View source for 2011 UNCO Math Contest II Answer Key

1) <math>41</math>

2) <math>\{22,28,30\}</math>

3) <math>65</math>

4) (a) <math>101</math>   (b)<math> \{101, 325, 2501\}</math>

5) <math>160</math>

6) <math>9</math>

7) <math>2^{502}-1504</math>

8) (a) <math>74</math>  (b) <math>45 \times 74</math>

9) (a) <math>T(n+1)+T(n)=\binom{n}{3}</math>  (b) <math>T(N) = \binom{N-1}{3}-\binom{N-2}{3}+\binom{N-3}{3}-\binom{N-4}{3}+\cdots</math>

10) First try <math>\{1, 2, 3, \ldots , n\}</math> for <math>n= 2, 3, 4, 5</math>. The crossing off process yields <math>\{5,23,119,719\}</math> each one being one less
than a factorial. So for general <math>n</math> you should end up with<math>(n+1)!-1</math>. Now look at <math>n=3</math> again and replace <math>1, 2, 3</math>
with <math>a,b,c</math> (order does not matter). Crossing off gives you
<cmath>(a+b+ab) + c + (a+b+ab)c  =a+b+c+ab+ac+bc+abc</cmath>
reminding one of the coefficients in
<cmath>(x-a)(x-b)(x-c)= x^3-(a+b+c)x^2+(ab+ac+bc)x-abc</cmath>
Now let <math>x=-1</math>, and watch what happens remember that <math>\{a,b,c\} = \{1,2,3\} </math>.
There are other approaches.

11) See solution to #2. Integers that are one less than a prime cannot be written in the form <math>m +n +m</math>.