MathDB

2

Part of 2023 Middle European Mathematical Olympiad

Problems(2)

Chords in a circle

Source: MEMO 2023 I2

8/24/2023
Find all positive integers n≥3n \geq 3, for which it is possible to draw nn chords on a circle, with their 2n2n endpoints being pairwise distinct, such that each chords intersects exactly kk others for:
(a) k=n−2k=n-2,
(b) k=n−3k=n-3.
combinatoricsMEMO2023geometry
4-variable inequality

Source: MEMO 2023 T2

8/25/2023
If a,b,c,d>0a, b, c, d>0 and abcd=1abcd=1, show that ab+1a+1+bc+1b+1+cd+1c+1+da+1d+1≥4.\frac{ab+1}{a+1}+\frac{bc+1}{b+1}+\frac{cd+1}{c+1}+\frac{da+1}{d+1} \geq 4. When does equality hold?
inequalitiesalgebra