2

Part of 2020 Durer Math Competition Finals

Problems(2)

tank of enemy at one of the 13 x 13 fields

Source: 2020 Dürer Math Competition Finals E1.2

We are given a map divided into

13\times 13

fields. It is also known that at one of the fields a tank of the enemy is stationed, which we must destroy. To achieve this we need to hit it twice with shots aimed at the centre of some field. When the tank gets hit it gets moved to a neighbouring field out of precaution. At least how many shots must we fire, so that the tank gets destroyed certainly? We can neither see the tank, nor get any other feedback regarding its position.

equal in different bases 11001_? = 54001_{10}

Source: 2020 Dürer Math Competition Finals Day2 E2 https://artofproblemsolving.com/community/c1622639_2020_

What number should we put in place of the question mark such that the following statement becomes true?

11001_? = 54001_{10}

A number written in the subscript means which base the number is in.