MathDB

p1. It is known that

f

is a function such that

f(x)+2f\left(\frac{1}{x}\right)=3x

for every

x\ne 0

. Find the value of

x

that satisfies

f(x) = f(-x)

.
p2. It is known that ABC is an acute triangle whose vertices lie at circle centered at point

O

. Point

P

lies on side

BC

so that

AP

is the altitude of triangle ABC. If

\angle ABC + 30^o \le \angle ACB

, prove that

\angle COP + \angle CAB < 90^o

.
p3. Find all natural numbers

a, b

, and

c

that are greater than

1

and different, and fulfills the property that

abc

divides evenly

bc + ac + ab + 2

.
p4. Let

A, B

, and

P

be the nails planted on the board

ABP

. The length of

AP = a

units and

BP = b

units. The board

ABP

is placed on the paths

x_1x_2

and

y_1y_2

so that

A

only moves freely along path

x_1x_2

and only moves freely along the path

y_1y_2

as in following image. Let

x

be the distance from point

P

to the path

y_1y_2

and y is with respect to the path

x_1x_2

. Show that the equation for the path of the point

P

\frac{x^2}{b^2}+\frac{y^2}{a^2}=1

. https://cdn.artofproblemsolving.com/attachments/4/6/d88c337370e8c3bc5a1833bc9588d3fb047bd0.png
p5. There are three boxes

A, B

, and

C

each containing

3

colored white balls and

2

red balls. Next, take three ball with the following rules: 1. Step 1 Take one ball from box

A

. 2. Step 2

\bullet

If the ball drawn from box

A

in step 1 is white, then the ball is put into box

B

. Next from box

B

one ball is drawn, if it is a white ball, then the ball is put into box

C

, whereas if the one drawn is red ball, then the ball is put in box

A

\bullet

If the ball drawn from box

A

in step 1 is red, then the ball is put into box

C

. Next from box

C

one ball is taken. If what is drawn is a white ball then the ball is put into box

A

, whereas if the ball drawn is red, the ball is placed in box

B

. 3. Step 3 Take one ball each from squares

A, B

, and

C

. What is the probability that all the balls drawn in step 3 are colored red?

Problem Statement