MathDB

p1. Given

N = 9 + 99 + 999 + ... +\underbrace{\hbox{9999...9}}_{\hbox{121\,\,numbers}}

. Determine the value of N.
p2. The triangle

ABC

in the following picture is isosceles, with

AB = AC =90

cm and

BC = 108

cm. The points

P

and

Q

are located on

BC

, respectively such that

BP: PQ: QC = 1: 2: 1

. Points

S

and

R

are the midpoints of

AB

and

AC

respectively. From these two points draw a line perpendicular to

PR

so that it intersects at

PR

at points

M

and

N

respectively. Determine the length of

MN

. https://cdn.artofproblemsolving.com/attachments/7/1/e1d1c4e6f067df7efb69af264f5c8de5061a56.png
p3. If eight equilateral triangles with side

12

cm are arranged as shown in the picture on the side, we get a octahedral net. Define the volume of the octahedron. https://cdn.artofproblemsolving.com/attachments/4/8/18cdb8b15aaf4d92f9732880784facf9348a84.png
p4. It is known that

a^2 + b^2 = 1

and

x^2 + y^2 = 1

. Continue with the following algebraic process.

(a^2 + b^2)(x^2 + y^2) – (ax + by)^2 = ...

a. What relationship can be concluded between

ax + by

and

1

? b. Why?
p5. A set of questions consists of

3

questions with a choice of answers True (

T

) or False (

F

), as well as

3

multiple choice questions with answers

A, B, C

, or

D

. Someone answer all questions randomly. What is the probability that he is correct in only

2

questions?

Problem Statement