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1968 Leningrad Math Olympiad - Grade 6

Source:

9/1/2024
6.1 The student bought a briefcase, a fountain pen and a book. If the briefcase cost 5 times cheaper, the fountain pen was 2 times cheaper, and the book was 2 1/2 times cheaper cheaper, then the entire purchase would cost 2 rubles. If the briefcase was worth 2 times cheaper, a fountain pen is 4 times cheaper, and a book is 3 times cheaper, then the whole the purchase would cost 3 rubles. How much does it really cost? ´
6.2. Which number is greater: 888...8819digits333...3368digitsor444...4419digits666...6768digits?\underbrace{888...88}_{19 \, digits} \cdot \underbrace{333...33}_{68 \, digits} \,\,\, or \,\,\, \underbrace{444...44}_{19 \, digits} \cdot \underbrace{666...67}_{68 \, digits} \, ?
6.3 Distance between Luga and Volkhov 194 km, between Volkhov and Lodeynoye Pole 116 km, between Lodeynoye Pole and Pskov 451 km, between Pskov and Luga 141 km. What is the distance between Pskov and Volkhov?
6.4 There are 44 objects in pairs of different weights. How to use a pan scale without weights Using five weighings, arrange all these objects in order of increasing weights?
6.5 . Several teams took part in the volleyball tournament. Team A is considered stronger than team B if either A beat B or there is a team C such that A beat C, and C beat B. Prove that if team T is the winner of the tournament, then it is the strongest the rest of the teams.
6.6 In task 6.1, determine what is more expensive: a briefcase or a fountain pen.
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988084_1968_leningrad_math_olympiad]here.
leningrad math olympiadalgebrageometrycombinatoricsnumber theory