Let's have a problem discussion of Geometry problems in AIME. (American Invitational Mathematics Competitions). Give them a try. In $ \Delta ABC, AB = 3, BC = 4 $, and CA = 5. Circle $ \omega $ intersects $ \overline{AB} at E and B, \overline{BC} $ at B and D, and $ \overline{AC} $ at […]
Here is a very interesting problem based on counterfeit coin. There are five coins three of them are good one of them is heavier and one of them is lighter. It is not given whether the amount of extra weight in the heavier coin is the same as the amount of lost weight in the […]
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 […]