How Cheenta works to ensure student success?
Explore the Back-Story

RMO 2010 | Regional Mathematics Olympiad Problems

In this post, there are problems from Regional Mathematics Olympiad, RMO 2010. Try out these problems.

  1. Let ABCDEF be a convex hexagon in which the diagonals AD, BE, CF are concurrent at O. Suppose the area of triangle OAF is the geometric mean of those of  OAB and OEF; and the area of the triangle OBC is the geometric mean of those of  OAB and OCD. Prove that the area of triangle OED is the geometric mean of those of  OCD and OEF.
  2. Let P_1(x)=ax^2-bx-c , P_2(x)=bx^2-cx-a, P_3(x)=cx^2-ax-b be three quadratic polynomials where a, b, c are non-zero real numbers. Suppose there exits a real number \alphasuch that P_1(\alpha)=P_2(\alpha)=P_3(\alpha). Prove that a=b=c.
  3. Find the number of  4-digit numbers (in base 10) having non-zero digits and which are divisible by 4 but not by 8.
  4. Find three distinct positive integers with the least possible sum such that the sum of the reciprocals of any two integers among them is an integral multiple of the reciprocal of the third integer.
  5. Let ABC be a triangle in which ∠A = 60^{\circ} Let BE and CF be the bisectors of the angles B and C with E on AC and F on AB. Let M be the reflection of  A in the line EF. Prove that M lies on BC.
  6. For each integer 1 ≤ n, define a_n=\frac{n}{\sqrt{n}}] where [x] denotes the largest integer not exceeding x, for any real numbers x. Find the number of all n in the set {1, 2, 3……, 2010} for which a_n > a_{n+1}.

Some Useful Links:

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

Our Math Olympiad Program

In this post, there are problems from Regional Mathematics Olympiad, RMO 2010. Try out these problems.

  1. Let ABCDEF be a convex hexagon in which the diagonals AD, BE, CF are concurrent at O. Suppose the area of triangle OAF is the geometric mean of those of  OAB and OEF; and the area of the triangle OBC is the geometric mean of those of  OAB and OCD. Prove that the area of triangle OED is the geometric mean of those of  OCD and OEF.
  2. Let P_1(x)=ax^2-bx-c , P_2(x)=bx^2-cx-a, P_3(x)=cx^2-ax-b be three quadratic polynomials where a, b, c are non-zero real numbers. Suppose there exits a real number \alphasuch that P_1(\alpha)=P_2(\alpha)=P_3(\alpha). Prove that a=b=c.
  3. Find the number of  4-digit numbers (in base 10) having non-zero digits and which are divisible by 4 but not by 8.
  4. Find three distinct positive integers with the least possible sum such that the sum of the reciprocals of any two integers among them is an integral multiple of the reciprocal of the third integer.
  5. Let ABC be a triangle in which ∠A = 60^{\circ} Let BE and CF be the bisectors of the angles B and C with E on AC and F on AB. Let M be the reflection of  A in the line EF. Prove that M lies on BC.
  6. For each integer 1 ≤ n, define a_n=\frac{n}{\sqrt{n}}] where [x] denotes the largest integer not exceeding x, for any real numbers x. Find the number of all n in the set {1, 2, 3……, 2010} for which a_n > a_{n+1}.

Some Useful Links:

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

Our Math Olympiad Program

Leave a Reply

Your email address will not be published. Required fields are marked *

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

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
Menu
Trial
Whatsapp
Math Olympiad Program
magic-wandrockethighlight