Embibe Experts Solutions for Chapter: Polynomials, Exercise 1: IOQM - PRMO and RMO 2018

$a, b, c$ are positive real numbers such that $a^2+b^2=c^2$ and $ab=c.$ Determine the value of $\left|\frac{(a+b+c)(a+b-c)(b+c-a)(c+a-b)}{c^2}\right|.$

Find the number of ordered triples $(x,y,z)$ of real numbers that satisfy the system of equation $x+y+z=7; x^2+y^2+z^2=27; xyz=5$

Each of the numbers $x_1, x_2, \dots , x_{101}$ is $\pm 1.$ What is the smallest positive value of $\sum _{1\le i\le j\le 101}{x}_i{x}_j ?$

If $x=\sqrt{2}+\sqrt{3}+\sqrt{6}$ is a root of $x^4+a{x}^3+b{x}^2+cx+d=0$, where $a, b, c, d$ are integers, what is the value of $\left|a+b+c+d\right|$ ?

Find the number of positive integers less than $101$ that cannot be written as the difference of two squares of integers.

Find the largest value of $a^b$ such that the positive integers $a, b>1$ satisfy $a^b{b}^a+a^b+b^a=5329$.

What is the smallest prime number $p$ such that $p^3+4p^2+4p$ has exactly $30$ positive divisors?

Let $P\left(x\right)=a_0+a_{1}x+a_2{x}^2+\dots +a_{\mathit{n}}{x}^{\mathit{n}}$ be a polynomial in which $a_{\mathit{i}}$ is a non-negative integer for each $i\in \left\{0,1,2,3,\dots ,n\right\}$. If $P\left(1\right)=4$ and $P\left(5\right)=136$, what is the value of $P\left(3\right)$?