More splitting
Consider the shape in the problem Splitting Them Up ; that is, the irregular hexagon given by the points , , , , , . Let be equal to the sum of the slopes of the lines through which divide the figure into equal areas ( , ). If , where and are co-prime, and , find the sum of the digits of q.
The answer is 767.
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1 solution
Define the piecewise functions x(t) and y(t) for the shape:
x ( t ) = ⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ t 6 t − 4 1 2 0 < = t < 6 6 < = t < 1 0 1 0 < = t < 1 6 1 6 < = t < 1 8
y ( t ) = ⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ 6 1 2 − t 2 1 8 − t 0 < = t < 6 6 < = t < 1 0 1 0 < = t < 1 6 1 6 < = t < 1 8
and let g ( t ) = ( x ( t ) , y ( t ) ) .
Integrate g ( t ) to find the area function:
a ( t ) = ⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ 3 t 3 t t + 2 0 6 t − 6 0 0 < = t < 6 6 < = t < 1 0 1 0 < = t < 1 6 1 6 < = t < 1 8
Inverted:
t ( a ) = ⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ 3 a 3 a a − 2 0 6 a + 6 0 0 < = a < 1 8 1 8 < = a < 3 0 3 0 < = a < 3 6 3 6 < = a < 4 8
f ( n ) = ∑ i = 1 n − 1 x ( t ( n 4 8 i ) ) y ( t ( n 4 8 i ) )