How many positive integers less than \(1000\) have the property that the sum of the digits of each such number is divisible by \(7\) and the number itself is divisible by \(3\) ? Suppose \(a,b\) are positive real numbers such that \(a\sqrt{a}+b\sqrt{b}=183\). \(a\sqrt{b}+b\sqrt{a}=182\). Find \(\frac{9}{5}(a+b)\). A contractor has two teams of workers: team A and […]
Here is the post for the Regional Mathematics Olympiad (India) RMO Number Theory Problems. These are problems from previous year papers. (This is a work in progress. More problems will be added soon). RMO Number Theory Problems: Find all triples (p, q, r) of primes such that pq = r + 1 and 2(p 2 […]
Problems Problem 1 Let \( a, b, c \) be positive real numbers such that $$ \frac{a}{1+a} + \frac{b}{1+b} + \frac{c}{1+c} = 1 $$ Prove that \( abc \leq \frac{1}{8} \)