An interesting math puzzle for you...?

Question

Let a, b, c be the lengths of the sides of a triangle. Show that:

(a/b+c) + (b/a+c) + (c/a+b) < 2

Now see if you can describe the shape of a triangle for which the above expression is very close to 2.

Any answers, and any explanations?

Have fun with this one.

Answer 1

Thanks for the 'wait', and the formula a/(b+c) ...

Using 3,4,5, I got 65/42 > 1.5, hmm... may be 1.5 could be a minimum...

So, using limits, try a triangle with one infinitely short side.
lim_a->0 (a/(b+c) + b/(a+c) + c/(a+b))
= lim_a->0 (0/(b+c) + b/c + c/b)
but in such a triangle, where a->0, b->c
it is actually lim_a->0_and_b->c (a/(b+c) + b/(a+c) + c/(a+b))
= lim_a->0_and_b->c (0/(b+c) + b/c + c/b)
= lim_a->0_and_b->c (0/(2c) + 2/c + c/c)
= 2

Ok, now for the main question, to prove < 2..
Thanks again for the a a/(b+c) < 1/2 because b+c > 2a
c/(a+b) < 1 because in a triangle, sum of lengths of two sides > the other side.

What about, b/(a+c)? ab but c 1/3 (Oh oh.. no use)

Probably have to consider the first two terms together, we have
a/(b+c) + b/(a+c) = (a^2+ac+b^2+bc)/(b+c)(a+c)
= (a^2+ac+b^2+bc)/(ab+ac+bc+c^2)
< 1
because ac=ac, bc=bc, a^2
So, (a/(b+c) + b/(a+c) + c/(a+b)) < 2.

Answer 2

you should apply a recursive approach. on the tip, the barrel contains below 9 liters of water, because of the fact that some water became into bumped off alongside with the wine the 2nd 2 cases. the 1st removal is 3 liters, leaving you with x-3 liters of wine. the 2nd removal is 3 liters of fluid with a concentration of (x-3)/x. In different words, you have: (x-3) - 3*(x-3)/x Dividing that by making use of x components your new concentration. The 0.33 removal is 3 liters of [(x-3) - 3* (x-3)/x]/x and could leave you with x/2 liters of wine in the barrel or: (x-3) - 3*(x-3)/x -3*[(x-3) - 3* (x-3)/x]/x = x/2. the rest is a few straightforward algebra to simplify the equation right into a cubic formulation with 3 roots. different than no longer so straightforward that i did no longer make an blunders someplace alongside the line and land up with 15.338 liters as a substitute of the main suitable volume of (re-edit:14.542 liters - no ask your self I had a typo in my Excel formulation). Microsoft Excel facilitates you to brute capability those sort of issues, in specific cases with much less possibility of blunders (provided you enter the formulation properly and don't constantly start up with 15 liters - heh).

Answer 3

A= 3
B=4
C=5

[3/4 + 5] + [4/3 + 5] + [5/3 + 4] = 17 3/4 > 2

I guess I just don't understand your question.

Answer 4

the above statement is dimensionally incorrect I feel you mean 1st term as a/(b+c) and so on

Though I cannot prove the above by symmetry the above is maximum when b=a = c and the result is 1.5
2 is very far off

Answer 5

wait...

please bear/bare with me... I don't know what I'm doing...

let c be the longest side
b be the longer side
a be the shorter side

therefore a < b < c

I don't know anymore sorry...

I have to guess 1,3,5

Answer 6

huh.. i failed my recent math paper.. how am i suppose to have fun with this?

Answer 7

i always thought my maths to be strong bu i was wrong!!!!!!

Answer 8

Have FUN with this one? Read: Please do my homework for me.

Answer 9

Fun !!!!!!!!!!! with all that