MathDB

1

Part of 2020 Tournament Of Towns

Problems(5)

TOT Problem 1 Spring O level

Source:

6/3/2020
The Quadrumland map is a 6 × 6 square where each square cell is either a kingdom or a disputed territory. There are 27 kingdoms and 9 disputed territories. Each disputed territory is claimed by those and only those kingdoms that are neighbouring with it (adjacent by an edge or a vertex). Is it possible that for each disputed territory the numbers of claims are different?
You can discuss your solutions here
ToT
positive integer divisible by 2020 with equal numbers of digits 0, 1, 2,...,9

Source: Tournament of Towns, Junior A-Level Paper, Spring 2020 , p1

6/10/2020
Does there exist a positive integer that is divisible by 20202020 and has equal numbers of digits 0,1,2,...,90, 1, 2, . . . , 9 ?
Mikhail Evdokimov
Digitsnumber theorydivisible
fill a 40 x 41 table with integers

Source: Tournament of Towns, Senior O-Level Paper, Spring 2020 , p1

6/4/2020
Is it possible to fill a 40×4140 \times 41 table with integers so that each integer equals the number of adjacent (by an edge) cells with the same integer?
Alexandr Gribalko
combinatoricsnumbers in a tabletable
line segment of length greater than 10^6 between y = x^2 and y = x^2 - 1

Source: Tournament of Towns, Senior A-Level Paper, Spring 2020 , p1

6/3/2020
Consider two parabolas y=x2y = x^2 and y=x2−1y = x^2 - 1. Let UU be the set of points between the parabolas (including the points on the parabolas themselves). Does UU contain a line segment of length greater than 10610^6 ?
Alexey Tolpygo
parabolaSegmentanalytic geometryalgebra
sequence, each is divisible by previous and by the sum of 2 previous numbers

Source: Tournament of Towns 2020 oral p1 (15 March 2020)

5/18/2020
20202020 positive integers are written in one line. Each of them starting with the third is divisible by previous and by the sum of two previous numbers. What is the smallest value the last number can take?
A. Gribalko
Sequencenumber theoryDivisibility