MathDB

p1. If

x + y + z = 2

, show that

\frac{1}{xy+z-1}+\frac{1}{yz+x-1}+\frac{1}{xz+y-1}=\frac{-1}{(x-1)(y-1)(z-1)}

.
p2. Determine the simplest form of

\frac{3}{1!+2!+3!}+\frac{4}{2!+3!+4!}+\frac{5}{3!+4!+5!}+...+\frac{100}{98!+99!+100!}

p3. It is known that

ABCD

and

DEFG

are two parallelograms. Point

E

lies on

AB

and point

C

lie on

FG

. The area of

​​ABCD

20

units.

H

is the point on

DG

so that

EH

is perpendicular to

DG

. If the length of

DG

5

units, determine the length of

EH

. https://cdn.artofproblemsolving.com/attachments/b/e/42453bf6768129ed84fbdc81ab7235e780b0e1.png
p4. Each room in the following picture will be painted so that every two rooms which is directly connected to the door is given a different color. If

10

different colors are provided and

4

of them can not be used close together for two rooms that are directly connected with a door, determine how many different ways to color the

4

rooms. https://cdn.artofproblemsolving.com/attachments/4/a/e80a464a282b3fe3cdadde832b2fd38b51a41a.png
5. The floor of a hall is rectangular

ABCD

with

AB = 30

meters and

BC = 15

meters. A cat is in position

A

. Seeing the cat, the mouse in the midpoint of

AB

ran and tried to escape from cat. The mouse runs from its place to point

C

at a speed of

3

meters/second. The trajectory is a straight line. Watching the mice run away, at the same time from point

A

the cat is chasing with a speed of

5

meters/second. If the cat's path is also a straight line and the mouse caught before in

C

, determine an equation that can be used for determine the position and time the mouse was caught by the cat.

Problem Statement