MathDB

8.1 On the median drawn from the vertex of the triangle to the base, point

A

is taken. The sum of the distances from

A

to the sides of the triangle is equal to

s

. Find the distances from

A

to the sides if the lengths of the sides are equal to

x

and

y

.
8.2 Fraction

0, abc...

is composed according to the following rule:

a

and

c

are arbitrary digits, and each next digit is equal to the remainder of the sum of the previous two digits when divided by

10

. Prove that this fraction is purely periodic.
8.3 Two convex polygons with

m

and

n

sides are drawn on the plane (

m>n

). What is the greatest possible number of parts, they can break the plane?
8.4 The sum of three integers that are perfect squares is divisible by

9

. Prove that among them, there are two numbers whose difference is divisible by

9

.
8.5 / 9.5 Given

k+2

integers. Prove that among them there are two integers such that either their sum or their difference is divisible by

2k

.
8.6 A right angle rotates around its vertex. Find the locus of the midpoints of the segments connecting the intersection points sides of an angle and a given circle.
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983460_1963_leningrad_math_olympiad]here.

Problem Statement