BTGmoderatorDC wrote:The participants in a race consisted of 3 teams with 3 runners on each team. A team was awarded 6 -n points if one of its runners finished in nth place, where 1 <= n <= 5. If all of the runners finished the race and if there were no ties, was each team awarded at least one point?
(1) No team was awarded more than a total of 6 points.
(2) No pair of teammates finished in consecutive places among the top five places.
Source: Princeton Review
\[A,B,C\,\,{\text{teams}}\,\,\left( {3\,\,{\text{people}}\,\,{\text{in}}\,\,{\text{each}}} \right)\]
\[?\,\,:\,\,{\text{all}}\,\,{\text{teams}}\,\,{\text{got}}\,\,{\text{points}}\]
\[\begin{array}{*{20}{l}}
{{\text{Place}}}&{{\text{Points}}} \\
1&5 \\
2&4 \\
3&3 \\
4&2 \\
5&1
\end{array}\]
\[15\,\,{\text{points}}\,\,{\text{given}}\,\,{\text{in}}\,\,{\text{total}}\,\,\,\left( * \right)\]
\[\left( 1 \right)\,\sum {{\text{any}}\,{\text{two}}\,\,{\text{teams}}} \,\,\, \leqslant \,\,12\,\,{\text{points}}\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \]
\[\left( 2 \right)\,\,\,\left\{ \begin{gathered}
\hfill \,{\text{Take}}\,\,A = \left\{ {1,3,5} \right\}\,\,{\text{and}}\,\,B = \left\{ {2,4} \right\}\,\,\,\left( {{\text{places}}} \right)\,\,\, \Rightarrow \,\,\,\left\langle {{\text{NO}}} \right\rangle \\
\hfill \,{\text{Take}}\,\,A = \left\{ {1,3} \right\}\,,\,B = \left\{ {2,4} \right\}\,\,{\text{and}}\,\,C = \left\{ 5 \right\}\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle \\
\end{gathered} \right.\]
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
