Hello, this is a discussion page for the college students who are in various prestigious colleges throughout India, and are keen to pursue Mathematics. Abstract Algebra plays a pivotal role in college mathematics, and it mainly focuses on three things GROUPS, RINGS, and FIELDS. Though Field is not in the course of some colleges, eventually […]
Let's discuss a Common but deadly question in Group theory. Question: Is it possible to get an infinite group which has elements of finite order? Discussion To pursue this discussion which is basically a very good concept for the students who are new in group theory, they must know first about the QUOTIENT GROUPS. Particularly […]
Try this problem from TOMATO Problem 7 based on the Parity of the terms of a sequence. Problem: Parity of the terms of a sequence If \( a_0 = 1 , a_1 = 1 \) and \( a_n = a_{n - 1} a_{n - 2} + 1 \) for \( n > 1 \), then: […]
This is a problem from TOMATO Problem number 2, useful for ISI and CMI entrance exam based on Men and Job. Problem: If m men can do a job in d days, then the number of days in which m+r men can do the job is (A) d+r; (B) $\frac{d}{m} (m+r)$ ; (C) $\frac {d}{m+r}$ […]
This is a problem number 3 from TOMATO based on Calculating Average Speed. Problem: Calculating Average Speed. A boy walks from his home to school at 6 kmph. He walks back at 2 kmph. His average speed, in kmph is (A) 3; (B) 4; (C) 5; (D) $\sqrt {12}$; Discussion: Suppose the distance from home […]
This is a problem number 95 from TOMATO based on finding the Number of factors of 1800. Problem The number of different factors of $1800$ equals: (A) $12$; (B) $210$; (C) $36$; (D) $18$; Discussion: We may factor $1800$ as $2^3 \times 3^2 \times 5^2 $ Then the number of factors is: $(3+1) \times (2+1) […]
This is an objective problem from TOMATO based on finding the Number of Positive Divisors. Problem: The number of positive integers which divide $240$ is- (A) $18$; (B) $20$; (C) $30$; (D) $24$; Discussion: We use the formula for computing number of divisors of a number: Step 1: Prime factorise the given number $240 = […]
Let us discuss about 'inequality' related problems - Minimum Perimeter Problem. All algebraic inequality problems can be traced back to two key ideas: Positive times positive is positive Square of a real number is nonnegative Though these two notions seem trivial and obvious in nature, they lead to a very rich and diverse theory of […]
A worker suffers a 20% cut in wages. He regains his original pay by obtaining a rise of (A) 20% (B) 22.50% (C) 25% (D) 27.50 % If \( \mathbf {m} \) men can do a job in \( \mathbf {d} \) days , then the number of days in which \( \mathbf {m+r} \) […]
This post contains ISI TOMATO Solutions of Objective Problems from 101 to 200. The number of ways of distributing 12 identical oranges among children so that every child gets at least one and no child more than 4 is (A) 31; (B) 52; (C) 35; (D) 42. The number of terms in the expansion of […]
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.
Try this beautiful problem from Geometry:Tetrahedron box from AMC-10A, 2011. You may use sequential hints to solve the problem
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1992 based on Largest Area of Triangle.
Try this beautiful problem from Geometry:cubical box from AMC-10A, 2010. You may use sequential hints to solve the problem
Try this beautiful problem from Algebra:Roots of cubic equation from AMC-10A, 2010. You may use sequential hints to solve the problem
There should be no such thing as boring mathematics. Edsger W. Dijkstra In one of our previous post, we have discussed on Mandelbrot Set. That set is one of the most beautiful piece of art and mystery. At the end of that post, I have said that we can calculate the value of $\pi $ […]
Author: Kazi Abu Rousan The pure mathematician, like the musician, is a free creator of his world of ordered beauty. Bertrand Russell Today we will be discussing one of the most fascinating idea of number theory, which is very simple to understand but very complex to get into. Today we will see how to find […]
Problem Let $f$:$\mathbb{R} \rightarrow \mathbb{R}$ be a continous function such that for all$x \in \mathbb{R}$ and all $t\geq 0$ f(x)=f(ktx) where $k>1$ is a fixed constant Hint Case-1 choose any 2 arbitary nos $x,y$ using the functional relationship prove that $f(x)=f(y)$ Case-2 when $x,y$ are of opposite signs then show that $$f(x)=f(\frac{x}{2})=f(\frac{x}{4})\dots$$ use continuity to […]
PROBLEM Let $f (0,\infty)\rightarrow \mathbb{R}$ be a continous function such that for all $x \in (0,\infty)$ $f(x)=f(3x)$ Define $g(x)= \int_{x}^{3x} \frac{f(t)}{t}dt$ for $x \in (0,\infty)$ is a constant function HINT Use leibniz rule for differentiation under integral sign SOLUTION using leibniz rule for differentiation under integral sign we get $g'(x)=f(3x)-f(x)$ $\Rightarrow g'(x)=0$ [ Because f(3x)=f(x)] […]
PROBLEM Show that there are exactly $2$ numbers $a$ in the set $\{1,2,3\dots9400\}$ such that $a^2-a$ is divisible by $10000$ HINT Use Modular arithmetic and concepts of coprime numbers SOLUTION we know $10000=2^4*5^4$ In order for $10000$ to divide $a^2-a$ both $2^4$ and $5^4$ must divide $ a^2-a $ Write $a^2-a=a(a-1)$ Note that $a$ and […]
Author: Kazi Abu Rousan $\pi$ is not just a collection of random digits. $\pi$ is a journey; an experience; unless you try to see the natural poetry that exists in $\pi$, you will find it very difficult to learn. Today we will see a python code to find the value of $\pi $ up to […]
Author: Kazi Abu Rousan Pi is not merely the ubiquitous factor in high school geometry problems; it is stitched across the whole tapestry of mathematics, not just geometry’s little corner of it. $\pi$ is truly one of the most fascinating things exist in mathematics. It's not just there in geometry, but it's also there in pendulum, […]
To some extent the beauty of number theory seems to be related to the contradiction between the simplicity of the integers and the complicated structure of the primes, their building blocks. This has always attracted people. A. Knauf from "Number theory, dynamical systems and statistical mechanics" This quote is indeed true. If you just think about the […]