Is the information sufficient 6?
For is defined at , and the function is a one-to-one relation. Also, for all such .
Evaluate:
For more problems like this, try answering this set .
The answer is 98475.
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1 solution
Let f ( 2 0 1 7 ) = a . Suppose x = 2 0 1 7 ⟹ a f ( a + 2 0 1 7 2 0 1 6 ) = 2 0 1 6 ⟹ f ( a + 2 0 1 7 2 0 1 6 ) = a 2 0 1 6
Suppose x = a + 2 0 1 7 2 0 1 6 ⟹ a 2 0 1 6 f ⎝ ⎜ ⎛ a 2 0 1 6 + a + 2 0 1 7 2 0 1 6 2 0 1 6 ⎠ ⎟ ⎞ = 2 0 1 6 ⟹ f ⎝ ⎜ ⎛ a 2 0 1 6 + a + 2 0 1 7 2 0 1 6 2 0 1 6 ⎠ ⎟ ⎞ = a
Since f ( x ) is one-to-one, ⟹ a 2 0 1 6 + 2 0 1 7 a + 2 0 1 6 2 0 1 6 ⋅ 2 0 1 7 = 2 0 1 7 ⟹ ( 2 0 1 6 ⋅ 2 0 1 7 a ) + 2 0 1 6 2 + ( 2 0 1 6 ⋅ 2 0 1 7 a ) = 2 0 1 7 2 a 2 + ( 2 0 1 7 ⋅ 2 0 1 6 a ) ⟹ 2 0 1 7 2 a 2 − ( 2 0 1 6 ⋅ 2 0 1 7 a ) − 2 0 1 6 2 = 0 ⟹ a = 2 ⋅ 2 0 1 7 2 ( 2 0 1 6 ⋅ 2 0 1 7 ) ± 2 0 1 6 2 2 0 1 7 2 + ( 4 ⋅ 2 0 1 7 2 ⋅ 2 0 1 6 2 ) = 2 0 1 7 1 0 0 8 ± 1 0 0 8 5
Since f ( x ) > 1 for all x > 1 , ⟹ a = f ( 2 0 1 7 ) = 2 0 1 7 1 0 0 8 + 1 0 0 8 5
∴ 6 5 ⋅ ( ( ( 2 0 1 7 f ( 2 0 1 7 ) − 1 0 0 8 ) 2 m o d 2 0 1 7 ) + 1 ) = 6 5 ⋅ ⎝ ⎛ ⎝ ⎛ ( 2 0 1 7 ⋅ 2 0 1 7 1 0 0 8 + 1 0 0 8 5 − 1 0 0 8 ) 2 m o d 2 0 1 7 ⎠ ⎞ + 1 ⎠ ⎞ = 6 5 ⋅ ( ( ( 1 0 0 8 5 ) 2 m o d 2 0 1 7 ) + 1 ) = 9 8 4 7 5 .