a) 2 white and 2 black cats are sitting on the line. The sum of distances from the white cats to one black cat is 4, to the other black cat is 8. The sum of distances from the black cats to one white cat is 3, to the other white cat is 9. What cats are sitting on the edges? b) 2 white and 3 black cats are sitting on the line. The sum of distances from the white cats to one black cat is 11, to another black cat is 7 and to the third black cat is 9. The sum of distances from the black cats to one white cat is 12, to the other white cat is 15. What cats are sitting on the edges? (Kyiv mathematical festival 2014)
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Immanuel Bonfils
13.05.2014 03:48
a) Let $ |W_1B_1|=a_1 , |W_2B_1|=a_2, |W_1B_2|=b_1, |W_2B_2|=b_2 $ . So $ a_1+a_2 = 4 \rightarrow a_1,a_2 <4 ,\;\; (*) b_1+b_2= 8\;\; b_1+a_1=9$ $ (**)b_2+a_2=3 \rightarrow b_2,a_2<3 $. Subtracting $(**)$ from $(*)$, we get $ b_1-a_2=5 \rightarrow b_1>5 $. Then, $b_1$ is greater than the other three distances, an the extremes are $ W_1 $ (first white cat) and $ B_2 $ (second black cat).
wcao9311
13.05.2014 04:24
Can we have line going though 5 dimensions ?
Immanuel Bonfils
14.05.2014 03:26
Forgot to say (rather write...) that the extremes can't be of the same color, otherwise the sums in the data should include a pair with the same value.