Fox 356

Tony García has sent this one.  The relationship may be simplified.  At least, a more concise expression may be found - with your help.

http://www.8foxes.com

The question is still open:

Can r be written as function of a only?

Fox 355

Roland Sampy submited this one.  Let's confirm his answer.
Happy new year to all, by the way!

Fox 353

Tony García‏ from Dominican Republic submitted this one.  It looks like it requires brute force.  Better than being idle I guess!  So enjoy...

The answer is not confirmed yet.

http://www.8foxes.com

Fox 343

x, y, and z are perpendicular to the chord.

What makes a man a man
is not the fire
nor the bullets,

but pure,
simple,
childish forgiveness.

-- Dervish Fox

Fox 330 - Solutions

Below are three solutions to Fox 330.

By Six:




By Bleaug:

By Henk Reuling:

http://www.8foxes.com/

Fox 330

http://www.8foxes.com/

Fox 319 - Solutions

Below are 2 distinct solutions for Fox 319...

A solution based on symmetry by Bleaug:


Trigonometry and calculus by Six:

x + y + z = 360 and sinx*siny*sinz

-> f(x,y) = sin(x)*sin(y)*sin(360-x-y)
-> f(x,y) = sin(x)*sin(y)*-sin(x+y)

Now to find the critical points of f(x,y), we just need to find the partial derivative with respect to x and y and solve for 0.

f_x(x,y) =

sin(x)sin(y)(-cos(x+y)-cos(x)sin(y)sin(x+y)

solve for 0.

sin(x)sin(y)(-cos(x+y)-cos(x)sin(y)sin(x+y)=0

-> tan(x) = -tan(x+y)

Since the function is symmetric, we should get the same partial derivative for y.

-> tan(y) = -tan(y+x)

-> tan(x)=tan(y)

-> x = y or they are opposites. However, if they are opposites, the original function just becomes 0. Thus, x = y.

Now substitute in x for y in the original equation and find its critical points.

Eventually, you will get sin(3x)=0

x = 120 degrees
y = 120 degrees
z = 120 degrees

Answer (D)

http://www.8foxes.com/

Fox 316 and 318 - Solutions

Both solutions to 316 and 318 are by Bleaug. Also César Lozada has commented out a very similar solution to 316. Obviously, the maximum area is achieved when P is the mid-point, which is also the case for 321. But for Fox 322, it is not that obvious. That's why it still remains as unsolved. Thank you all for the good work.

Fox 316 Solution:
Bleaug:
"I couldn't get a pure geometric demonstration of these two problems. At best, with some lazy observations you are able to show that for both #316 and #318, maximum area is achieved for P midpoint of BC:
a) angle(QPR) in P remains constant when P varies
b) PR/PB is constant
c) PQ/PC is constantHence area(PQR) is proportional to PB.PC which is maximized when P is the midpoint of BC.
The rest is obtained by calculus."



Fox 318 Solution:

http://www.8foxes.com/

Fox 321

In line with Foxie 316 and 318 ...
http://www.8foxes.com/

Fox 319

http://www.8foxes.com/

Fox 318

http://www.8foxes.com/

Dervish Fox: I see some beauty between this one and Fox 316.

Red Fox: Would you please help me see that beauty?

Dervish Fox: The term "sin(B)+sin(C)-sin(A)" has evolved into "sin(B+C-A)" in this one. And the rest is identical.

Red Fox: And you found that beautiful?

Dervish Fox: Both answers are very different and very similar at the same time. Don't you feel anything about that?

Red Fox: I am a rational guy. I depend on my intelligence only. I can easily prove both and there is nothing magical or irrational about that. Everything's explainable. Pure and simple!

Dervish Fox: And is that all you can be? Is that the only direction you can grow? Is that who you are, or are you more than that? Why walk when you can fly? Why mutter when you can sing? Why scribble when you can paint?

Red Fox: I think I am starting to feel little bit of annoyance.

Dervish Fox: Not bad for a start my friend, not bad at all !

Fox 317

Based on our earlier analysis, the sample picture below will never happen.

By "equal-separation" we mean 120-120-120 split.
Otherwise, some may consider 12:00:00 as "equally-separated".

Peace out!

http://www.8foxes.com/

Fox 316

http://www.8foxes.com/

Fox 315

http://www.8foxes.com/

Polar Fox: I have a problem with this one!

Red Fox: What's wrong with it?

Polar Fox: Nothing moves continuously?

Red Fox: Come again?!

Polar Fox: The universe is discrete! THERE IS NO CONTINUUM.

Red Fox: Then how do the things move, flow, or slide?

Polar Fox: Nothing moves! Absolutely nothing moves!!

Red Fox: But the time ticks away, no? Clock arms DO advance.

Polar Fox: No, they don't! But they die in one instance and resurrect in the next one. In between no measurable time passes. Matter oscillates between existence and non-existence continuously.

Red Fox: Continuously? Isn't that ironic?

Polar Fox: Between any two existence, there is nothing but emptiness.

Red Fox: I am having the feeling that your intelligence fall into non-existing state just now.

Polar Fox: Hold on. I think I am about to jump back into the reality. Wait a sec.

Red Fox: Take your time dude. Just take your time!

Fox 311 - Solution

This good geometric solution to Fox 311 is by Newzad.

http://www.8foxes.com/

Fox 311

A natural extension of Fox 3.

The claim may inspire purely-geometric solutions, but looks hard.

http://www.8foxes.com/

Fox 300 - Solutions

Bleaug:

Newzad:

Giannno:

http://www.8foxes.com/

Fox 300

We have been delaying this for a while, but Ajit and Vihaan have already solved it. So there is no point of keeping it a secret :)
Note that this may be a general case for Foxes: 296, 297, and 298. It is also related to Fox 301.

It would be great if there are purely geometric solutions!!

http://www.8foxes.com/

Fox 182 - Solution

Here are two solutions to Fox 182 by Giannno.

Analytic Geometry:

Pythagoras and quadratics:

http://www.8foxes.com/

Fox 295

Let the one who lives, live with an evidence,

Let the one who dies, perish with an evidence.

http://www.8foxes.com/