Olympiad Corner 2

We note that: $3x + 4y = \dfrac{3}{2}x + \dfrac{3}{2}x + \dfrac{4}{3}y + \dfrac{4}{3}y + \dfrac{4}{3}y$ and that:

$\begin{aligned} \frac{3}{2}x + \frac{3}{2}x + \frac{4}{3}y + \frac{4}{3}y + \frac{4}{3}y & \ge 5 \sqrt[5]{\left(\frac{3}{2}x\right)^2 \left(\frac{4}{3}y\right)^3} \\ \Rightarrow 5 & \ge 5 \sqrt[5]{\frac{16x^2y^3}{3}} \\ 1 & \ge \frac{16x^2y^3}{3} \\ \Rightarrow x^2y^3 & \le \frac{3}{16} = \boxed{0.1875} \end{aligned}$

Note that the same result is obtained using Lagrange multiplier method, which does not need $x$ and $y$ to be positive.

$\color{#3D99F6}{\text{Using Lagrange Multiplier -- as requested by }}$ Shivam Mishra

Let $F(x,y,\lambda) = x^2y^3 - \lambda(3x+4y - 5)$ , where $\lambda$ is the Lagrange multiplier and $3x+4y-5$ is the constraint. Then, we have:

\(\begin{array} {} \dfrac{\partial F}{\partial x} = 0 & \Rightarrow 2xy^3 = 3 \lambda & ... (1) \\ \dfrac{\partial F}{\partial y} = 0 & \Rightarrow 3x^2y^2 = 4 \lambda & ... (2) \\ \dfrac{\partial F}{\partial \lambda} = 0 & \Rightarrow 3x+4y=5 & ... (3) \end{array} \)

\(\begin{array} {} \dfrac{(1)}{(2)}: & \dfrac{2y}{3x} = \dfrac{3}{4} & \Rightarrow y = \dfrac{9}{8}x \\ (3): & 3x + 4 \times \dfrac{9}{8}x = 5 & \Rightarrow x = \frac{2}{3} & \Rightarrow y = \frac{3}{4} \end{array} \)

$\Rightarrow x^2y^3 \le \left(\frac{2}{3}\right)^2 \left(\frac{3}{4}\right)^3 = \frac{3}{16} \\ \Rightarrow A + B = 3+16 = \boxed{19}$

Let $x = a/3$ and $y = b/4$ , then we have $b = 5 - a$ and we maximize $f(a) = \frac {a^2\ (5-a)^3}{3^2\cdot 4^3}.$

The maximum corresponds to a derivative of zero: $f'(a) = \frac1{3^2\cdot 4^3}\cdot \left(2a(5-a)^3-3a^2(5-a)^2\right) = \frac 1{3^2\cdot 4^3}\cdot \left(2(5-a)-3a\right)\cdot a(5-a)^2 = 0.$ The only non-trivial solution (which corresponds to a maximum) is $2(5-a)-3a=0$ or $a = 2$ .

Finishing up: $b = 5 - a = 3$ , and $f(a) = \frac{2^2\cdot 3^3}{3^2 \cdot 4^3} = \frac{3}{4^2},$ so that the solution is $3 + 16 = \boxed{19}$ .

Derivative also yields same result

Please mention that x and y are positive real numbers