4
Part of 2016 Tournament Of Towns
Problems(4)
Painting a Cube
Source: Tournament of Towns Spring 2016 Junior A-Level
2/23/2017
A designer took a wooden cube , divided each face into unit squares and painted each square black, white or red so that any two squares with a common side have different colours. What is the least possible number of black squares? (Squares with a common side may belong to the same face of the cube or to two different faces.)
(8 points)Mikhail Evdokimov
combinatoricsgeometry3D geometry
Optimally determining connectivity of a graph.
Source: Tournament of the Towns 2016, Spring A level, Senior.
9/30/2016
There are towns in a country and some pairs of towns are connected by roads but we do not know these pairs. We may choose any pair of towns and find out whether they are connected or not. Our aim is to determine whether it is possible to travel from any town to any other by a sequence of roads. Prove that there is no algorithm which enables us to do so in less than questions. (Proposed by Konstantin Knop)
combinatoricsalgorithmConnected graphsgraph theory
Tennis players meeting
Source: Tournament of Towns oral round p4
3/21/2016
30 masters and 30 juniors came onto tennis players meeting .Each master played with one master and 15 juniors while each junior played with one junior and 15 masters.Prove that one can find two masters and two juniors such that these masters played with each other ,juniors -with each other ,each of two masters played with at least one of two juniors and each of two juniors played with at least one of two masters.
graph theorycombinatorics
Distinguishing sum-set from product-set
Source: Tournament of Towns 2016 Fall Tour, A Senior, Problem #4
4/22/2017
There are red and blue cards each having a number written on it. For some distinct positive real numbers, it is known that the set of numbers on cards of a particular color happens to be the set of their pairwise sums and the other happens to be the set of their pairwise products. Can we necessarily determine which color corresponds to sum and which to product?(B. Frenkin)(Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.)
algebra