Fox 298 - Solutions

Solutions to Fox 298.

by Giannno:
by Newzad:

http://www.8foxes.com/

Fox 297 - Solutions

by Giannno:

by Bleaug:

Start from an arbitrary AOC triangle such that OC > OA.
Set length unit as (OC-OA)/2 and OA as x.

Reasoning is identical to Fox 296. See its solution.

Polar Fox adds:

By definition OB=x+1 => OBH is an isosceles triangle (1).
Also known that, m(BHO)=90 degrees (2).

Nope, aint gonna happen.

i.e. (1) and (2) can't happen at the same time.

http://www.8foxes.com/

Fox 296 - Solutions

Among similar ones, Bleaug finds a contradiction for Fox 296:Take an arbitrary AOC triangle such that OC=3OA. Build B as intersection of triangle's circumscribed circle and AOC internal bisector. B is such that ABC is isoceles, i.e. AB=BC. Symmetry in OABI implies AB=BI, hence IBC is isoceles. Orthogonal projection of B on OC coincides with H such that IH= HC=OA. OHB is rectangle, therefore
OB^2 = (2OA)^2 + BH^2 => OB > 2OA iff angle(AOC) > 0.

Strictly speaking, if angle(AOC)=0 there is a solution if we admit that a straight line is a circle with center at infinity (e.g. in projective plane).

And Giannno solves too:

http://www.8foxes.com/

Fox 225

http://www.8foxes.com/

Himalayan Fox: Sooooooo, this is the one with several lengths that will never exist?

Polar Fox: Yep, neat, isn't it?
Himalayan Fox: Kinda reminds me of "life" itself.
Polar Fox: Come on, this is just a simple problem. You must be over-reading something.
Himalayan Fox: Well, look at it this way; components are meaningless, but together they are telling a story!

Polar Fox: I have no idea what you're talking about, but you make more sense than the Red Fox.

Fox 224

http://www.8foxes.com/

Polar Fox: Dude what's up with that "getting inspired by our foxes" business?
Red Fox: A proper reference should be given when an idea is being used.
Polar Fox: Oh come on, you are becoming paranoic. Who's gonna steal your stupid ideas? And you are not getting inspired by others work, right?
Red Fox: Well sometimes yes.
Polar Fox: And the commercial logos you've messing around with permission, right?
Red Fox: So what?
Polar Fox: Whoever wants to use, let them use, man, AND don't forget your own "public service" crap.
Red Fox: Yeah, whatever. I guess you're right.

Fox 178 - revisited - 1

ORIGINAL PROBLEM:
http://www.8foxes.com/Home/178

The above geometric solution was submitted by Janos of Budapest/Hungary.

Impressive! Enjoy!