All non-trivial proper subgroups of (R, +) are cyclic. False Discussion: There is a simple counter example: (Q, +) (the additive group of rational numbers). We also note that every additive subgroup of integers is cyclic (in fact they are of the for nZ). Cyclic groups have exactly one generator. We can construct numerous counter […]
The equation $latex x^3 + 10x^2 - 100x + 1729 $ has at least one complex root α such that |α| > 12. False ** Discussion: A fun fact : 1729 is the Ramanujan Number; it is the smallest number expressible as the sum of two cubes in two different ways We conduct normal extrema tests. First […]
The equation $latex x^3 + 3x - 4 $ has exactly one real root. True Discussion: Consider the derivative of the function $latex f(x) = x^3 + 3x - 4 = 0 $ . It is $latex 3x^2 + 3 $ . Note that the derivative is strictly positive ( positive times square + positive is […]
Problem: Every differentiable function f: (0, 1) --> [0, 1] is uniformly continuous. Discussion; False Note that every differentiable function f: [0,1] --> (0, 1) is uniformly continuous by virtue of uniform continuity theorem which says every continuous map from closed bounded interval to R is uniformly continuous. However in this case the domain is […]
Problem: Let f: R --> R be defined by $latex f(x) = sin (x^3) $. Then f is continuous but not uniformly continuous. Discussion: True It is sufficient to show that there exists an $latex epsilon > 0 $ such that for all $latex \delta > 0 $ there exist $latex x_1 , x_2 \in […]
This post contains a problem from TIFR 2013 Math paper D based on Inequality of square root function. The inequality $ \sqrt {n+1} - \sqrt n < \frac {1}{\sqrt n } $ is false for all in n such that $ 101 \le n \le 2000 $ False Discussion: $ \sqrt {n+1} - \sqrt n […]
Any automorphism of the group Q under addition is of the form x → qx for some q ∈ Q. True Discussion: Suppose f is an automorphism of the group Q. Let f(1) = m (of course 'm' will be different for different automorphisms). Now $f(x+y) = f(x) + f(y)$ implies $f(x) = mx$ where m […]
Problem 1 . A shop sells two kind of products A and B. One day a salesman sold both A and B at the same price, $2100$ to a customer. Suppose A makes a profit of 20% and B makes a loss of 20%. Then the deal(A) make a profit of $70$; (B) make a […]
What motivates research in Non-Linear Partial Differential Equation? Swarnendu Sil, presently a Ph.D. student in Ecole polytechnique de federale de lausannee (one of the leading universities of the world located in Switzerland), delivered a talk (through video conference) on this topic this Sunday in the reunion of Cheenta. The seminar began with an analysis of […]
Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2010 based on Probability of divisors.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2015 from Geometry based on Area of Equilateral Triangle.
Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2015 based on Probability. You may use sequential hints.
Try this beautiful problem from American Invitational Mathematics Examination I, AIME I, 2009 based on geometric sequence. Use hints to solve the problem.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2009 based on Coordinate Geometry.
Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2010 based on Exponents and Equations.
Try this beautiful problem from the Pre-RMO, 2019 based on Two Arrangements. You may use sequential hints to solve the problem.
Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2012 based on Arrangement of Digits. You may use sequential hints.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2012 based on Complex Numbers and prime.
Try this beautiful problem from the Pre-RMO, 2019 based on Trigonometry Problem. You may use sequential hints to solve the problem.