MathDB

p1. Given

f(x)=\frac{1+x}{1-x}

, for

x \ne 1

. Defined

p @ q = \frac{p+q}{1+pq}

for all positive rational numbers

p

and

q

. Note the sequence with

a_1,a_2,a_3,...

with

a_1=2 @3

a_{n}=a_{n-1}@ (n+2)

for

n \ge 2

. Determine

f(a_{233})

and

a_{233}

p2. It is known that

a

and

b

are positive integers with

a > b > 2

. Is

\frac{2^a+1}{2^b-1}

an integer? Write down your reasons.
p3. Given a cube

ABCD.EFGH

with side length

1

dm. There is a square

PQRS

on the diagonal plane

ABGH

with points

P

HG

and

Q

AH

as shown in the figure below. Point

T

is the center point of the square

PQRS

. The line

HT

is extended so that it intersects the diagonal line

BG

N

. Point

M

is the projection of

N

BC

. Determine the volume of the truncated prism

DCM.HGN

. https://cdn.artofproblemsolving.com/attachments/f/6/22c26f2c7c66293ad7065a3c8ce3ac2ffd938b.png
4. Nine pairs of husband and wife want to take pictures in a three-line position with the background of the Palembang Ampera Bridge. There are

4

people in the front row,

6

people in the middle row, and

8

people in the back row. They agreed that every married couple must be in the same row, and every two people next to each other must be a married couple or of the same sex. Specify the number of different possible arrangements of positions.
p5. p5. A hotel provides four types of rooms with capacity, rate, and number of rooms as presented in the following table.
type of room, capacity of persons/ room, day / rate (Rp.), / number of rooms https://cdn.artofproblemsolving.com/attachments/3/c/e9e1ed86887e692f9d66349a82eaaffc730b46.jpg A group of four families wanted to stay overnight at the hotel. Each family consists of husband and wife and their unmarried children. The number of family members by gender is presented in the following table.
family / man / woman/ total https://cdn.artofproblemsolving.com/attachments/4/6/5961b130c13723dc9fa4e34b43be30c31ee635.jpg The group leader enforces the following provisions. I. Each husband and wife must share a room and may not share a room with other married couples. II. Men and women may not share the same room unless they are from the same family. III. At least one room is occupied by all family representatives (“representative room”) IV. Each family occupies at most

3

types of rooms. V. No rooms are occupied by more than one family except representative rooms. You are asked to arrange a room for the group so that the total cost of lodging is as low as possible. Provide two possible alternative room arrangements for each family and determine the total cost.

Problem Statement