The equation $latex x^3 + 10x^2 - 100x + 1729 $ has at least one complex root α such that |α| > 12. False ** Discussion: A fun fact : 1729 is the Ramanujan Number; it is the smallest number expressible as the sum of two cubes in two different ways We conduct normal extrema tests. First […]
The equation $latex x^3 + 3x - 4 $ has exactly one real root. True Discussion: Consider the derivative of the function $latex f(x) = x^3 + 3x - 4 = 0 $ . It is $latex 3x^2 + 3 $ . Note that the derivative is strictly positive ( positive times square + positive is […]
Problem: Every differentiable function f: (0, 1) --> [0, 1] is uniformly continuous. Discussion; False Note that every differentiable function f: [0,1] --> (0, 1) is uniformly continuous by virtue of uniform continuity theorem which says every continuous map from closed bounded interval to R is uniformly continuous. However in this case the domain is […]
Problem: Let f: R --> R be defined by $latex f(x) = sin (x^3) $. Then f is continuous but not uniformly continuous. Discussion: True It is sufficient to show that there exists an $latex epsilon > 0 $ such that for all $latex \delta > 0 $ there exist $latex x_1 , x_2 \in […]
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 […]