# Difference between revisions of "2011 UNCO Math Contest II Answer Key"

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8) (a) <math>74</math> (b) <math>45 \times 74</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 | + | 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 | 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 )! | + | 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 | 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> | <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> | <cmath>(x-a)(x-b)(x-c)= x^3-(a+b+c)x^2+(ab+ac+bc)x-abc</cmath> | ||

− | Now let <math>x= | + | Now let <math>x=-1</math>, and watch what happens remember that <math>\{a,b,c\} = \{1,2,3\} </math>. |

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

## Latest revision as of 01:03, 6 November 2015

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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. Now look at again and replace with (order does not matter). Crossing off gives you reminding one of the coefficients in Now let , 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 .