Let ABCD be a convex quadrilateral with positive integer side lengths, angle A = angle B = 120 degrees, the absolute value of AD - BC = 42, and CD = 98. Find the maximum possible value of AB.

The maximum possible value of AB is 181.

Expert Answers

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Denote `A B = x , ` `A D = y ` and suppose that `B C ` is smaller than `A D , ` that is, `B C = y - 42 . ` Then apply the cosine law to the triangles `A B C ` and `A D B` and use the fact that `cos ( 120^@ ) = - 1 / 2 :`

`x^2 + ( y - 42 )^2 + 2 x ( y-42 ) / 2 = AC^2 = y^2 + ( 98 )^2 + 2 y * 9 8 / 2 ,`

so `x^2 - 84 y + ( 42 )^2 + x y - 42 x = ( 98 )^2 + 9 8 y ,`

or `x^2 - 182 y + x y - 42 x = ( 98 )^2 - 42^2 = 56 * 140 = 7840 .`

This way, `y = ( -x^2 + 42 x + 7 8 4 0 ) / ( x - 1 8 2 ) . ` It is positive (for positive `x `) only when `112 lt x lt 182 , ` so the maximum possible integer `x ` is `1 8 1 , ` for which `y ` is also an integer .

You can analyze the situation when `B C ` is greater than `A D , ` that is, `BC = y + 42 .`

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