MathDB

P3

Part of 2019 Dürer Math Competition (First Round)

Problems(2)

Game on 3X3 table

Source: 12th Dürer Math Competition ,First Round, Category E , P3

8/18/2020
a) We are playing the following game on this table: In each move we select a row or a column of the table, reduce two neighboring numbers in that row or column by 11 and increase the third one by 11. After some of these moves can we get to a table with all the same entries?
b) This time we have the choice to arrange the integers from 11 to 99 in the3×3 3 \times3 table. Still using the same moves now our aim is to create a table with all the same entries, maximising the value of the entries. What is the highest possible number we can achieve?
tablecombinatorics
Anne and bob play a game on R^2

Source: 12th Dürer Math Competition ,First Round, Category E+, P3

8/18/2020
Anne has thought of a finite set A⊆R2A \subseteq \mathbb{R}^2 . Bob does not know how many elements AA has, but his goal is to completely determine AA.
To achieve this, Bob can chooseany point b∈R2b \in \mathbb{R}^2 and ask Anne how far it is fromA A . Anne replies with the distance, defined as min{d(a,b)∣a∈A}min \{d(a, b) | a \in A\}. (Here d(a,b)d(a, b) denotes the distance between points a,b∈Ra, b \in \mathbb{R} .)
Bob can ask as many questions of this type as he wants, until he can determine A with certainty. a) Can Bob achieve his goal with finitely many questions? b) What if Anne tells Bob in advance that all points of A have both coordinates in the interval [0,1] \ [0, 1]\ ? Note: R2\mathbb{R}^2 is the set of points in the plane.
combinatoricsPlane