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grade 8

Part of 1962 Leningrad Math Olympiad

Problems(1)

1962 Leningrad Math Olympiad - Grade 8

Source:

9/1/2024
8.1 Four circles are placed on planes so that each one touches the other two externally. Prove that the points of tangency lie on one circle. https://cdn.artofproblemsolving.com/attachments/9/8/883a82fb568954b09a4499a955372e2492dbb8.png
8.2. Let the integers aa and bb be represented as x2−5y2x^2-5y^2, where xx and yy are integer numbers. Prove that the number abab can also be presented in this form.
8.3 Solve the equation x(x+d)(x+2d)(x+3d)=ax(x + d)(x + 2d)(x + 3d) = a.
8.4 / 9.1 Let a+b+c=1a+b+c=1, m+n+p=1m+n+p=1 . Prove that −1≤am+bn+cp≤1-1 \le am + bn + cp \le 1
8.5 Inscribe a triangle with the largest area in a semicircle.
8.6 Three circles of the same radius intersect at one point. Prove that the other three points intersections lie on a circle of the same radius. https://cdn.artofproblemsolving.com/attachments/4/7/014952f2dcf0349d54b07230e45a42c242a49d.png
8.7 Find the circle of smallest radius that contains a given triangle.
8.8 / 9.2 Given a polynomial x2n+a1x2n−2+a2x2n−4+...+an−1x2+an,x^{2n} +a_1x^{2n-2} + a_2x^{2n-4} + ... + a_{n-1}x^2 + a_n, which is divisible by x−1 x-1. Prove that it is divisible by x2−1x^2-1.
8.9 Prove that for any prime number pp other than 22 and from 55, there is a natural number kk such that only ones are involved in the decimal notation of the number pkpk..
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983459_1962_leningrad_math_olympiad]here.
leningrad math olympiadalgebrageometrycombinatoricsnumber theory