1965 Leningrad Math Olympiad - Grade 7
Source:
August 30, 2024
leningrad math olympiadalgebrageometrycombinatoricsnumber theory
Problem Statement
7.1 Prove that a natural number with an odd number of divisors is a perfect square.
7.2 In a triangle with area , medians and are drawn, intersecting at the point . Find the area of the quadrilateral .
https://cdn.artofproblemsolving.com/attachments/0/f/9cd32bef4f4459dc2f8f736f7cc9ca07e57d05.png7.3 . The front tires of a car wear out after kilometers, and the rear tires after kilometers. When you need to swap tires so that the car can travel the longest possible distance with the same tires?
7.4 A rectangle is divided by lines parallel to it sides, into unit squares. How many parts will this rectangle be divided into if you also draw a diagonal in it?
7.5 / 8.4 Let denote the largest integer not greater than . Solve the equation: .
7.6 Black paint was sprayed onto a white surface. Prove that there are two points of the same color, the distance between which is meters.
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988081_1965_leningrad_math_olympiad]here.