MathDB

P6. It is given the integer

Y

with

Y = 2018 + 20118 + 201018 + 2010018 + \cdots + 201 \underbrace{00 \ldots 0}_{\textrm{100 digits}} 18.

Determine the sum of all the digits of such

Y

. (It is implied that

Y

is written with a decimal representation.)
P7. Three groups of lines divides a plane into

D

regions. Every pair of lines in the same group are parallel. Let

x, y

and

z

respectively be the number of lines in groups 1, 2, and 3. If no lines in group 3 go through the intersection of any two lines (in groups 1 and 2, of course), then the least number of lines required in order to have more than 2018 regions is ....
P8. It is known a frustum

ABCD.EFGH

where

ABCD

and

EFGH

are squares with both planes being parallel. The length of the sides of

ABCD

and

EFGH

respectively are

6a

and

3a

, and the height of the frustum is

3t

. Points

M

and

N

respectively are intersections of the diagonals of

ABCD

and

EFGH

and the line

MN

is perpendicular to the plane

EFGH

. Construct the pyramids

M.EFGH

and

N.ABCD

and calculate the volume of the 3D figure which is the intersection of pyramids

N.ABCD

and

M.EFGH

.
P9. Look at the arrangement of natural numbers in the following table. The position of the numbers is determined by their row and column numbers, and its diagonal (which, the sequence of numbers is read from the bottom left to the top right). As an example, the number

19

is on the 3rd row, 4th column, and on the 6th diagonal. Meanwhile the position of the number

26

is on the 3rd row, 5th column, and 7th diagonal.
(Image should be placed here, look at attachment.)
a) Determine the position of the number

2018

based on its row, column, and diagonal. b) Determine the average of the sequence of numbers whose position is on the "main diagonal" (quotation marks not there in the first place), which is the sequence of numbers read from the top left to the bottom right: 1, 5, 13, 25, ..., which the last term is the largest number that is less than or equal to

2018

.
P10. It is known that

A

is the set of 3-digit integers not containing the digit

0

. Define a gadang number to be the element of

A

whose digits are all distinct and the digits contained in such number are not prime, and (a gadang number leaves a remainder of 5 when divided by 7. If we pick an element of

A

at random, what is the probability that the number we picked is a gadang number?

Problem Statement