Let's try to find the solution to Duke Math Meet 2008 Problem 8. This question is from the individual round of that meet. Problem: Duke Math Meet 2008 Problem 8 Find the last two digits of $ \sum_{k=1}^{2008} k {{2008}\choose{k}} $ Discussion: $ (1+x)^n = \sum_{k=0}^{n} {{n}\choose{k}}x^k $ We differentiate both sides to have $ […]
Try this Remainder problem from Duke Math Meet 2009 Problem 9. This problem is from the Team Round of the meet. Problem: Duke Math Meet 2009 Problem 9 What is the remainder when $ 5^{5^{5^5}} $ is divided by 13 ? By Fermat's Little Theorem $ 5^{12} = 1 \mod 13 $ Now if we […]
Try this problem from Duke Math Meet 2009 Problem 7 based on the number of ordered triples. This problem was asked in the team round. How many ordered triples of integers (a, b, c) are there such that $ 1 \le a, b, c \le 70 $ and $ a^2 + b^2 + c^2 $ […]
Try this problem from Duke Math Meet 2009 Problem 6 based on Count of Sparse Subsets. This problem was asked in the team round. Call a set S sparse if every pair of distinct elements of S differ by more than 1. Find the number of sparse subsets (possibly empty) of {1, 2, . . […]
This post contains problems from the first relay round of the Duke Math Meet 2009. Try to solve these problems.
Try this problem from Duke Math Meet 2009 Problem 7 based on Area of Ellipse. This problem was asked in the individual round.
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 […]