Inspired by Khang Nguyen Thanh

Geometry Level pending

Let $ABCD$ be a cyclic quadrilateral. Denote the area by $A$ and the perimeter by $P$ . Given that $\frac{A}{P} = 2000$ , find the minimum value of $P$ .

The answer is 32000.

2 solutions

Kenny Lau
Sep 9, 2015

The minimum is when it is a square (pure guess).

Then let the side of the square be $x$ .

$A=x^2$ and $P=4x$ .

$\dfrac{x^2}{4x}=2000 \implies x=8000 \implies P=32000$

APPENDIX: Guessing is only for the lazy people! xd

Let me ask you a counter-question, what is the greatest volume of a cylinder inscribed in a sphere of radius 5? Round to the nearest integer. Can you guess it?

Alan Yan - 5 years, 9 months ago

I would anti-spin the figure to obtain a rectangle in a circle. Then the rectangle would be guessed to a square.

Kenny Lau - 5 years, 9 months ago

Unfortunately, that is incorrect. The maximum volume is not when it is a square. Let $h$ be the height and $r$ be the radius of this cylinder. You know that $V = h\pi r^2$ and $(\frac{h}{2})^2+r^2 = 25$ . Substituting, you get that $V = h\pi(25 - (\frac{h}{2})^2)$ which implies $V$ is a function on the height. We can find the maximum for the interval $h \in [0 , 10]$ by taking the derivative or you can check by graphing to get a number about $302$ .

If you assume it is a square, you only get a number about $277$ .

Although guessing can be a good way to get the answer quickly, sometimes it doesn't work.

And at least don't post a straight guessing solution. Anyone else could've assumed it was a square and still have gotten the right answer.

Alan Yan - 5 years, 9 months ago

@Alan Yan – Come on. Nobody is saying that guessing works every time. But this is what I do in competitions. And who said that guessing solutions are not allowed?

Kenny Lau - 5 years, 9 months ago

@Kenny Lau – I'm not saying they are not allowed and I also using guessing at competitions, but in this scenario where you are given unlimited time to fully understand and try to solve a question, a striaght on guessing solution is unneeded to be written down and in this case, undesired.

Alan Yan - 5 years, 9 months ago

@Alan Yan – Satisfied?

Kenny Lau - 5 years, 9 months ago

@Kenny Lau – Uh...not really. You really didn't need to do that. I just wanted to get my point across. But we are here for problem solving and not arguing, so just keep on solving problems!

Alan Yan - 5 years, 9 months ago

@Alan Yan – Thank you. I understand your point.

Kenny Lau - 5 years, 9 months ago

Alan Yan
Sep 8, 2015

$A = \sqrt{(s-a)(s-b)(s-c)(s-d)}$ $P = 2s$ $(s-a) + (s-b) + (s-c) + (s-d) \geq 4\sqrt[4]{(s-a)(s-b)(s-c)(s-d)}$ $2s = P \geq 4\sqrt{A}$ $P^2 \geq 16A$ $P \geq 16 \cdot \frac{A}{P} = 16 \cdot 2000 = \boxed{32000}$

The minimum is obtained when $a = b = c = d = 8000$ .

Inspired by Khang Nguyen Thanh

The answer is 32000.

2 solutions

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