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.
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