MathDB

8.1

x

and

y

are the roots of the equation

t^2-ct-c=0

. Prove that holds the inequality

x^3 + y^3 + (xy)^3 \ge 0.

8.2. Two circles touch internally at point

A

. Through a point

B

of the inner circle, different from

A

, a tangent to this circle intersecting the outer circle at points C and

D

. Prove that

AB

is a bisector of angle

CAD

. https://cdn.artofproblemsolving.com/attachments/2/8/3bab4b5c57639f24a6fd737f2386a5e05e6bc7.png
8.3 Prove that

2^{3^{100}} + 1

is divisible by

3^{101}

.
8.4 / 7.5 An entire arc of circle is drawn through the vertices

A

and

C

of the rectangle

ABCD

lying inside the rectangle. Draw a line parallel to

AB

intersecting

BC

at point

P

AD

at point

Q

, and the arc

AC

at point

R

so that the sum of the areas of the figures

AQR

and

CPR

is the smallest. https://cdn.artofproblemsolving.com/attachments/1/4/9b5a594f82a96d7eff750e15ca6801a5fc0bf1.png
8.5 In a certain group of people, everyone has one enemy and one Friend. Prove that these people can be divided into two companies so that in every company there will be neither enemies nor friends.
8.6 Numbers

a_1, a_2, . . . , a_{100}

are such that

a_1 - 2a_2 + a_3 \le 0

a_2-2a_3 + a_ 4 \le 0

...

a_{98}-2a_{99 }+ a_{100} \le 0

and at the same time

a_1 = a_{100}\ge 0

. Prove that all these numbers are non-negative.
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988083_1967_leningrad_math_olympiad]here.

Problem Statement