MathDB

Problem 1

Part of 2024 Kyiv City MO Round 1

Problems(5)

Square cutting

Source: Kyiv City MO 2024 Round 1, Problem 7.1

1/28/2024
Square ABCDABCD is cut by a line segment EFEF into two rectangles AEFDAEFD and BCFEBCFE. The lengths of the sides of each of these rectangles are positive integers. It is known that the area of the rectangle AEFDAEFD is 3030 and it is larger than the area of the rectangle BCFEBCFE. Find the area of square ABCDABCD.
Proposed by Bogdan Rublov
geometryrectangle
Sum and product equal to 8

Source: Kyiv City MO 2024 Round 1, Problem 8.1

1/28/2024
Find the number of positive integers for which the product of digits and the sum of digits are the same and equal to 88.
number theoryDigits
Irreducible difference

Source: Kyiv City MO 2024 Round 1, Problem 9.1

1/28/2024
The difference of fractions 20242023−20232024\frac{2024}{2023} - \frac{2023}{2024} was represented as an irreducible fraction pq\frac{p}{q}. Find the value of pp.
number theoryalgebra
Strange divisibility

Source: Kyiv City MO 2024 Round 1, Problem 10.1

1/28/2024
Find all pairs of positive integers (a,b)(a, b) such that 4b−14b - 1 is divisible by 3a+13a + 1 and 3a−13a - 1 is divisible by 2b+12b + 1.
number theoryalgebraDivisibility
Twice meaner

Source: Kyiv City MO 2024 Round 1, Problem 11.1

1/28/2024
Four positive integers a,b,c,da, b, c, d satisfy the condition: a<b<c<da < b < c < d. For what smallest possible value of dd could the following condition be true: the arithmetic mean of numbers a,b,ca, b, c is twice smaller than the arithmetic mean of numbers a,b,c,da, b, c, d?
algebramean