This is problem from Chennai Mathematical Institute, CMI 2015 based on Divisibility of product of consecutive numbers. Try it out! a be a positive integer from set {2, 3, 4, … 9999}. Show that there are exactly two positive integers in that set such that 10000 divides a*a-1. Put $ n^2 -1 $ in place of 9999. […]
In a circle, AB be the diameter.. X is an external point. Using straight edge construct a perpendicular to AB from X If X is inside the circle then how can this be done Discussion: Teacher: What fascinates me about CMI problems is that they are at once fundamental and beautiful in nature. This problem […]
This post contains Chennai Mathematical Institute, CMI, 2015 Objective, and Subjective Problems and Solutions. Please contribute problems and solutions in the comments. Objective For all finite word strings comprising A and B only, A string is arranged by dictionary order. eg. ABAA For any arbitrary string w, with another string y<w, there cannot always exist […]
This post contains solutions of Indian Statistical Institute, ISI 2015 BStat - BMath Objective Problems. Try to solve them. This is a work in progress. Candidates, please submit objective problems in the comments section (even if you partially remember them). If you do not remember the options, that is fine too. We can work with […]
Here, you will find all the questions of ISI Entrance Paper 2014 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems. this is a work in progress. post problems, solutions and correction in the comment section Problem 1: Let $ y = x^2 + ax […]
This is a problem from the Indian Statistical Institute, ISI BStat 2006 Subjective Problem 3 based on Sophie Germain Identity. Try to solve it. Problem: Prove that $\mathbf{n^4 + 4^{n}}$ is composite for all values of $n$ greater than $1$. Discussion: Teacher: This problem uses an identity that has a fancy name: Sophie Germain identity. […]
Problem: In the figure below, $E$ is the midpoint of the arc $ABEC$ and the segment $ED$ is perpendicular to the chord $BC$ at $D$. If the length of the chord $AB$ is $\mathbf{\ell_1} $, and that of the segment $BD$ is $\mathbf{\ell_2} $, determine the length of $DC$ in terms of $\mathbf{\ell_1, \ell_2} $. […]
Problem: Find all real solutions of the equation $sin^{5}x+cos^{3}x=1$ . Discussion: Teacher: Notice that $|\sin x| \leq 1 , |\cos x | \leq 1 $ . So if you raise $\sin x$ and $\cos x$ to higher powers you necessarily lower the value. Take for example the number $\frac{1}{2}$. If you raise that to the […]
This is problem number 5 from Indian Statistical Institute, ISI BStat 2005 based on Geometric inequality. Consider an acute angled triangle $PQR$ such that $C,I$ and $O$ are the circumcentre, incentre and orthocentre respectively. Suppose $\displaystyle{ \angle QCR, \angle QIR } $ and $ \displaystyle{ \angle QOR } $, measured in degrees, are $ \displaystyle{ […]
Gauss called it the 'fundamental theorem' and published 6 proofs of it. Since then quadratic reciprocity has been an obsession of the mathematical community. Over 200 proofs has been published. I encountered a very simple and elegant proof. Here is a pdf file with a simple 2-page proof. quadratic reciprocity
Try this beautiful problem from the Pre-RMO, 2019 based on Sum of digits. You may use sequential hints to solve the problem.
Try this beautiful problem from Geometry: Sectors in Circle from AMC-10A, 2012. You may use sequential hints to solve the problem
Try this beautiful problem from Algebra: Sum of whole numbers from AMC-10A, 2012. You may use sequential hints to solve the problem
Try this beautiful problem from Singapore Mathematics Olympiad, SMO, 2009 based on Trigonometry Simplification. You may use sequential hints.
Try this beautiful problem from Geometry: Area of quadrilateral from AMC-10A, 2020. You may use sequential hints to solve the problem.
Try this beautiful problem from the Pre-RMO, 2019 based on Sum of digits. You may use sequential hints to solve the problem.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1993 based on Smallest positive Integer.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1993 based on Equation of X and Y.
Try this beautiful problem from PRMO, 2018 based on Algebra: Inequality You may use sequential hints to solve the problem.
Try this beautiful problem from PRMO, 2019, problem-4, based on Geometry: Direction & Angles. You may use sequential hints to solve the problem.