This is a problem from Regional Mathematics Olympiad, RMO 2011 Problem 1 based on the angles of a triangle. Try to solve it out!
This is a problem from Regional Mathematics Olympiad, RMO 2011 Problem 1 based on the angles of a triangle. Try to solve it out!
This post contains RMO 2007 problems. Try to solve these problems Let ABC be an acute-angled triangle; AD be the bisector of angle BAC with D on BC, and BE be the altitude from B on AC. Show that $ \angle CED > 45^\circ $ . [weightage 17/100] Let a, b, c be three natural […]
Overview of Math Olympiads in United States The American Mathematics Competitions (AMC) are the first of a series of competitions in middle school and high school mathematics that lead to the United States team for the International Mathematical Olympiad (IMO). AMC has three levels: AMC 8 - grade 8 and below AMC 10 - grades 10 and […]
Day 1 - 03 January 2014 1 Let be two positive sequences defined by and for all . Prove that they are converges and find their limits. 2 Given the polynomial where is a positive integer. Prove that can't be written as a product of non-constant polynomials with integer coefficients. 3 Given a regular […]
Problem 1In a triangle $A B C,$ let $D$ be a point on the segment $B C$ such that $A B+B D=A C+C D .$ Suppose that the points $B, C$ and the centroids of triangles $A B D$ and $A C D$ lie on a circle. Prove that $A B=A C .$Solution […]
This is a problem from Indian National Mathematics Olympiad, INMO, 2013 based on Orthocenter on perpendicular bisector. Try out this problem. Problem: Orthocenter on perpendicular bisector In an acute angled triangle ABC with AB < AC the circle $latex \Gamma $ touches AB at B and passes through C intersecting AC again at D. Prove […]
Find the number of 4-tuples (a,b,c,d) of natural numbers with $latex a \le b \le c $ and $latex a! + b! + c! = 3^d $ Discussion: Number of 4-tuples The basic idea is: factorial function is faster than the exponential function in the long run. Note that all three of a, b, c […]
Find the number of 8 digit numbers sum of whose digits is 4. Discussion: Suppose the number is $latex a_1 a_2 a_3 ... a_8 $.The possible values of $latex a_1 $ are 1, 2, 3, 4. We consider these four cases. If $latex a_1 = 4 $ then all other digits are 0 (since sum […]
In this post, there are questions from Regional Math Olympiad 2013. Try out the problems. Find the number of 8 digit numbers sum of whose digits are 4.Discussion Find the number of 4-tuples (a,b,c,d) of natural numbers with $latex a \le b \le c $ and $latex a! + b! + c! = 3^d […]