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ISI 2021 | PCB Mathematics | Question: 2

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Consider three real numbers $a \geq b \geq c>0$. If $\left(a^{x}-b^{x}-c^{x}\right)(x-2)>0$ for any rational number $x \neq 2$, show that

  1. $a, b$ and $c$ can be the lengths of the three sides of a triangle $A B C;$
  2. $A B C$ is a right-angled triangle.
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ISI 2021 | PCB Mathematics | Question: 6
Let $p$ be an odd prime and let $n=(p-1)(p+1)$. Show that $p$ divides $n 2^{n}+1$. Show that there are infinitely many integers $m$ such that $p$ divides $m 2^{m}+1$.
Let $p$ be an odd prime and let $n=(p-1)(p+1)$.Show that $p$ divides $n 2^{n}+1$.Show that there are infinitely many integers $m$ such that $p$ divides $m 2^{m}+1$.
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