In math olympiads around the world, number theory problems have many recurring themes. One such theme is the 'LTE' or lifting the exponent. Selective Logarithm We define a function callled Selective Logarithm (SL) which works on all integers (positive, negative or zero). Suppose N is any integer and p be a prime then $latex SL_p […]
Let's discuss a problem based on Arithmetic of Remainders and understand the concept behind it. Consider the two number: 37 and 52 What is the remainder when we divide 37 by 7? 2 of course. And 52 produces remainder 3 when divided by 7. Suppose we want to know the remainder when the product of […]
Let's understand one-one function and differentiability with the help of a problem. Try it yourself before reading the solution. Let f be real valued, differentiable on (a, b) and $ f'(x) \ne 0 $ for all $ x \in (a, b) $. Then f is 1-1. True Discussion: Suppose f is not 1-1. Then there […]
Lets understand direct product of two subgroups with the help of a problem. This problem is useful for College Mathematics.
What is RMO? RMO or Regional Math Olympiad is the second round of Mathematics Contest in India after the Pre-regional Mathematics Olympiad (Pre-RMO or PRMO) leading to the prestigious International Mathematics Olympiad (IMO). It is held in the month of December (the first Sunday of December). The test is conducted in each of the 19 […]
Let's understand Fixed Point of continuous bounded function with the help of a problem. This problem is useful for College Mathematics.
Given any integer $n \ge 2 $ , we can always find an integer m such that each of the n-1 consecutive integers m + 2, m + 3,..., m + n are composite. True Discussion: Take m=n!. Then the consecutive integers n! + 2 , n! + 3 , ... n! + n are […]
Let's discuss a problem based on Least Value of a Sum of Complex Numbers. Try to solve it yourself before reading the solution. Problem: Least Value of a Sum of Complex Numbers If $ z_1 , z_2 , z_3 , z_4 \in \mathbb{C} $ satisfy $ z_1 + z_2 + z_3 + z_4 = 0 […]
Problem: Continuous Functions and open sets Suppose f be a continuous function from X to Y (where X and Y are domain and range). If Y is a closed set (closed interval if we working in $ R^1 $ ) then can we say that the domain is also closed? There is a simple counter […]
(This is a series of discussions on Homothety. It is largely derived from the Math Olympiad Classroom Discussion in Cheenta - cheenta.com) Homothety is a geometric transformation. It has a couple of synonyms: dilation and central similarity. A geometric transformation is a function. It can be thought of as a machine which takes in a […]