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

Find the number of ordered triples $\left(a,b,c\right)$ of positive integers such that $30a+50b+70c\le 343.$

What is the greatest integer not exceeding the sum $\sum _{n=1}^{1599}\frac{1}{\sqrt{n}}?$

Determine the sum of all possible positive integers $n,$ the product of whose digits equals $n^2-15n-27.$ 

What is the largest positive integer $n$ such that $\frac{a^2}{{\displaystyle \frac{b}{29}}+{\displaystyle \frac{c}{31}}}+\frac{b^2}{{\displaystyle \frac{c}{29}}+{\displaystyle \frac{a}{31}}}+\frac{c^2}{{\displaystyle \frac{a}{29}}+{\displaystyle \frac{b}{31}}}\ge n\left(a+b+c\right)$ holds for all positive real numbers $a, b, c$