Problem: Let ( \ k) be a fixed odd positive integer.Find the minimum value of ( \ x^2+y^2),where ( \ x,y) are non-negative integers and ( \ x+y=k). Solution: According to Cauchy Schwarz's inequality, we can write, ( \ (x^2+y^2)\times(1^2+1^2) \ge)(\ (x\times1+y\times1)^2) =>( \ 2(x^2+y^2)\ge)(\ (x+y)^2) =>( \ x^2+y^2\ge) (\frac{k^2}{2}) Therefore,the minimum value of ( \ x^2+y^2) is […]
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 \( \mathbf {m+r} \) […]
Try this beautiful problem from PRMO, 2019, problem-12, based on Integer Problem. You may use sequential hints to solve the problem.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1995 based on Right Rectangular Prism.
Try this beautiful problem from the Pre-RMO, 2019 based on Greatest Integer. You may use sequential hints to solve the problem.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1996 based on Parallelogram Problem.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1995 based on Pyramid with Square base.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1995 based on Repeatedly Flipping a Fair Coin.
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.