MathDB

p1. Circle

M

is the incircle of ABC, while circle

N

is the incircle of

ACD

. Circles

M

and

N

are tangent at point

E

. If side length

AD = x

cm,

AB = y

cm,

BC = z

cm, find the length of side

DC

(in terms of

x, y

, and

z

). https://cdn.artofproblemsolving.com/attachments/d/5/66ddc8a27e20e5a3b27ab24ff1eba3abee49a6.png
p2. The address of the house on Jalan Bahagia will be numbered with the following rules:

\bullet

One side of the road is numbered with consecutive even numbers starting from number

2

\bullet

The opposite side is numbered with an odd number starting from number

3

\bullet

In a row of even numbered houses, there is some land vacant house that has not been built.

\bullet

The first house numbered

2

has a neighbor next door. When the RT management ordered the numbers of the house, it is known that the cost of making each digit is

12.000

Rp. For that, the total cost to be incurred is

1.020.000

Rp. It is also known that the cost of all even-sided house numbers is

132.000

Rp. cheaper than the odd side. When the land is empty later a house has been built, the number of houses on the even and odd sides is the same. Determine the number of houses that are now on Jalan Bahagia .
p3. Given the following problem: Each element in the set

A = \{10, 11, 12,...,2008\}

multiplied by each element in the set

B = \{21, 22, 23,...,99\}

. The results are then added together to give value of

X

. Determine the value of

X

. Someone answers the question by multiplying

2016991

with

4740

. How can you explain that how does that person make sense?
p4. Let

P

be the set of all positive integers between

0

and

2008

which can be expressed as the sum of two or more consecutive positive integers . (For example:

11 = 5 + 6

90 = 29 + 30 + 31

100 = 18 + 19 +20 + 21 + 22

. So

11, 90, 100

are some members of

P

.) Find the sum of of all members of

P

.
p5. A four-digit number will be formed from the numbers at

0, 1, 2, 3, 4, 5

provided that the numbers in the number are not repeated, and the number formed is a multiple of

3

. What is the probability that the number formed has a value less than

3000

Problem Statement