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Algebra Level 4

Let a , b , c a,b,c and d d be real numbers in a geometric progression . Which of the following is equal to ( a 2 + b 2 + c 2 ) ( b 2 + c 2 + d 2 ) (a^2+b^2+c^2)(b^2+c^2+d^2) ?

0 ( a b + b c + c d ) 2 (ab+bc+cd)^2 a b + b c + c d ab+bc+cd ( a b + c d + a d ) 2 (ab+cd+ad)^2

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1 solution

展豪 張
Jun 1, 2016

Let a b = b c = c d = r \dfrac ab=\dfrac bc=\dfrac cd=r

( a 2 + b 2 + c 2 ) ( b 2 + c 2 + d 2 ) \;\;\;\;(a^2+b^2+c^2)(b^2+c^2+d^2)
= ( 1 r ( a 2 + b 2 + c 2 ) ) ( r ( b 2 + c 2 + d 2 ) ) =(\dfrac 1r(a^2+b^2+c^2))(r(b^2+c^2+d^2))
= ( a a r + b b r + c c r ) ( b ( b r ) + c ( c r ) + d ( d r ) ) =(a\dfrac ar+b\dfrac br+c\dfrac cr)(b(br)+c(cr)+d(dr))
= ( a b + b c + c d ) ( b a + c b + d c ) =(ab+bc+cd)(ba+cb+dc)
= ( a b + b c + c d ) 2 =(ab+bc+cd)^2

Perfect Solution , did it in a similar manner \text {Perfect Solution , did it in a similar manner} + 1 ! +1!

Rishabh Tiwari - 5 years ago

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Thank you!

展豪 張 - 5 years ago

The third option works as well.

Greg Grapsas - 1 year, 10 months ago

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