MathDB

8

Part of 2022 IFYM, Sozopol

Problems(5)

Problem 8 of First round

Source: XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade

9/9/2022
A subset of the set A=1,2,…,nA={1,2,\dots ,n} is called connected, if it consists of one number or a certain amount of consecutive numbers. Find the greatest kk (defined as a function of nn) for which there exists kk different subsets A1,A2,…,AkA_1,A_2,…,A_k of AA the intersection of each two of which is a connected set.
set theory
Problem 8 of Second round

Source: XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade

9/9/2022
Let xx be a real number. Find the greatest possible value of the following expression:
47x43+43x47−2021x\frac{47^x}{\sqrt{43}}+\frac{43^x}{\sqrt{47}}-2021^x.
algebra
Problem 8 of Third round

Source: XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade

9/9/2022
Determine the number of ordered quadruples of integers (a,b,c,d)(a,b,c,d) for which 0≤a,b,c,d≤360\leq a,b,c,d\leq 36 and 37∣a2+b2−c3−d337|a^2+b^2-c^3-d^3.
number theorymodulo
magician with 15 hats

Source: IFYM - XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade, 4th round p8

11/12/2022
A magician wants to demonstrate the following trick to an audience of n≥16n \ge 16 people. He gives them 1515 hats and after giving instructions to his assistant (which the audience does not hear), leaves the hall. Some 1515 people in the audience put on one of the hats. The assistant tags in front of everyone, one of the hats with a marker and then the person with an unmarked hat takes it off. The magician then returns back to the hall and after surveying the situation, knows who in the audience has taken off his hat. For what nn is this possible?
[hide=original wording]Магьосник иска да покаже следния фокус пред публика от n≥16n \ge 16 души. Той им дава 1515 шапки и след като даде инструкции на помощника си (които публиката не чува), напуска залата. Някои 1515 души от публиката си слагат по една от шапките. Асистентът маркира пред всички една от шапките с маркер и след това човек с немаркирана шапка си я сваля. След това магьосникът се връща обратно в залата и след оглед на ситуацията познава кой от публиката си е свалил шапката. За кои nn е възможно това?
combinatoricsnumber theory
switching places of numbers in any permutation if their difference is p, or q

Source: IFYM - XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade,finals p8

11/13/2022
Let pp and qq be mutually prime natural numbers greater than 11. Starting with the permutation (1,2,...,n)(1, 2, . . . , n), in one move we can switch the places of two numbers if their difference is pp or qq. Prove that with such moves we can get any another permutation if and only if n≥p+q−1n \ge p + q - 1.
combinatoricspermutations