MathDB

7.1 / 6.1 There are

8

rooks on the chessboard such that no two of them they don't hit each other. Prove that the black squares contain an even number of rooks.
7.2 The sides of triangle

ABC

are extended as shown in the figure. At this

AA' = 3 AB

BB' = 5BC

CC'= 8 CA

. How many times is the area of the triangle

ABC

less than the area of the triangle

A'B'C'

? https://cdn.artofproblemsolving.com/attachments/9/f/06795292291cd234bf2469e8311f55897552f6.png
[url=https://artofproblemsolving.com/community/c893771h1860178p12579333]7.3 Prove the equality

\frac{2}{x^2-1}+\frac{4}{x^2-4} +\frac{6}{x^2-9}+...+\frac{20}{x^2-100} =\frac{11}{(x-1)(x+10)}+\frac{11}{(x-2)(x+9)}+...+\frac{11}{(x-10)(x+1)}

[url=https://artofproblemsolving.com/community/c893771h1861966p12597273]7.4* / 8.4 * (asterisk problems in separate posts)
7.5 . The collective farm consists of

4

villages located in the peaks of square with side

10

km. It has the means to conctruct 28 kilometers of roads . Can a collective farm build such a road system so that was it possible to get from any village to any other?
7.6 / 6.6 Two brilliant mathematicians were told in natural terms number and were told that these numbers differ by one. After that they take turns asking each other the same question: “Do you know my number?" Prove that sooner or later one of them will answer positively.
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988085_1969_leningrad_math_olympiad]here.

Problem Statement