MathDB

Problem 1

Part of 2022 Kyiv City MO Round 1

Problems(5)

Present fraction as a difference of smaller fractions

Source: Kyiv City MO 2022 Round 1, Problem 7.1

1/23/2022
Represent 12021\frac{1}{2021} as a difference of two irreducible fractions with smaller denominators.
(Proposed by Bogdan Rublov)
constructionFractionalgebra
Relating mean and gcd

Source: Kyiv City MO 2022 Round 1, Problem 8.1

1/23/2022
Consider 55 distinct positive integers. Can their mean be
a)Exactly 33 times larger than their largest common divisor?
b)Exactly 22 times larger than their largest common divisor?
number theoryalgebragreatest common divisor
Easiest algebra ever

Source: Kyiv City MO 2022 Round 1, Problem 9.1

1/23/2022
What's the smallest possible value of (x+y+∣x−y∣)2xy\frac{(x+y+|x-y|)^2}{xy} over positive real numbers x,yx, y?
algebrainequalities
Fractions everywhere

Source: Kyiv City MO 2022 Round 1, Problem 10.1

1/23/2022
Does there exist a quadratic trinomial ax2+bx+cax^2 + bx + c such that a,b,ca, b, c are odd integers, and 12022\frac{1}{2022} is one of its roots?
algebraquadratic trinomial
Same multisets of sums

Source: Kyiv City MO 2022 Round 1, Problem 11.1

1/23/2022
The teacher wrote 55 distinct real numbers on the board. After this, Petryk calculated the sums of each pair of these numbers and wrote them on the left part of the board, and Vasyl calculated the sums of each triple of these numbers and wrote them on the left part of the board (each of them wrote 1010 numbers). Could the multisets of numbers written by Petryk and Vasyl be identical?
algebra