This a problem from from National Board for Higher Mathematics (NBHM) 2013 based on Cyclic Group Problem for M.Sc Students. Which of the following statements are true? Every group of order 11 is cyclic. Every group of order 111 is cyclic. Every group of order 1111 is cyclic. Discussion: Every group of order 11 is […]
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 […]
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, $ […]
12. In $ (\triangle ABC, AB=AC=28)$ and BC=20. Points D,E, and F are on sides $ (\overline{AB}, \overline{BC})$, and $ (\overline{AC})$, respectively, such that $ (\overline{DE})$ and $ (\overline{EF})$ are parallel to $ (\overline{AC})$ and $ (\overline{AB})$, respectively. What is the perimeter of parallelogram ADEF?$ (\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }56\qquad\textbf{(D) }60\qquad\textbf{(E) }72\qquad )$ Solution: Perimeter = […]
4. Let N be an integer greater than 1 and let $ (T_n)$ be the number of non empty subsets S of ({1,2,.....,n}) with the property that the average of the elements of S is an integer. Prove that $(T_n - n)$ is always even. Sketch of the Proof: $ (T_n )$ = number of […]
3 Let $ (a,b,c,d \in \mathbb{N})$ such that $ (a \ge b \ge c \ge d)$. Show that the equation $ (x^4 - ax^3 - bx^2 - cx -d = 0)$ has no integer solution. Sketch of the Solution: Claim 1: There cannot be a negative integer solution. Suppose other wise. If possible $x= […]