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Algebra Level 5

Fix the constant A A . The greatest value of ( A + X ) 3 × ( A − X ) 4 (A+X)^3 \times (A-X)^4 over the domain X ≤ A X \leq A can be expressed as

p a × q b × A c r d \frac{ p^a \times q^b \times A ^c} { r ^ d }

for prime numbers p , q , p,q, and r r and positive integers a , b , c , d a, b, c, d .

Find p + q + r + a + b + c + d p + q + r + a + b+c+d .


The answer is 44.

Abhinav Jha by Abhinav Jha , 24, India

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2 solutions

Prince Loomba
Jun 17, 2016

Differentiate to get X = A , − A , − A 7 X=A,-A,\frac {-A}{7} . At X = − A 7 X=\frac {-A}{7} , the value is 2 15 3 3 A 7 7 7 \frac {2^{15}3^{3}A^{7}}{7^{7}} .The answer is 44 (2+15+3+3+7+7+7)

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Abhinav Jha
May 11, 2016

The given expression is greatest when [(A + X)/3]^3 * [(A - X)/4]^4 is greatest.,but the sum of factors of this expression is 3[(A + X)/3] + 4[(A - X)/4] that is 2A. Hence the expression is greatest when (A + X)/3 = (A - X)/4 , or X = -A/7. Thus the greatest value is (6^3 * 8^4 * A^7) / 7^7. As per question demands answer is 28.

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In order to apply AM-GM, we require that all of the terms be positive. Do you need A > 0 A> 0 and x > 0 x > 0 too?

Calvin Lin Staff - 5 years, 1 month ago

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