MathDB

3

Part of 2018 VJIMC

Problems(2)

Cyclic inequality with real numbers close to each other

Source: VJIMC 2018, Category I, Problem 3

4/14/2018
Let nn be a positive integer and let x1,…,xnx_1,\dotsc,x_n be positive real numbers satisfying ∣xi−xj∣≤1\vert x_i-x_j\vert \le 1 for all pairs (i,j)(i,j) with 1≤i<j≤n1 \le i<j \le n. Prove that x1x2+x2x3+⋯+xn−1xn+xnx1≥x2+1x1+1+x3+1x2+1+⋯+xn+1xn−1+1+x1+1xn+1.\frac{x_1}{x_2}+\frac{x_2}{x_3}+\dots+\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1} \ge \frac{x_2+1}{x_1+1}+\frac{x_3+1}{x_2+1}+\dots+\frac{x_n+1}{x_{n-1}+1}+\frac{x_1+1}{x_n+1}.
inequalitiesalgebra
Colouring points in R^3

Source: VJIMC 2018, Category II, Problem 3

4/14/2018
In R3\mathbb{R}^3 some nn points are coloured. In every step, if four coloured points lie on the same line, Vojtěch can colour any other point on this line. He observes that he can colour any point P∈R3P \in \mathbb{R}^3 in a finite number of steps (possibly depending on PP). Find the minimal value of nn for which this could happen.
Coloring3D geometrygeometry