Find all triples $latex (p, x, y) $ such that $latex p^x = y^4 + 4 $, where $latex p $ is a prime and $latex x, y $ are natural numbers. Hint 1: p cannot be 2. For if p is 2 then $latex p^x $ is even which implies $latex y^4 + 4 […]
Find all triples $latex (p, x, y) $ such that $latex p^x = y^4 + 4 $, where $latex p $ is a prime and $latex x, y $ are natural numbers. Hint 1: p cannot be 2. For if p is 2 then $latex p^x $ is even which implies $latex y^4 + 4 […]
This post contains Indian National Mathematical Olympiad, INMO 2010 questions. Try to solve these problems and share it in the comments. Let ABC be a triangle with circum-circle $ \Gamma $.Let M be a point in the interior of the triangle ABC which is also on the bisector of $ \angle A $. Let AM, […]
This post contains the problems from Indian National Mathematics Olympiad, INMO 2009 Question Paper. Do try to find their solutions. Indian National Mathematics Olympiad (INMO) 2009 Question Paper: Let ABC be a triangle and P be a interior point such that $ \angle BPC $=$ 90^0 $, $ \angle BAP $ = $ \angle BCP […]
Let $ABC$ be a triangle, $I$ its in-centre; $ A_1, B_1, C_1 $ be the reflections of $I$ in BC, CA, AB respectively. Suppose the circumcircle of triangle $ A_1 B_1 C_1 $ passes through A. Prove that $ B_1, C_1, I, I_1 $ are concyclic, where $ I_1 $ is the incentre of triangle […]
This post contains the problems from Indian National Mathematics Olympiad, INMO 2008 Question Paper. Do try to find their solutions. Let $ABC$ be a triangle, $I$ its in-centre; $ A_1, B_1, C_1 $ be the reflections of $I$ in BC, CA, AB respectively. Suppose the circum-circle of triangle $ A_1 B_1 C_1 $ passes through […]
This post contains a problem from TIFR 2013 Math paper D based on Inequality of square root function. The inequality $ \sqrt {n+1} - \sqrt n < \frac {1}{\sqrt n } $ is false for all in n such that $ 101 \le n \le 2000 $ False Discussion: $ \sqrt {n+1} - \sqrt n […]
Any automorphism of the group Q under addition is of the form x → qx for some q ∈ Q. True Discussion: Suppose f is an automorphism of the group Q. Let f(1) = m (of course 'm' will be different for different automorphisms). Now $f(x+y) = f(x) + f(y)$ implies $f(x) = mx$ where m […]
Test of Mathematics Solution Subjective 35 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance. Also see: Cheenta I.S.I. & C.M.I. Entrance Course Problem (a) Prove that, for any odd integer n, $ […]
Problem 1 . A shop sells two kind of products A and B. One day a salesman sold both A and B at the same price, $2100$ to a customer. Suppose A makes a profit of 20% and B makes a loss of 20%. Then the deal(A) make a profit of $70$; (B) make a […]