Cheenta
How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?
Learn More

RMO 2009

In this post, there are problems from Regional Mathematics Olympiad, RMO 2009. Try out these problems.

  1. Let $ABC$ be a triangle in which $AB = AC$ and let $I$ be its in-centre. Suppose $BC = AB + AI$. Find $∠BAC$.
    Discussion
  2. Show that there is no integer a such that $ a^2-3a-19 $ is divisible by 289.
  3. Show that $ 3^{4008}+4^{2009} $ can be written as product of two positive integers each of which is larger than $ 2009^{182} $.
  4. Find the sum of all $3$-digit natural numbers which contain at least one odd digit and at least one even digit.
  5. A convex polygon $\Gamma $ is such that the distance between any two vertices of $ \Gamma $ does not exceed 1.
    • Prove that the distance between any two points on the boundary of $ \Gamma $ does not exceed $1$.
    • If X and Y are two distinct points inside $ \Gamma $, prove that there exists a point Z on the boundary of $ \Gamma $ such that $ XZ+YZ \le 1 $.
  6. In a book with page numbers from $1$ to $100$, some pages are torn off. The sum of the numbers on the remaining pages is $4949$. How many pages are torn off?

Some Useful Links:

RMO 2002 Problem 2 – Fermat’s Last Theorem as a guessing tool– Video

Our Math Olympiad Program

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy
Cheenta

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com