# 2011 UNCO Math Contest II Answer Key

1)

2)

3)

4) (a) (b)

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8) (a) (b)

9) (a) (b) $T(N) = \binom{N − 1}{3} − !\binom{N− 2}{3} + \binom{N − 3}{3} − \binom{N − 4}{3} +\cdots$ (Error compiling LaTeX. ! Package inputenc Error: Unicode character − (U+2212))

10) First try for . The crossing off process yields each one being one less than a factorial. So for general you should end up with$(n+ 1 )! − 1$ (Error compiling LaTeX. ! Package inputenc Error: Unicode character − (U+2212)). Now look at again and replace with (order does not matter). Crossing off gives you

reminding one of the coefficients in

Now let $x= −1$ (Error compiling LaTeX. ! Package inputenc Error: Unicode character − (U+2212)), and watch what happens remember that . There are other approaches.

11) See solution to #2. Integers that are one less than a prime cannot be written in the form .