MathDB

Problem 8

Part of 1998 Ukraine National Math Olympiad

Problems(3)

combo-geo in 2000-gon (Ukraine 1998 Grade 9 P8)

Source:

6/5/2021
A convex 20002000-gon is given in the plane. Show that one can select 19981998 points in the plane such that every triangle with the vertices at vertices of the 20002000-gon contains exactly one of the selected points in its interior (excluding the boundary).
combinatoricsgeometrycombinatorial geometry
P(xy)=P(x)+Q(x)P(y)

Source: Ukraine 1998 Grade 10 P8

6/5/2021
Find all pairs of polynomials f(x),g(x)f(x),g(x) that satisfy f(xy)=f(x)+g(x)f(y)f(xy)=f(x)+g(x)f(y) for all x,yx,y.
Polynomialsfealgebrafunctional equationFunctional Equations
x_n is nondecreasing and floor(x_n)=n

Source: Ukraine 1998 Grade 11 P8

6/6/2021
The sequence (xn)(x_n) is given by x1=1x_1=1 and xn+1=n2xn+xnn2+2for n≥1.x_{n+1}=\frac{n^2}{x_n}+\frac{x_n}{n^2}+2\enspace\text{for }n\ge1. (a) Prove that xn+1≥xnx_{n+1}\ge x_n for all n≥4n\ge4. (b) Prove that ⌊xn⌋=n\lfloor x_n\rfloor=n for all n≥4n\ge4.
floor functionSequencesalgebra