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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 $ \alpha $such 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

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