7
Part of 2016 Tournament Of Towns
Problems(3)
Minimizing connections of batteries
Source: Tournament of the Towns
12/11/2016
a.) There are () batteries. We don't know which batteries are good and which are bad but we know that the number of good batteries is greater by than the number of bad batteries. A lamp uses two batteries, and it works only if both of them are good. What is the least number of attempts sufficient to make the lamp work?b.) The same problem but the total number of batteries is () and the numbers of good and bad batteries are equal.Proposed by Alexander Shapovalov
combinatoricsgraph theoryalgorithms
Trains on a Planet
Source: Tournament of Towns Spring 2016
2/22/2017
A spherical planet has the equator of length . On this planet, circular roads of length each are to be built and used for several trains each. The trains must have the same constant positive speed and never stop or collide. What is the greatest possible sum of lengths of all the trains? The trains are arcs of zero width with endpoints removed (so that if only endpoints of two arcs have coincided then it is not a collision). Solve the problem for :
(a) (4 points)
(b) (6 points)Alexandr Berdnikov
combinatorics
Equal number of left and right jumps
Source: Tournament of Towns 2016 Fall Tour, A Senior, Problem #7
4/22/2017
Several frogs are sitting on the real line at distinct integer points. In each move, one of them can take a -jump towards the right as long as they are still in on distinct points. We calculate the number of ways they can make moves in this way for a positive integer . Prove that if the jumps were all towards the left, we will still get the same number of ways.(F. Petrov)(Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.)
combinatorics