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grade 8

Part of 1966 Leningrad Math Olympiad

Problems(1)

1966 Leningrad Math Olympiad - Grade 8

Source:

9/1/2024
8.1 / 7.4 What number needs to be put in place * so that the next the problem had a unique solution: “There are n straight lines on the plane, intersecting at * points. Find n.” ?
8.2 / 7.3 Prove that for any natural number nn the number n(2n+1)(3n+1)...(1966n+1) n(2n+1)(3n+1)...(1966n + 1) is divisible by every prime number less than 19661966.
8.3 / 7.6 There are nn points on the plane so that any triangle with vertices at these points has an area less than 11. Prove that all these points can be enclosed in a triangle of area 44.
8.4 Prove that the sum of all divisors of the number n2n^2 is odd.
8.5 A quadrilateral has three obtuse angles. Prove that the larger of its two diagonals emerges from the vertex of an acute angle.
8.6 Numbers x1,x2,...x_1, x_2, . . . are constructed according to the following rule: x1=2,x2=(x15+1)/5x1,x3=(x25+1)/5x2,...x_1 = 2, x_2 = (x^5_1 + 1)/5x_1, x_3 = (x^5_2 + 1)/5x_2, ... Prove that no matter how much we continued this construction, all the resulting numbers will be no less 1/51/5 and no more than 22.
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988082_1966_leningrad_math_olympiad]here.
leningrad math olympiadalgebrageometrycombinatoricsnumber theory