MathDB

3

Part of 2018 India IMO Training Camp

Problems(3)

Sequence gets beyond 2018

Source: IMOTC PT1 P3 2018, India

7/18/2018
Let an,bna_n, b_n be sequences of positive reals such that,an+1=an+12bna_{n+1}= a_n + \frac{1}{2b_n} bn+1=bn+12anb_{n+1}= b_n + \frac{1}{2a_n} for all n∈Nn\in\mathbb N. Prove that, max(a2018,b2018)>44\text{max}\left(a_{2018}, b_{2018}\right) >44.
algebraSequenceinequalities
Triangulating three-coloured polygon

Source: IMOTC PT 2 2018 P3, India

7/18/2018
A convex polygon has the property that its vertices are coloured by three colors, each colour occurring at least once and any two adjacent vertices having different colours. Prove that the polygon can be divided into triangles by diagonals, no two of which intersect in the interior of the polygon, in such a way that all the resulting triangles have vertices of all three colours.
combinatoricsconvex polygon
Functional Equation

Source: India TST 2018 D1 P3

7/18/2018
Find all functions f:R↦Rf: \mathbb{R} \mapsto \mathbb{R} such that f(x)f(yf(x)−1)=x2f(y)−f(x),f(x)f\left(yf(x)-1\right)=x^2f(y)-f(x),for all x,y∈Rx,y \in \mathbb{R}.
algebrafunctional equation