[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 304

[Sceptic Philozoff]

Let F(x,y) = min( x, y + 1/x, 1/y ), then
F(1/y, 1/x) = min( 1/y, 1/x + y, x ) = F(x,y).
Hence, the minimum if you replace y with 1/x and x with 1/y will not change. Y = 1/x.
So F(x,1/x) = min( x, 2/x, x ) = min( x, 2/x ). It equals x if x < sqrt(2), and 2/x otherwise.
Draw both curves y=x and y=2/x. Obviously, the maximum is exactly at the point of their intersection and equals sqrt(2).
The solution in the problem book is rather vague, I don't like it:


Next (8th):

This part is trivially constructed. Let's leave the intrigue behind.
The second part of the problem (also 8th):

[Sceptic Philozoff]

Another follow-up is geometric (8th):

[richie]

Simplified the figure:

[Sceptic Philozoff]

Richie, how come there are equal angles in a shaded triangle?

[richie]

Mathemat писал(а) >>
Richie, how come there are equal angles in a shaded triangle?

In the central one? It's obvious.

[Sceptic Philozoff]

Go on, then. Prove it.
By the way, the problem statement doesn't say anything about the original triangle being equilateral. Although it is drawn like an equilateral one.

[richie]

Mathemat писал(а) >>
Come on. Prove it.

I haven't figured out how yet. I'll think about it.
-
Well, anyway, the idea was this:
The area of a 4x triangle is equal to one third of the difference of the area of the large triangle and the 4 small triangles, i.e. 4 sq.cm.
To find the area of the large triangle you need to find its side (in the figure - A).
Find the side of the central triangle by area, knowing that it is equilateral is not a problem, it is equal to sqr(4*S/sqr(3)).

[михаил потапыч]

Richie >>:
В центральном? Это очевидно.

Only if the three triangles (apart from the central one) are the same
But that's not a fact according to the conditions

[Sceptic Philozoff]

That's not a fact.
There has to be something to hold on to. There is one lead, but I don't know what to do with it yet.

[richie]

Mischek писал(а) >>

Only if the three triangles (apart from the central one) are identical.
But that's not a fact.

Well you've got me completely confused.
I thought the big triangle was equilateral. The small 3 triangles are equilateral, so as a consequence they are similar.