MathDB

5

Part of 2019 Saint Petersburg Mathematical Olympiad

Problems(3)

2019 Saint Petersburg Grade 11 P5

Source: Saint Petersburg 2019

4/14/2019
Baron Munchhausen has a collection of stones, such that they are of 10001000 distinct whole weights, 210002^{1000} stones of every weight. Baron states that if one takes exactly one stone of every weight, then the weight of all these 10001000 stones chosen will be less than 210102^{1010}, and there is no other way to obtain this weight by picking another set of stones of the collection. Can this statement happen to be true?
(М. Антипов)
Thanks to the user Vlados021 for translating the problem.
combinatoricsnumber theory
candies for those who solve correctly problems

Source: St. Petersburg 2019 10.5

5/1/2019
A class has 2525 students. The teacher wants to stock NN candies, hold the Olympics and give away all NN candies for success in it (those who solve equally tasks should get equally, those who solve less get less, including, possibly, zero candies). At what smallest NN this will be possible, regardless of the number of tasks on Olympiad and the student successes?
combinatoricsminimum
improvement of a positive number its replacement by a power of two

Source: St. Petersburg 2019 9.5

5/2/2019
Call the improvement of a positive number its replacement by a power of two. (i.e. one of the numbers 1,2,4,8,...1, 2, 4, 8, ...), for which it increases, but not more than than 33 times. Given 21002^{100} positive numbers with a sum of 21002^{100}. Prove that you can erase some of them, and improve each of the other numbers so that the sum the resulting numbers were again 21002^{100}.
combinatoricsSumalgebranumber theory