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1961 Leningrad Math Olympiad - Grade 8

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August 30, 2024
leningrad math olympiadalgebrageometrycombinatoricsnumber theory

Problem Statement

8.1 Construct a quadrilateral using side lengths and distances between the midpoints of the diagonals.
8.2 It is known that a,ba,b and a+b\sqrt{a}+\sqrt{b} are rational numbers. Prove that then a\sqrt{a}, b\sqrt{b} are rational.
8.3 / 9.2 Solve equation x3−[x]=3x^3 - [x]=3
8.4 Prove that if in a triangle the angle bisector of the vertex, bisects the angle between the median and the altitude, then the triangle either isosceles or right. .
8.5 Given nn numbers x1,x2,...,xnx_1, x_2, . . . , x_n, each of which is equal to +1+1 or −1-1. At the same time x1x2+x2x3+...+xn−1xn+xnx1=0.x_1x_2 + x_2x_3 + . . . + x_{n-1}x_n + x_nx_1 = 0 . Prove that nn is divisible by 44.
8.6 There are nn points marked on the circle, and it is known that for of any two, one of the arcs connecting them has a measure less than 1200120^0.Prove that all points lie on an arc of size 1200120^0.
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983442_1961_leningrad_math_olympiad]here.
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