MathDB

2

Part of 2016 India IMO Training Camp

Problems(3)

Powers with same first digit

Source: India IMO Training Camp 2016, Practice Test 1 Problem 2

7/22/2016
Given that nn is a natural number such that the leftmost digits in the decimal representations of 2n2^n and 3n3^n are the same, find all possible values of the leftmost digit.
number theory
Another FE

Source: India IMO Training Camp 2016, Practice test 2, Problem 2

7/22/2016
Find all functions f:R→Rf:\mathbb R\to\mathbb R such that f(x2+xf(y))=xf(x+y)f\left( x^2+xf(y)\right)=xf(x+y) for all reals x,yx,y.
functionfunctional equationalgebra
Functional Equation

Source: India TST 2016 Day 4 Problem 2

7/22/2016
Find all functions f:R→Rf:\mathbb{R}\to\mathbb{R} such that f(x3+f(y))=x2f(x)+y,f\left(x^3+f(y)\right)=x^2f(x)+y,for all x,y∈R.x,y\in\mathbb{R}. (Here R\mathbb{R} denotes the set of all real numbers.)
functional equationfunctionalgebra