This post contains problems from Indian National Mathematics Olympiad, INMO 2012. Try them and share your solution in the comments. Problem 1 Let ABCD be a quadrilateral inscribed in a circle. Suppose AB =$latex \sqrt {2 + \sqrt {2} } $ and AB subtends 1350 at the center of the circle. Find the maximum possible […]
This post contains problem from Indian National Mathematics Olympiad, INMO 2011. Try them out and share your solution in the comments. Let D, E, F be points on the sides BC, CA, AB respectively of a triangle ABC such that BD = CE = AF and ∠BDF = ∠CED = ∠AFE. Prove that ABC […]
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 […]
View the other sections of this test. Algebra || Geometry Try a online trial session of Cheenta I.S.I. M.Math, IIT JAM, TIFR Entrance Program. Mail us at helpdesk@cheenta.com View the other sections of this test. Algebra || Geometry Try a online trial session of Cheenta I.S.I. M.Math, IIT JAM, TIFR Entrance Program. Mail us at […]
Section 3: Geometry View the other sections of this test. Algebra || Analysis Try an online trial session of Cheenta I.S.I. M.Math, IIT JAM, TIFR Entrance Program. Mail us at helpdesk@cheenta.com Find the reflection of the point (2, 1) with respect to the line x=y in the xy-plane. Find the area of the circle in […]
Section 1: Algebra View the other sections of this test. Geometry || Analysis Try a online trial session of Cheenta I.S.I. M.Math, IIT JAM, TIFR Entrance Program. Mail us at helpdesk@cheenta.com Which of the following statements are true? Every group of order 11 is cyclic. Every group of order 111 is cyclic. Every group of […]
Any normal subgroup of order 2 is contained in the center of the group. True Discussion: Center of a group Z(G) is the sub group of elements that commute with all members of the group. A subgroup of order two has two elements: identity element and another element, say x, which is self inverse. Since […]
There is an element of order 51 in the multiplicative group (Z/103Z) True Discussion: First note that (Z/103Z) has 102 elements as 103 is a prime (in fact one of the twin primes of 101, 103 pair). Also 102 = 2317. So it has Sylow-3 subgroup of order 3 (prime order hence it is cyclic […]
All non-trivial proper subgroups of (R, +) are cyclic. False Discussion: There is a simple counter example: (Q, +) (the additive group of rational numbers). We also note that every additive subgroup of integers is cyclic (in fact they are of the for nZ). Cyclic groups have exactly one generator. We can construct numerous counter […]