Existence of a function as a self composition of another function

This a very interesting problem that shows that there does not always exist a function which can be represented as a self composition of another function. Problem: a) Let be a function. Does there always exists a function such that ?b) Let be a continuous function. Does there always exist a continuous function as defined […]

Existence of a root of a function

The following problem uses the intermediate value property of a continuous function. Problem: Let be a continuously differentiable function with . Prove that for some . Solution: , i.e., is strictly increasing. Then , , i.e., . Now , , i.e., . If , we are done. So let us assume that . Then either […]

Inequality of Integral-2

This problem appeared in the Day 1 of International Mathematics Competition 2019 for undergraduate and graduate students. Problem: Let be a twice differentiable function such that for . Prove that . Analysis of the problem: Generally, in these kind of problems where you are given a differential in-equation of the form check if the differential […]

Convergence of a fixed-point iteration

This problem appeared in the category 2 of the Vojtěch Jarník International Mathematical Competition 1999. Problem: Let where , , be a function such that for every . Choose and define for all . Prove that the sequence converges to some fixed point of . Analysis of the problem: First of all observe that for […]

Finding a function

This problem recently appeared in the Madhava Mathematics Competition 2020 for undergraduate students of India. Problem : Let be a continuous function satisfying and for all positive real numbers . a) Find . b) Show that . c)Find one example of such a function. Solution: a)Let be a function defined by . Then is continuous. […]

A self composed combinatorics problem

This problem is proposed by me to the problem section of the Mathematical Reflections journal of Awesomemath.org. You can find the link to the journal here. The problem is in the olympiad section of the journal. Problem: Let be an ordered set of positive integers. A transformation of is the sequence of positive integers , […]

A Problem of Real Analysis

This is a problem of real analysis in which we use Lagrange’s Mean Value Theorem and the concept of Cauchy Sequence to obtain the desired result. Problem: Let be a differentiable function such that , . Show that the sequence converges in . Solution: Consider the interval . By Lagrange’s Mean Value Theorem, such that […]