RMO 2009

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

Let $ABC$ be a triangle in which $AB = AC$ and let $I$ be its in-centre. Suppose $BC = AB + AI$ . Find $∠BAC$ .
Discussion
Show that there is no integer a such that $a^2-3a-19$ is divisible by 289.
Show that $3^{4008}+4^{2009}$ can be written as product of two positive integers each of which is larger than $2009^{182}$ .
Find the sum of all $3$ -digit natural numbers which contain at least one odd digit and at least one even digit.
A convex polygon is such that the distance between any two vertices of 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 $XZ+YZ \le 1$ .
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

Related

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

Let $ABC$ be a triangle in which $AB = AC$ and let $I$ be its in-centre. Suppose $BC = AB + AI$ . Find $∠BAC$ .
Discussion
Show that there is no integer a such that $a^2-3a-19$ is divisible by 289.
Show that $3^{4008}+4^{2009}$ can be written as product of two positive integers each of which is larger than $2009^{182}$ .
Find the sum of all $3$ -digit natural numbers which contain at least one odd digit and at least one even digit.
A convex polygon is such that the distance between any two vertices of 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 $XZ+YZ \le 1$ .
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

Related

Leave a ReplyCancel reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

One comment on “RMO 2009”

neelanjan mondal says:

December 1, 2014 at 3:55 am

Please tell me the solutions of the problems 5,rmo-2009

Reply

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy

Cheenta Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.

HALL OF FAME BLOG OFFLINE CLASS

CHEENTA ON DEMAND BOSE OLYMPIAD

Cheenta Academy Private Limited |

support@cheenta.com

Menu

Math Olympiad Program