42 lines
1.1 KiB
TeX
42 lines
1.1 KiB
TeX
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\problemdef{Combinatorics}{1}{
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\statement{
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A real estate investor asked a developer to paint 2017 houses so that at least 1000 are green and 1000 are red. What is the maximum number of colors the developer can use?
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}
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\answer{19}
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}
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\problemdef{Combinatorics}{2}{
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\statement{
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How many ways are there to cut one $3 \times 5$ rectangle into five $1 \times 3$ rectangles?
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}
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\answer{4}
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}
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\problemdef{Combinatorics}{3}{
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\statement{
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How many different integral solutions $(x, y, z)$ are there to $x+y+z = 20$?
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}
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\answer{$C_{19}^2$}
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}
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\problemdef{Combinatorics}{4}{
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\statement{
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Consider an uncolored $8 \times 8$ board. How many ways are there to paint the squares black or white so that we end up with exactly 31 black squares, none of which share an edge?
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}
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\answer{68}
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}
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\problemdef{Combinatorics}{5}{
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\statement{
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Let $A$ be the set of four-digit integers in which the first digit is equal to the sum of the other three. \par
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Let $B$ be the set of four-digit integers in which the last digit is equal to the sum of the other three. \par
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Which set is larger, and by how many elements?
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}
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\answer{$A$, by 54.}
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}
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