2011 UNCO Math Contest II Answer Key
1) 
2) 
3) 
4) (a) 
(b)
5) 
6) 
7) 
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. Unknown error_msg)
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. Unknown error_msg). Now look at 
again and replace 
with 
(order does not matter). Crossing off gives you
![\[(a+b+ab) + c + (a+b+ab)c =a+b+c+ab+ac+bc+abc\]](http://latex.artofproblemsolving.com/d/f/3/df32bf0089ce03bf07b1428033a04eff3c24eb46.png)
reminding one of the coefficients in
![\[(x-a)(x-b)(x-c)= x^3-(a+b+c)x^2+(ab+ac+bc)x-abc\]](http://latex.artofproblemsolving.com/9/7/c/97c99449bfa0c5bc6f98d38dc412d010d8d84de0.png)
Now let $x= −1$ (Error compiling LaTeX. Unknown error_msg), 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 
.
