Embibe Experts Solutions for Chapter: Sequence and Series, Exercise 1: IOQM - PRMO and RMO 2019 (11-08-2019)

Let $x_1$ be a positive real number and for every integer $n\ge 1$ let $x_{n+1}=1+x_1{x}_2\dots x_{n-1}{x}_n.$ If $x_5=43,$ what is the sum of digits of the largest prime factor of $x_6$?

A natural number $k>1$ is called good if there exist natural numbers $a_1<a_2<\dots <a_k$ such that $\frac{1}{\sqrt{a_1}}+\frac{1}{\sqrt{a_2}}+\dots +\frac{1}{\sqrt{a_k}}=1$. Let $f\left(n\right)$ be the sum of the first $n$ good numbers, $n\ge 1$. Find the sum of all values of $n$ for which $\frac{f\left(n+5\right)}{f\left(n\right)}$ is an integer.

Consider the set $E$ of all positive integers $n$ such that when divided by $9,10,11$, respectively, the remainders (in that order) are all $>1$ and form a non-constant geometric progression. If $N$ is the largest element of $E$, then find $\frac{N}{4}$.

Let $N=6+66+666.+\dots +666\dots 66,$ where there are hundred $6 \text{'}s$ in the last term in the sum. How many times does the digit $7$ occur in the number $N$ ?