Let a,b,c,d be positive real numbers such that a+b+c+d=11.

Question

Let $a,b,c,d$ be positive real numbers such that $a+b+c+d=11$. If the maximum value of $a^5b^3c^2d$ is $3750\beta$, then the value of $\beta$ is:

Accepted Answer

By weighted AM--GM, split $a+b+c+d$ into $5+3+2+1=11$ equal unit parts: the maximum occurs at $a=5,b=3,c=2,d=1$.

$$a^5b^3c^2d = 5^5\times3^3\times2^2\times1 = 3125\times27\times4 = 337500 = 3750\times90.$$

So $\beta = 90$.

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