RMO 2011 Re- Test Paper | RMO Problems

1. In this post, here are problems from Regional Mathematics Olympiad, RMO 2011 Re-Test Paper.
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3. Let ABC be an acute angled scalene triangle with circumcenter O and orthocenter H. If M is the midpoint of BC, then show that AO and HM intersect at the circumcircle of ABC.
4. Let n be a positive integer such that 2n + 1 and 3n + 1 are both perfect squares. Show that 5n + 3 is a composite numbers.
5. Let a, b, c > 0. If 1/a , 1/b and 1/c are in arithmetic progression, and if $latex a^2+b^2 , b^2+c^2 , c^2+a^2 $ are in geometric progression, prove that a=b=c.
6. Find the number of 4 digit numbers with distinct digits chosen from the set {0, 1, 2, 3, 4, 5} in which no two adjacent digits are even.
7. Let ABCD be a convex quadrilateral. Let E, F, G, H be midpoints of AB, BC, CD, DA respectively. If AC, BD, EG, FH concur at a point O, prove that ABCD is a parallelogram.
8. Find the largest real constant $latex \gamma $ such that $latex \frac{{\gamma}abc}{a+b+c} \le (a+b)^2 +(a+b+4c)^2 $ for all positive real numbers a, b and c.

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RMO 2002 Problem 2 - Fermat's Last Theorem as a guessing tool- Video

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