A combinatorial proof of Lindstr”om’s theorem on unions of subsets of a finite set

Cosmin Pohoata, A combinatorial proof of Lindstr”om’s theorem on unions of subsets of a finite set. (Link to the PDF file)

In this note, we give a combinatorial proof of Bernt Lindstr\”om’s theorem that if A_{1}, A_{2},…, A_{n+1} are nonempty subsets of an n-element set, then we can find two disjoint and nonempty groups of indices \left\{i_{1},\ i_{2},\ \ldots,\ i_{k}\right\} and \left\{j_{1},\ j_{2},\ \ldots,\ j_{m}\right\} such that
A_{i_{1}} \cup A_{i_{2}} \cup \ldots \cup A_{i_{k}} = A_{j_{1}} \cup A_{j_{2}} \cup \ldots \cup A_{j_{m}}.

Boole’s formula as a consequence of Lagrange’s Interpolating Polynomial Theorem

Cosmin Pohoata, Boole’s formula as a consequence of Lagrange’s interpolating polynomial theorem, INTEGERS: The Electronic Journal of Combinatorial Number Theory, 8 (2008), #A23. (Link to the PDF file).
Another variant of the paper can be downloaded from http://arxiv.org/abs/0807.1133.)

We present a slightly more general version of Boole’s additive formula for factorials as a simple consequence of Lagrange’s Interpolating Polynomial theorem.

Cezar Lupu, Cosmin Pohoata, About a nice inequality, Mathematical Reflections, 1 (2007). (Link to the journal’s PDF file)

We consider two proofs for Darij Grinberg’s a^2+b^2+c^2 + 2abc +1 \geq 2(ab+bc+ca), which lead us to solving several other three-variable inequalities: one from the Romanian National Mathematics Olympiad from 2004, one from the Romanian BMO 2005 Team Selection Tests, one from the Asian Pacific Mathematics Olympiad from 2004 and also one from the USA IMO 2000 Team Selection Tests.