4

Part of 2013 239 Open Mathematical Olympiad

Problems(2)

Sqrt inequality with a+b+c<2

Source: 239 2013 S4

For positive numbers

a, b, c

satisfying condition

a+b+c<2

\sqrt{a^2 +bc}+\sqrt{b^2 +ca}+\sqrt{c^2 + ab}<3.

inequalitiesalgebrainequalities proposed

Lesser degree on the edges

Source: 239 2013 J4

We are given a graph

G

n

edges. For each edge, we write down the lesser degree of two vertices at the end of that edge. Prove that the sum of the resulting

n

numbers is at most

100n\sqrt{n}

combinatoricsgraph theory