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Number Theory Level 3

If $x$ and $y$ are two non-negative integers. And $x + y = n$ , where $n$ is a non-negative odd integers, find the maximum value of $x \times y$ .

\dfrac{n^2}{4}

\dfrac{3{n^2}}{4}

\dfrac{n^2-1}{4}

\dfrac{n^2-4}{4}

1 solution

Abhay Tiwari
May 5, 2016

A number $n$ if even, when divided into two halfs then they produce the maximum product.

Whereas, in case $n$ is odd, we slit it up as $\frac{n+1}{2}$ and $\frac{n-1}{2}$ .

Then the product in this case will be maximum and will be $\frac{n+1}{2} \times \frac{n-1}{2} =\boxed{ \frac{{n}^{2}-1}{4}}$

if Every Variable is a Positive Integer , why can't we apply $\text{A.M - G.M}$ , from that condition i got $\implies \dfrac{x+y}2=(xy)^{\frac12} \\ \implies x\times{y}=\dfrac{n^2}4$

Sabhrant Sachan - 5 years, 1 month ago

Sambrant this is a AM-GM inequality, you cannot use it here. Although the AM makes a little bit of sense here as it splits up the series into two half and your answer would have been correct but only in the case of even set of integers(here it is odd ). But Geometric mean is something else. It does not relate the question given at hand. For an explicit example take the two numbers $2, 4$ it's arithmetic mean is $3$ , while the geometric mean is $2\sqrt{2}$ . You can just interpret that AM>GM but nothing about the product.

Abhay Tiwari - 5 years, 1 month ago

$n$ is odd.

Mateo Matijasevick - 5 years, 1 month ago

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