4 distinct reals
Algebra
Level
2
Do there exist distinct reals which satisfy the above equation?
Yes
No
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1 solution
Multiplying by 2, we have 2 ( a 2 + b 2 + c 2 + d 2 ) − 2 ( a b + b c + c d + d a ) = 0 . Group the terms to get ( a 2 − 2 a b + b 2 ) + ( b 2 − 2 b c + c 2 ) + ( c 2 − 2 c d + d 2 ) + ( d 2 − 2 d a + a 2 ) = 0 . Note that all the terms are squares. Thus the equation can be written as ( a − b ) 2 + ( b − c ) 2 + ( c − d ) 2 + ( d − a ) 2 = 0 ⟹ a − b = b − c = c − d = d − a = 0 . This implies that a = b = c = d , contradicting the fact that a , b , c , d are distinct .