BTGmoderatorDC wrote:Of the 66 people in a certain auditorium, at most 6 people have their birthdays in any one given month. Does at least one person in the auditorium have a birthday in January?

(1) More of the people in the auditorium have their birthday in February than in March.
(2) Five of the people in the auditorium have their birthday in March.
Source: Official Guide

$$0\,\, \leqslant \,\,J,F,M, \ldots ,N,D\,\, \leqslant \,\,6\,\,\,\left( * \right)\,\,\,\,\,\,\left( {{\text{ints}}} \right)$$
$$J + F + M + \ldots + N + D = 66\,\,\,\,\,\,\, \Rightarrow \,\,\,{\mu _{{\text{all}}}} = 5.5\,\,\frac{{{\text{birthdays}}}}{{{\text{month}}}}$$
$$?\,\,\,:\,\,\,J\,\,\mathop \geqslant \limits^? \,\,1$$

$$\left( 1 \right)\,\,F > M\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\left\langle {{\text{YES}}} \right\rangle $$
$$\left( {**} \right)\,\,\,J = 0\,\,\,\, \Rightarrow \,\,\,{\mu _{{\text{all}}\, - \,\left\{ J \right\}}} = 6\,\,\frac{{{\text{birthdays}}}}{{{\text{month}}}}\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\underline {F = M} = \ldots = N = D = 6\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\underline {{\text{impossible}}} $$

$$\left( 2 \right)\,\,\,M = 5\,\,\,\,\mathop \Rightarrow \limits^{\left( {***} \right)} \,\,\,\left\langle {{\text{YES}}} \right\rangle $$
$$\left( {***} \right)\,\,\,J = 0\,\,\,\, \Rightarrow \,\,\,{\mu _{{\text{all}}\, - \,\left\{ J \right\}}} = 6\,\,\frac{{{\text{birthdays}}}}{{{\text{month}}}}\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,F = \underline M = \ldots = N = D = \underline 6 \,\,\,\,\, \Rightarrow \,\,\,\,\,\,\underline {{\text{impossible}}} $$


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.