geometry # 3

Source:

3/3/2024

In a cube with side length

6

, what is the volume of the tetrahedron formed by any vertex and the three vertices connected to that vertex by edges of the cube?

geometry

1999 Calculus #3: Interesting Integral with Irrational Bound

Source:

6/21/2012

Find

\int_{-4\pi\sqrt{2}}^{4\pi\sqrt{2}}\left(\dfrac{\sin x}{1+x^4}+1\right)dx.

calculusintegrationtrigonometryfunction

1999 HMMT Team #3

Source:

3/8/2024

How many non-empty subsets of

\{1, 2, 3, 4, 5, 6,7,8\}

have exactly

k

elements and do not contain the element

k

for some

k = 1, 2,...,8

.

combinatorics

1999 Oral #3: Distinct Positive Integers

Source:

10/20/2012

If

x

,

y

, and

z

are distinct positive integers such that

x^2+y^2=z^3

, what is the smallest possible value of

x+y+z

?

3

Problems(6)