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
This is an objective problem 151 from TOMATO based on Consecutive composites, useful for Indian Statistical Institute Entrance Exam. Let $n = 51! + 1$. Then the number of primes among $n+1, n+2, ... , n+50$ is (A) $0$; (B) $1$; (C) $2$; (D) more than $2$; Discussion: $51!$ is divisible by $2, 3,... 51$. […]
If three prime numbers, all greater than $3$, are in A.P. , then their common difference (A) must be divisible by $2$ but not necessarily by $3$; (B) must be divisible by $3$ but not necessarily by $2$; (C) must be divisible by both $2$ and $3$; (D) need not be divisible by any of […]
Let N be a positive integer not equal to 1. Then note that none of the numbers 2, 3, ... , N is a divisor of (N! -1). From this we can conclude that: (A) (N! - 1) is a prime number; (B) at least one of the numbers N+1 , N+2 , ...., N! […]
The number $1000! = 1.2.3...1000$ ends exactly with (A) $249$ zeroes; (B) $250$ zeroes; (C) $240$ zeroes; (D) $200$ zeroes; Discussion: To find the number of zeroes at the end of n! we just need to figure out the number of 5's occurring in prime factorization of it. Why? Because there are much more 2's […]
This post consists of Problems and solutions from TIFR 2013 Paper. Try to solve them and then read their solutions. TIFR 2013 Paper PART A (Linear and Abstract Algebra) Problem 1 Problem 2 - Automorphism of the Additive Group of Rationals Problem 3 - Existence of Real Root Problem 4 - Existence of Complex Root […]
This is an (ever-growing and ever-changing) list of books, useful for school and college mathematics students. If you are working toward Math Olympiad, I.S.I., C.M.I. entrance programs or intense college mathematics, these books may prove to be your best friend. If you are taking a Cheenta Advanced Math Program, chances are that you will referred […]
Let's discuss this objective problem number 14 from ISI BStat BMath. Try to solve the problem and then read their solution. Problem 14 f(x) = tan(sinx) (x > 0) To understand the graph of a function, easiest and the most proper method is to apply techniques from calculus. We will quickly compute, derivative and second […]
This post contains the six Indian National Maths Olympiad, INMO 2015 problems. Try to solve these problems. Let ABC be a right-angled triangle with $ \angle{B}=90^{\circ} $. Let BD is the altitude from B on AC. Let P, Q and Ibe the incenters of triangles ABD, CBD, and ABC respectively. Show that circumcenter of triangle […]
Let's understand the factor method of Diophantine equations step-by-step. Aso, try the question related to it. Diophantine Equations Consider an equation for which we seek only integer solutions. There is no standard technique of solving such a problem, though there are some common heuristics that you may apply. A simple example is $ x^2 - […]
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.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1994 based on Length and Inequalities.
Try this beautiful problem from the Pre-RMO, 2017 based on Trigonometry & natural numbers. 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 Right angled triangle.
Try this beautiful problem from Probability: positive factors AMC-10A, 2003. You may use sequential hints to solve the problem
Try this beautiful problem from AMC 10A, 2007 based on Numbers on cube. You may use sequential hints to solve the problem.
Let's discuss a problem from CMI Entrance Exam 2019 based on the Inscribed Angle Theorem or Central Angle Theorem and Transformation Geometry. The Problem: Let $A B C D$ be a parallelogram. Let 'O' be a point in its interior such that $\angle A D B+\angle D O C=180^{\circ}$. Show that $\angle O D C=\angle […]
Problems and Solutions of ISI MStat Entrance 2020 of Indian Statistical Institute.
In this post, we will be learning about the Rational Root Theorem Proof. It is a great tool from Algebra and is useful for the Math Olympiad Exams and ISI and CMI Entrance Exams. So, here is the starting point.... $a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{2} x^{2}+a_{1} x+a_{0}$ This polynomial has certain properties. 1. The coefficients are all […]
This is a solution to a problem from American Mathematics Competition (AMC) 8 2020 Problem 18 based on Geometry. AMC 8 2018 Problem 24 In the cube $ABCDEFGH$ with opposite vertices $C$ and $E,$ $J$ and $I$ are the midpoints of edges $\overline{FB}$ and $\overline{HD},$ respectively. Let $R$ be the ratio of the area of […]
This is a solution to a problem from American Mathematics Competition (AMC) 8 2020 Problem 18 based on Geometry. AMC 8 2020 Problem 18 Rectangle $A B C D$ is inscribed in a semicircle with diameter $\overline{F E}$ as shown in the figure. Let $D A=16$, and let $F D=A E=9 .$ What is the […]
Mann Shah is a Gold Awardee in HKIMO (Hong Kong International Mathematical Olympiad), AIMO (Asia International Mathematical Olympiad), and SASMO (Singapore and Asian Schools Math Olympiad) 2021. He is a student of Class 7 and also a proud Young Achiever of Cheenta. Cheenta is happy to share the success story of Mann! Mann says, "I […]
Aryabhata is considered the "Father of Mathematics" in India. He is the first ancient Mathematician- Astronomer, whose important work includes "Aryabhatiya" and "Arya-Siddhanta". Today, let's learn 3 lessons from Aryabhata - the 𝗙𝗮𝘁𝗵𝗲𝗿 𝗼𝗳 𝗠𝗮𝘁𝗵𝗲m𝗮𝘁𝗶𝗰𝘀. 𝟭. 𝗛𝗮𝘃𝗲 𝗖𝗼𝘂𝗿𝗮𝗴𝗲 𝘁𝗼 𝗤𝘂𝗲𝘀𝘁𝗶𝗼𝗻 When eclipses were seen as something to be feared, and the concepts of "Rahu" and […]
An interesting problem based on complex numbers and their inversion. This is a Subjective Problem 89 from the Test of Mathematics Book, highly recommended for the ISI and CMI Entrance Exams. Let's check out the problem and solutions in two episodes: Useful Resources Previous Year Problems for ISI and CMI How to use invariance in […]
This post discusses the solutions of Problems from RMO 1994 Question Paper. You may find to solution to some of these. RMO 1994 Problem 1: A leaf is torn from a paperback novel. The sum of the numbers on the remaining pages is 15000. What are the page numbers on the torn leaf. RMO 1994 Problem2: […]
This is the solution to the real analysis from ISI MStat 2020 PSB Problem 8 with designed food for thoughts on hypothesis testing and probability theory.