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Olympiad workbook

Exam: Olympiad workbook
Subject: IOQM
Chapter: Quadratic Equation

Olympiad workbook · IOQM · Quadratic Equation

Let a, b be positive integers satisfying a^3-b^3-a b=25 . Find the...

Let

a, b

be positive integers satisfying

a^3-b^3-a b=25

. Find the largest possible value of

a^2+b^3

.

Correct answer

43

Step-by-step solution

a^3-b^3-a b=25

for

a=4

and

b=3

Because for any greater number

a^3-b^3-a b>25

To prove this if

a>b

, then

a^3-b^3-a b

\begin{aligned} & =(b+t)^3-b^3-b(b+t), t>0 \\ & =(3 t-1) b^2+\left(3 t^2-t\right) b+t^3 \text { is always greater than } 4 \end{aligned}

then

b \geq 3

So,

a^2+b^3=4^2+3^3=43

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