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grade 7

Part of 1969 Leningrad Math Olympiad

Problems(1)

1969 Leningrad Math Olympiad - Grade 7

Source:

9/1/2024
7.1 / 6.1 There are 88 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 ABCABC are extended as shown in the figure. At this AA′=3ABAA' = 3 AB,, BB′=5BCBB' = 5BC , CC′=8CACC'= 8 CA. How many times is the area of the triangle ABCABC less than the area of the triangle A′B′C′A'B'C' ? https://cdn.artofproblemsolving.com/attachments/9/f/06795292291cd234bf2469e8311f55897552f6.png
[url=https://artofproblemsolving.com/community/c893771h1860178p12579333]7.3 Prove the equality 2x2−1+4x2−4+6x2−9+...+20x2−100=11(x−1)(x+10)+11(x−2)(x+9)+...+11(x−10)(x+1)\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 44 villages located in the peaks of square with side 1010 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.
leningrad math olympiadgeometryalgebranumber theorycombinatorics