2011 UNCO Math Contest II Answer Key

Revision as of 01:01, 6 November 2015 by Timneh (talk | contribs) (Created page with "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)...")
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1) $41$

2) $\{22,28,30\}$

3) $65$

4) (a) $101$ (b)$\{101, 325, 2501\}$

5) $160$

6) $9$

7) $2^{502}-1504$

8) (a) $74$ (b) $45 \times 74$

9) (a) $T(n+1)+T(n)=\binom{n}{3}$ (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 $\{1, 2, 3, \ldots , n\}$ for $n= 2, 3, 4, 5$. The crossing off process yields $\{5,23,119,719\}$ each one being one less than a factorial. So for general $n$ you should end up with$(n+ 1 )! − 1$ (Error compiling LaTeX. ! Package inputenc Error: Unicode character − (U+2212)). Now look at $n=3$ again and replace $1, 2, 3$ with $a,b,c$ (order does not matter). Crossing off gives you \[(a+b+ab) + c + (a+b+ab)c  =a+b+c+ab+ac+bc+abc\]

reminding one of the coefficients in

\[(x-a)(x-b)(x-c)= x^3-(a+b+c)x^2+(ab+ac+bc)x-abc\] Now let $x= −1$ (Error compiling LaTeX. ! Package inputenc Error: Unicode character − (U+2212)), and watch what happens remember that $\{a,b,c\} = \{1,2,3\}$. There are other approaches.

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

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