GMO 2015 problem
Find the number of fractions that can be written simultaneously in the forms and for some integers .
The answer is 8.
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1 solution
4 l − 3 6 l − 1 = 5 k − 3 7 k − 5 After cross multiplication we get 8 k + l + l k = 6 Adding 8 to both sides 8 k + 8 + l + l k = 1 4 8 ( k + 1 ) + l ( 1 + k ) = 1 4 ( 8 + l ) ( 1 + k ) = 1 4 Since l , k are integers therefore the solutions of ( l , k ) = ( − 1 , 1 ) , ( − 6 , 6 ) , ( 6 , 0 ) , ( − 7 , 1 3 ) , ( − 1 5 , − 3 ) , ( − 2 2 , − 2 ) , ( − 1 0 , − 8 ) , ( − 9 , − 1 5 ) Therefore the required fractions are 1 , 3 1 4 3 , 3 5 , 2 7 3 7 , 9 1 3 , 1 3 1 9 , 4 3 6 1 , 3 9 5 5