Fun with triangles

Geometry Level pending

Let $a,b,c$ be the three sides of a scalene triangle such that $a+b+c<1729$ .Now the special property that $a,b,c$ posseses is that for every n (Natural number), $a^{n},b^{n},c^{n}$ also forms sides of a triangle.Find the total number of all such triangles.

The answer is 0.

1 solution

Rajdeep Brahma
Jun 19, 2018

Let,WLOG $a>b>c$ .So $a^{n}>b^{n}>c^{n}$ .Use the trianle inequality $b^{n}+c^{n}>a^{n}$ ,divide each side by $b^{n}$ .So, $\frac{c^n}{b^n}$ +1> $\frac{a^n}{b^n}$ .Now as n tends to $\infty$ ,the L.H.S tends to 1,whereas the RHS tends to $\infty$ ,leading to a contradiction. !!NOTE:THIS IS AN ISI 2018 SUBJECTIVE QUESTION SO NO CLAIM OF ORIGINALITY IS MADE!!

Fun with triangles

The answer is 0.

1 solution

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