MathDB

Problem 3

Part of 2024 Kyiv City MO Round 2

Problems(4)

Dividing circle into parts with equal sums

Source: Kyiv City MO 2024 Round 2, Problem 7.3

2/4/2024
20242024 ones and 20242024 twos are arranged in a circle in some order. Is it always possible to divide the circle into
a) two (contiguous) parts with equal sums? b) three (contiguous) parts with equal sums?
Proposed by Fedir Yudin
combinatoricsconstruction
(Angle) Chasing good geometry

Source: Kyiv City MO 2024 Round 2, Problem 8.3

2/4/2024
Let ω\omega denote the circumscribed circle of an acute-angled △ABC\triangle ABC with AB≠BCAB \neq BC. Let A′A' be the point symmetric to the point AA with respect to the line BCBC. The lines AA′AA' and A′CA'C intersect ω\omega for the second time at points DD and EE, respectively. Let the lines AEAE and BDBD intersect at point PP. Prove that the line A′PA'P is tangent to the circumscribed circle of △A′BC\triangle A'BC.
Proposed by Oleksii Masalitin
geometrycircumcircle
Diameter configuration

Source: Kyiv City MO 2024 Round 2, Problem 10.3

2/4/2024
Let AHA,BHB,CHCAH_A, BH_B, CH_C be the altitudes of the triangle ABCABC. Points A1A_1 and C1C_1 are the projections of the point HBH_B onto the sides ABAB and BCBC, respectively. B1B_1 is the projection of BB onto HAHCH_AH_C. Prove that the diameter of the circumscribed circle of △A1B1C1\triangle A_1B_1C_1 is equal to BHBBH_B.
Proposed by Anton Trygub
circumcirclegeometry
Interval fasting

Source: Kyiv City MO 2024 Round 2, Problem 11.3

2/4/2024
For a given positive integer nn, we consider the set MM of all intervals of the form [l,r][l, r], where the integers ll and rr satisfy the condition 0≤l<r≤n0 \leq l < r \leq n. What largest number of elements of MM can be chosen so that each chosen interval completely contains at most one other selected interval?
Proposed by Anton Trygub
combinatorics