What is the relationship
If , and are distinct numbers, which of the following relationships holds between and ?
Based on NMTC Sub-Junior Question, Final Level.
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1 solution
Way to approach, rather than method, which can be very short.
First look:
Relationship cannot be determined looks appealing.
Try substituting 0, 1, and -1:
A > B
Rules out A = B and A < B
To eliminate relationship cannot be determined, we need to solve algebraically:
2 ( 2 0 1 2 x y + 3 y z + 3 z x ) = 4 0 2 4 x y + 6 y z + 6 z x
The connection between x^2, y^2 and xy is given by the formula for (x + y)^2:
2 0 1 5 x 2 + 2 0 1 5 y 2 = 2 0 1 5 ( x 2 + y 2 ) = 2 0 1 5 ( x 2 + y 2 − 2 x y + 2 x y ) = 2 0 1 5 [ ( x − y ) 2 + 2 x y ] = 2 0 1 5 ( x − y ) 2 + 4 0 3 0 x y
First term is positive:
2 0 1 2 ( x − y ) 2 + 4 0 2 4 x y > 4 0 2 4 x y
Similar process for remaining two terms
This makes A > B correct.