# Problem of the week #89 - December 9th, 2013

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#### Jameson

Staff member
Three candidates A, B, C are contesting an election. In an opinion poll fraction $a$ of voters prefer A to B, fraction $b$ prefer B to C and fraction $c$ prefer C to A. then which of the following preferences are impossible for $(a,b,c)$?

1. (0.51,0.51,0.51)
2. (0.61,0.71,0.71)
3. (0.68,0.68,0.68)
4. (0.49,0.49,0.49)
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#### Jameson

Staff member
No one answered this week's problem

Solution (from Stackexchange):
There are six possible preference orders for the candidates:

$d$: A>B>C
$e$: A>C>B
$f$: B>A>C
$g$: B>C>A
$h$: C>A>B
$i$: C>B>A

From this, $a = d + e + f$, $b = f + g + i$, and $c = e + h + i$.

$a + b +c = d + 2e + 2f + g + h + 2i \le 2(d + e + f + g + h + i) = 2(\text{# of voters})$.

In scenario (3), $a + b + c = 204\%$, which is impossible.

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