MathDB

1

Part of 2016 Tournament Of Towns

Problems(3)

Donald and cards

Source: Tournament of Towns oral round 2016 p1

3/21/2016
On a blackboard the product log()[]×⋯×log()[]log_{( )}[ ]\times\dots\times log_{( )}[ ] is written (there are 50 logarithms in the product). Donald has 100100 cards: [2],[3],…,[51][2], [3],\dots, [51] and (52),…,(101)(52),\dots,(101). He is replacing each ()() with some card of form (x)(x) and each [][] with some card of form [y][y]. Find the difference between largest and smallest values Donald can achieve.
Tournament of Townsalgebralogarithm
Writing Numbers on a Tape

Source: Tournament of Towns Spring 2016

2/22/2017
All integers from 11 to one million are written on a tape in some arbitrary order. Then the tape is cut into pieces containing two consecutive digits each. Prove that these pieces contain all two-digit integers for sure, regardless of the initial order of integers.(4 points) Alexey Tolpygo
combinatorics
Children avoiding candies

Source: Tournament of Towns 2016 Fall Tour, A Senior, Problem #1

4/22/2017
100100 children stand in a line each having 100100 candies. In one move, one of them may take some of their candies and distribute them to a non-empty set of the remaining children. After what least number of moves can it happen that no two children have the same number of candies?
(N. Chernyatevya)
(Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.)
combinatorics