MathDB

6

Part of 2016 Tournament Of Towns

Problems(4)

Center of disk

Source: ToT 2016 Junior A

2/22/2017
Q. An automatic cleaner of the disc shape has passed along a plain floor. For each point of its circular boundary there exists a straight line that has contained this point all the time. Is it necessarily true that the center of the disc stayed on some straight line all the time? (99 marks)
geometryToT
Word and Palindromes

Source: Tournament of Towns Spring 2016

2/22/2017
Recall that a palindrome is a word which is the same when we read it forward or backward.
(a) We have an infinite number of cards with words {abc,bca,cab}\{abc, bca, cab\}. A word is made from them in the following way. The initial word is an arbitrary card. At each step we obtain a new word either gluing a card (from the right or from the left) to the existing word or making a cut between any two of its letters and gluing a card between both parts. Is it possible to obtain a palindrome this way? (4 points)
(b) We have an infinite number of red cards with words {abc,bca,cab}\{abc, bca, cab\} and of blue cards with words {cba,acb,bac}\{cba, acb, bac\}. A palindrome was formed from them in the same way as in part (a). Is it necessarily true that the number of red and blue cards used was equal? (6 points)
Alexandr Gribalko, Ivan Mitrofanov
combinatorics
Numbers on blackboard

Source: Tournament of Towns 2016 oral round p6

3/21/2016
NN different numbers are written on blackboard and one of these numbers is equal to 00.One may take any polynomial such that each of its coefficients is equal to one of written numbers ( there may be some equal coefficients ) and write all its roots on blackboard.After some of these operations all integers between āˆ’2016-2016 and 20162016 were written on blackboard(and some other numbers maybe). Find the smallest possible value of NN .
Polynomialsnumber theoryalgebra
Game of Polynomials

Source: Tournament of Towns 2016 Fall Tour, A Senior, Problem #6

4/22/2017
Petya and Vasya play the following game. Petya conceives a polynomial P(x)P(x) having integer coefficients. On each move, Vasya pays him a ruble, and calls an integer aa of his choice, which has not yet been called by him. Petya has to reply with the number of distinct integer solutions of the equation P(x)=aP(x)=a. The game continues until Petya is forced to repeat an answer. What minimal amount of rubles must Vasya pay in order to win?
(Anant Mudgal)
(Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.)
algebrapolynomialnumber theoryGame TheorycombinatoricsCombinatorial games