MathDB

Problem 4

Part of 2022 Kyiv City MO Round 1

Problems(5)

Magic country and businessman Victor

Source: Kyiv City MO 2022 Round 1, Problem 7.4

1/23/2022
In some magic country, there are banknotes only of values 33, 2525, 8080 hryvnyas. Businessman Victor ate in one restaurant of this country for 20242024 days in a row, and each day (except the first) he spent exactly 11 hryvnya more than the day before (without any change). Could he have spent exactly 10000001000000 banknotes?
(Proposed by Oleksii Masalitin)
number theory
Sums not divisible by differences

Source: Kyiv City MO 2022 Round 1, Problem 8.4

1/23/2022
What's the largest number of integers from 11 to 20222022 that you can choose so that no sum of any two different chosen integers is divisible by any difference of two different chosen integers?
(Proposed by Oleksii Masalitin)
number theory
Selecting divisors of square-free integers

Source: Kyiv City MO 2022 Round 1, Problem 9.4

1/23/2022
Let's call integer square-free if it's not divisible by p2p^2 for any prime pp. You are given a square-free integer n>1n>1, which has exactly dd positive divisors. Find the largest number of its divisors that you can choose, such that a2+ab−na^2 + ab - n isn't a square of an integer for any a,ba, b among chosen divisors.
(Proposed by Oleksii Masalitin)
number theory
Nobody solved this inequality!

Source: Kyiv City MO 2022 Round 1, Problem 10.4

1/23/2022
For any nonnegative reals x,yx, y show the inequality x2y2+x2y+xy2≤x4y+x+y4x^2y^2 + x^2y + xy^2 \le x^4y + x + y^4.
inequalities
Pairwise products form arithmetic progression

Source: Kyiv City MO 2022 Round 1, Problem 11.4

1/23/2022
You are given n≥4n\ge 4 positive real numbers. It turned out that all n(n−1)2\frac{n(n-1)}{2} of their pairwise products form an arithmetic progression in some order. Show that all given numbers are equal.
(Proposed by Anton Trygub)
algebraArithmetic Progressionarithmetic sequence