8

Part of 2013 239 Open Mathematical Olympiad

Problems(2)

Chords of a circle with 10^100 points

Source: 239 2013 S8

Prove that if you choose

10^{100}

points on a circle and arrange numbers from

1

10^{100}

on them in some order, then you can choose

100

pairwise disjoint chords with ends at the selected points such that the sums of the numbers at the ends of all of them are equal to each other.

5 variable inequality with product of 1

Source: 239 2013 J8

The product of the positive numbers

a, b, c, d

e

1

\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{d^2}+\frac{d^2}{e^2}+\frac{e^2}{a^2} \geq a+b+c+d+e .

algebrainequalities