MathDB

Problem 2

Part of 2024 Kyiv City MO Round 1

Problems(4)

Parity disparity

Source: Kyiv City MO 2024 Round 1, Problem 7.2

1/28/2024
Is it possible to write the numbers from 11 to 100100 in the cells of a of a 10×1010 \times 10 square so that: 1. Each cell contains exactly one number; 2. Each number is written exactly once; 3. For any two cells that are symmetrical with respect to any of the perpendicular bisectors of sides of the original 10×1010 \times 10 square, the numbers in them must have the same parity.
The figure below shows examples of such pairs of cells, in which the numbers must have the same parity.
https://i.ibb.co/b3P8t36/Kyiv-MO-2024-7-2.png
Proposed by Mykhailo Shtandenko
arrangingsquareParitycombinatorics
Intersection of bisector and altitude

Source: Kyiv City MO 2024 Round 1, Problem 9.2/10.2

1/28/2024
Let BL,ADBL, AD be the bisector and the altitude correspondingly of an acute triangle ABC. They intersect at point TT. It turned out that the altitude LKLK of △ALB\triangle ALB is divided in half by the line ADAD. Prove that KT⊥BLKT \perp BL.
Proposed by Mariia Rozhkova
geometryangle bisector
Square prime

Source: Kyiv City MO 2024 Round 1, Problem 8.2

1/28/2024
Write the numbers from 11 to 1616 in the cells of a of a 4×44 \times 4 square so that: 1. Each cell contains exactly one number; 2. Each number is written exactly once; 3. For any two cells that are symmetrical with respect to any of the perpendicular bisectors of sides of the original 4×44 \times 4 square, the sum of numbers in them is a prime number
The figure below shows examples of such pairs of cells, sums of numbers in which have to be prime.
https://i.ibb.co/fqX05dY/Kyiv-MO-2024-Round-1-8-2.png
Proposed by Mykhailo Shtandenko
number theoryprime numbers
It's a trapezoid

Source: Kyiv City MO 2024 Round 1, Problem 11.2

1/28/2024
ABCDABCD is a trapezoid with BC∥ADBC\parallel AD and BC=2ADBC = 2AD. Point MM is chosen on the side CDCD such that AB=AMAB = AM. Prove that BM⊥CDBM \perp CD.
Proposed by Bogdan Rublov
geometrytrapezoid