MathDB

grade 7

Part of 1967 Leningrad Math Olympiad

Problems(1)

1967 Leningrad Math Olympiad - Grade 7

Source:

9/1/2024
7.1 Construct a trapezoid given four sides.
7.2 Prove that (1+x+x2+...+x100)(1+x102)−102x101≥0.(1 + x + x^2 + ...+ x^{100})(1 + x^{102}) - 102x^{101} \ge 0 .
7.3 In a quadrilateral ABCDABCD, MM is the midpoint of AB, NN is the midpoint of CDCD. Lines ADAD and BC intersect MNMN at points PP and QQ, respectively. Prove that if ∠BQM=∠APM\angle BQM = \angle APM , then BC=ADBC=AD. https://cdn.artofproblemsolving.com/attachments/a/2/1c3cbc62ee570a823b5f3f8d046da9fbb4b0f2.png
7.4 / 6.4 Each of the eight given different natural numbers less than 1616. Prove that among their pairwise differences there is at least at least three are the same.
7.5 / 8.4 An entire arc of circle is drawn through the vertices AA and CC of the rectangle ABCDABCD lying inside the rectangle. Draw a line parallel to ABAB intersecting BCBC at point PP, ADAD at point QQ, and the arc ACAC at point RR so that the sum of the areas of the figures AQRAQR and CPRCPR is the smallest. https://cdn.artofproblemsolving.com/attachments/1/4/9b5a594f82a96d7eff750e15ca6801a5fc0bf1.png
7.6 / 6.5 The distance AB is 100 km. From A and B , cyclists simultaneously ride towards each other at speeds of 20 km/h and 30 km/hour accordingly. Together with the first A, a fly flies out with speed 50 km/h, she flies until she meets the cyclist from B, after which she turns around and flies back until she meets the cyclist from A, after which turns around, etc. How many kilometers will the fly fly in the direction from A to B until the cyclists meet?
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988083_1967_leningrad_math_olympiad]here.
leningrad math olympiadalgebracombinatoricsnumber theorygeometry