Art of Problem Solving

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2008 UNCO Math Contest II Answer Key

1) $21$

2) $\{1,4,9,10\}$ or $\{2,3,8,11\}$

3) $20$

4) (a) $3\binom{10}{2}$ (b) $3\binom{n+2}{2}=\frac{3(n+1)(n+2)}{2}$

5) (a) $3^{668}\cdot 2^2$ (b) $3^k$ if $n=3k$ ; $2^2\cdot 3^{k-1}$ if $n=3k+1$ ; $2\cdot 3^{k}$ if $n=3k+2$

6) $18$

7) $a=8$

8) (a) $157$ (b) $4n-3$ (c) $116$

9) (a) $\frac{10^n-1}{9}$ (b) $112,225$ (c) $11,122,225$

10) (a) $f(3,2)=15$ (b) $f(n,2)=(2n-1)!!$ (c) $f(2,3)=\binom{5}{2} ; f(3,3)=\binom{8}{2} \binom{5}{2} ;f(n,3)=\frac{(3n)!}{6^n \cdot n!}$

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