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2014 UNCO Math Contest II Answer Key

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1) The youngest will be $16$ and the perfect square is $100$ .

2) $145$

3) $x =-2$ and $y = \tfrac{-3}{4}$

4) (a) $210$ (b) $4010$ (c) $\frac{n(n^2+1)}{2}$

5) (a) $\frac{1}{5}$ (b) $\frac{(1-x)^2}{1+x^2}$

6) (a) $11$ (b) $76$

7) $\frac{112}{125}$

8) (a) $\frac{2}{5}$ (b) $\frac{8}{31}$

9) Zero probability for $n=1$ and $\frac{F_{n-1}}{2^{2n}}$ for $n\ge 2$ , where $F_{n-1}$ is the $(n-1)^{st}$ Fibonacci number.

10)(a) $\tfrac{1}{11}$ (b) $\tfrac{2}{11}$ (c) $\tfrac{1023}{5120} = \left(\tfrac{1}{m+1}\right) \tfrac{2^{m+1}-1}{2^m}$ ,

where there are $m$ guessers, not including Tweedledee and Tweedledum. Here $m = 9$ .

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