3

Part of 1991 Romania Team Selection Test

Problems(2)

min (q_r(C)+q_a(C)) \le \frac{1}{32} {n \choose 4}, coloring inequality

Source: Romania IMO TST 1991 p3

C

be a coloring of all edges and diagonals of a convex

n

−gon in red and blue (in Romanian, rosu and albastru). Denote by

q_r(C)

q_a(C)

) the number of quadrilaterals having all its edges and diagonals red (resp. blue). Prove:

\underset{C}{min} (q_r(C)+q_a(C)) \le \frac{1}{32} {n \choose 4}

inequalitiesColoringcombinatoricspolygon

A nice identity

Source: Romania TST 1991

Prove the following identity for every

n\in N

: \sum_{j\plus{}h\equal{}n,j\geq h}\frac{(\minus{}1)^h2^{j\minus{}h}\binom{j}{h}}{j}\equal{}\frac{2}{n}

inductionLaTeXlogarithmscombinatorics unsolvedcombinatorics