6

Part of 2019 Tournament Of Towns

Problems(4)

Source:

Given is a isosceles triangle ABC so that AB=BC. Point K is in ABC, so that CK=AB=BC and

card game with product of 5 variables from x_1,..., x_{10}

Source: Tournament of Towns, Senior A-Level , Spring 2019 p6

For each five distinct variables from the set

x_1,..., x_{10}

there is a single card on which their product is written. Peter and Basil play the following game. At each move, each player chooses a card, starting with Peter. When all cards have been taken, Basil assigns values to the variables as he wants, so that

0 \le x_1 \le ... \le x_10

. Can Basil ensure that the sum of the products on his cards is greater than the sum of the products on Peter's cards?
(Ilya Bogdanov)

combinatoricsProduct

100 ruble notes for buying books

Source: Tournament of Towns, Junior A-Level , Fall 2019 p6

Peter has several

100

ruble notes and no other money. He starts buying books; each book costs a positive integer number of rubles, and he gets change in

1

ruble coins. Whenever Peter is buying an expensive book for

100

rubles or higher he uses only

100

ruble notes in the minimum necessary number. Whenever he is buying a cheap one (for less than

100

rubles) he uses his coins if he has enough, otherwise using a

100

ruble note. When the

100

ruble notes have come to the end, Peter has expended exactly a half of his money. Is it possible that he has expended

5000

rubles or more?
(Tatiana Kazitsina)

each unit cube is pierced by at least one needle, (2N)^3 unit cubes in a cube

Source: Tournament of Towns, Senior A-Level , Fall 2019 p6

A cube consisting of

(2N)^3

unit cubes is pierced by several needles parallel to the edges of the cube (each needle pierces exactly

2N

unit cubes). Each unit cube is pierced by at least one needle. Let us call any subset of these needles “regular” if there are no two needles in this subset that pierce the same unit cube. a) Prove that there exists a regular subset consisting of

2N^2

needles such that all of them have either the same direction or two different directions. b) What is the maximum size of a regular subset that does exist for sure?
(Nikita Gladkov, Alexandr Zimin)

combinatoricscombinatorial geometrygeometry3D geometry