Two Similar Fractions
If , the value of can be written as where and are coprime integers. Find .
The answer is 4029.
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1 solution
We have
2 0 1 5 = ( q − r ) ( s − p ) ( p − q ) ( r − s ) = q s − r s − p q + p r p r − q r − p s + q s
So,
2 0 1 4 = q s − r s − p q + p r p r − q r − p s + q s − 1 = q s − r s − p q + p r p q + r s − q r − p s = ( q − r ) ( s − p ) ( p − r ) ( q − s )
Therefore,
( p − q ) ( r − s ) ( p − r ) ( q − s ) = ( q − r ) ( s − p ) ( p − r ) ( q − s ) × ( p − q ) ( r − s ) ( q − r ) ( s − p ) = 2 0 1 4 × 2 0 1 5 1 = 2 0 1 5 2 0 1 4 .
Thus, a + b = 2 0 1 4 + 2 0 1 5 = 4 0 2 9