MathDB

Problem 2

Part of 2022 Kyiv City MO Round 1

Problems(3)

Sticks forming non-degenerate triangles

Source: Kyiv City MO 2022 Round 1, Problem 7.2

1/23/2022
There are nn sticks which have distinct integer length. Suppose that it's possible to form a non-degenerate triangle from any 33 distinct sticks among them. It's also known that there are sticks of lengths 55 and 1212 among them. What's the largest possible value of nn under such conditions?
(Proposed by Bogdan Rublov)
combinatoricsgeometry
Hardest algebra ever

Source: Kyiv City MO 2022 Round 1, Problem 9.2

1/23/2022
For any reals x,yx, y, show the following inequality:
(x+4)2+(y+2)2+(x−5)2+(y+4)2≤(x−2)2+(y−6)2+(x−5)2+(y−6)2+20\sqrt{(x+4)^2 + (y+2)^2} + \sqrt{(x-5)^2 + (y+4)^2} \le \sqrt{(x-2)^2 + (y-6)^2} + \sqrt{(x-5)^2 + (y-6)^2} + 20
(Proposed by Bogdan Rublov)
algebrainequalities
Selecting pairs with composite sums

Source: Kyiv City MO 2022 Round 1, Problem 10.2, 11.2

1/23/2022
You are given 2n2n distinct integers. What's the largest integer CC such that you can always form at least CC pairs from them, so that no integer is in more than one pair, and the sum of integers in each pair is a composite number?
(Proposed by Anton Trygub)
number theory