This is Problem number 7 from the ISI Subjective Entrance Exam based on the Arithmetic Sequence of reciprocals. Try to solve the problem. Let $ m_1, m_2 , ... , m_k $ be k positive numbers such that their reciprocals are in A.P. Show that $ k< m_1 + 2 $ . Also find such […]
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{ […]
Try this beautiful problem from AMC-10A, 2004 based on ratio of two triangles.You may use sequential hints to solve the problem.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1994 based on Remainders and Functions.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1992 based on Rational Numbers.
Try this beautiful problem from Algebra based on real numbers from PRMO 2017. You may use sequential hints to solve the problem.
Try this beautiful problem from AMC 10A, 2004 based on Mensuration: Cylinder. You may use sequential hints to solve the problem.
Try this beautiful problem from AMC-10A, 2004 based on Triangle. You may use sequential hints to solve the problem.
Try this beautiful problem from Number Theory based on largest possible value from AMC-10A, 2004. You may use sequential hints to solve the problem.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1994 based on Points of Equilateral triangle.
Try this beautiful problem from the Pre-RMO, 2017 based on ratio and proportion. You may use sequential hints to solve the problem.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1994 based on Complex roots and equations.