Instinctive answer

Geometry Level 3

$\large \left (1+\frac{b-c}{a} \right)^{a}+\left (1+\frac{c-a}{b}\right)^{b}+\left (1+\frac{a-b}{c}\right)^{c}=1$

Let $a,b,c$ be sides of a triangle. then find the number of triangles possible such that the equation above is fulfilled.

The answer is 0.

1 solution

Arko Bhaumik
May 26, 2014

We know that in any triangle the sum of the lengths of two sides is greater than the third side. Here, for one case, (a+c)>b. Thus, rearranging, we get than (b-c)/a<1. Similarly, in other two cases, (c-a)/b and (a-b)/c are also less than unity. Hence each of the terms without the exponents turns out to be a fraction greater than 1. Thus there cannot exist any set of exponents a,b,c such that the sum of each term raised to the power of a,b,c respectively equals 1. In other words, the number of such triangles is 0.

my instings didnt fail me

math man - 6 years, 8 months ago

Instinctive answer

The answer is 0.

1 solution

0 pending reports