MathDB

1

Part of 2019 Tournament Of Towns

Problems(6)

sequence with sum 2020, no number and no sum of set of consecutives =3

Source: Tournament of Towns, Junior O-Level , Spring 2019 p1

5/9/2020
Consider a sequence of positive integers with total sum 2020 such that no number and no sum of a set of consecutive numbers is equal to 33. Is it possible for such a sequence to contain more than 1010 numbers?
(Alexandr Shapovalov)
combinatoricsSumconsecutiveTournament of TownsToT
2 wizards game, 7 positive integers with sum 100, 4th largest no spoken

Source: Tournament of Towns, Junior A-Level , Spring 2019 p1

5/9/2020
The King gives the following task to his two wizards. The First Wizard should choose 77 distinct positive integers with total sum 100100 and secretly submit them to the King. To the Second Wizard he should tell only the fourth largest number. The Second Wizard must figure out all the chosen numbers. Can the wizards succeed for sure? The wizards cannot discuss their strategy beforehand.
(Mikhail Evdokimov)
gamegame strategycombinatoricsTournament of TownsToT
regular hexagon side wanted given given 3 distances of point from 3 vertices

Source: Tournament of Towns, Senior O-Level , Spring 2019 p1

5/11/2020
The distances from a certain point inside a regular hexagon to three of its consecutive vertices are equal to 1,11, 1 and 22, respectively. Determine the length of this hexagon's side.
(Mikhail Evdokimov)
hexagondistancegeometry
Combinatorics

Source: International Mathematical Tournament of towns

12/2/2019
The magician puts out hardly a deck of 5252 cards and announces that 5151 of them will be thrown out of the table, and there will remain three of clubs. The viewer at each step says which count from the edge the card should be thrown out, and the magician chooses to count from the left or right edge, and ejects the corresponding card. At what initial positions of the three of clubs can the success of the focus be guaranteed?
combinatorics
complexity inequality, no of factors in prime decomposition

Source: Tournament of Towns, Junior A-Level , Fall 2019 p1

4/20/2020
Let us call the number of factors in the prime decomposition of an integer n>1n > 1 the complexity of nn. For example, complexity of numbers 44 and 66 is equal to 22. Find all nn such that all integers between nn and 2n2n have complexity a) not greater than the complexity of nn. b) less than the complexity of nn.
(Boris Frenkin)
inequalitiesnumber theoryprimenumber of divisorsprime factorizationfactorsDivisors
Degree of polynomial P(x,x)

Source: Tournament of towns

11/30/2019
The polynomial P(x,y) is such that for every integer n >= 0 each of the polynomials P(x,n) and P(n,y) either is a constant zero or has a degree not greater than n. Is it possible that P(x,x) has an odd degree?
algebrapolynomialtwo variables