MathDB

Problem 3

Part of 2022 Kyiv City MO Round 1

Problems(3)

Subsets with sum equal to one

Source: Kyiv City MO 2022 Round 1, Problem 7.3, 8.2

1/23/2022
You are given nn not necessarily distinct real numbers a1,a2,…,ana_1, a_2, \ldots, a_n. Let's consider all 2n−12^n-1 ways to select some nonempty subset of these numbers, and for each such subset calculate the sum of the selected numbers. What largest possible number of them could have been equal to 11?
For example, if a=[−1,2,2]a = [-1, 2, 2], then we got 33 once, 44 once, 22 twice, −1-1 once, 11 twice, so the total number of ones here is 22.
(Proposed by Anton Trygub)
algebracombinatorics
Geometry from legendary author

Source: Kyiv City MO 2022 Round 1, Problem 8.3

1/23/2022
In triangle ABCABC ∠B>90∘\angle B > 90^\circ. Tangents to this circle in points AA and BB meet at point PP, and the line passing through BB perpendicular to BCBC meets the line ACAC at point KK. Prove that PA=PKPA = PK.
(Proposed by Danylo Khilko)
geometry
Bisector and circles

Source: Kyiv City MO 2022 Round 1, Problem 9.3

1/23/2022
Let ALAL be the inner bisector of triangle ABCABC. The circle centered at BB with radius BLBL meets the ray ALAL at points LL and EE, and the circle centered at CC with radius CLCL meets the ray ALAL at points LL and DD. Show that AL2=AE×ADAL^2 = AE\times AD.
(Proposed by Mykola Moroz)
geometry