MathDB

1999 Advanced Topics #5: Sum of Subsets

Source:

6/21/2012

For any finite set

S

, let

f(S)

be the sum of the elements of

S

(if

S

is empty then

f(S)=0

). Find the sum over all subsets

E

of

S

of

\dfrac{f(E)}{f(S)}

for

S=\{1,2,\cdots,1999\}

.

1999 Algebra #5: Hallways and Land Mines

Source:

6/21/2012

You are trapped in a room with only one exit, a long hallway with a series of doors and land mines. To get out you must open all the doors and disarm all the mines. In the room is a panel with

3

buttons, which conveniently contains an instruction manual. The red button arms a mine, the yellow button disarms two mines and closes a door, and the green button opens two doors. Initially

3

doors are closed and

3

mines are armed. The manual warns that attempting to disarm two mines or open two doors when only one is armed/closed will reset the system to its initial state. What is the minimum number of buttons you must push to get out?

1999 Calculus #5: Continuous Fraction

Source:

6/21/2012

Let

f(x)=x+\cfrac{1}{2x+\cfrac{1}{2x+\cfrac{1}{2x+\cdots}}}

. Find

f(99)f^\prime (99)

.

calculusquadraticsalgebraquadratic formula

1999 HMMT Geometry # 5

Source:

3/3/2024

In triangle

BEN

shown below with its altitudes intersecting at

X

,

NA = 7

,

EA = 3

,

AX = 4

, and

NS = 8

. Find the area of

BEN

. https://cdn.artofproblemsolving.com/attachments/5/7/7e6dcbe6aa220821cb5020824b8aa6d4fc597d.png

geometry

1999 HMMT Team #5

Source:

3/8/2024

If

a

and

b

are randomly selected real numbers between

0

and

1

, find the probability that the nearest integer to

\frac{a-b}{a+b}

is odd.

combinatorics

1999 Oral #5: Always the Side Lengths of a Triangle

Source:

10/20/2012

Let

r

be the inradius of triangle

ABC

. Take a point

D

on side

BC

, and let

r_1

and

r_2

be the inradii of triangles

ABD

and

ACD

. Prove that

r

,

r_1

, and

r_2

can always be the side lengths of a triangle.

geometryinradius

5

Problems(6)