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 […]
"How do I involve my child in challenging mathematics? He gets good marks in school tests but I think he is smarter than school curriculum." "My daughter is in 4th grade. What competitions in mathematics and science can she participate in? How do I help her to perform well in those competitions?" "I have a […]
What is RMO? RMO or Regional Math Olympiad is the second round of Mathematics Contest in India after the Pre-regional Mathematics Olympiad (Pre-RMO or PRMO) leading to the prestigious International Mathematics Olympiad (IMO). It is held in the month of December (the first Sunday of December). The test is conducted in each of the 19 […]
Problem: Let ABCDE be a regular pentagon inscribed in a circle. P be any point in the minor arc AE. Prove that PA + PC + PE = PB + PD Proof: Suppose length of each side is 's' and each diagonal is 'x'. Apply Ptolemy's Theorem in PABC. We have PA . s + […]