MathDB

2

Part of 2000 Vietnam Team Selection Test

Problems(2)

$x_n = [an]$ for all $n$

Source: Vietnam TST 2000

4/3/2007
Let kk be a given positive integer. Define x1=1x_{1}= 1 and, for each n>1n > 1, set xn+1x_{n+1} to be the smallest positive integer not belonging to the set {xi,xi+ik∣i=1,...,n}\{x_{i}, x_{i}+ik | i = 1, . . . , n\}. Prove that there is a real number aa such that xn=[an]x_{n}= [an] for all n∈Nn \in\mathbb{ N}.
searchnumber theory unsolvednumber theory
ineq function

Source: Vietnam TST 2000

4/3/2007
Let a>1a > 1 and r>1r > 1 be real numbers. (a) Prove that if f:R+→R+f : \mathbb{R}^{+}\to\mathbb{ R}^{+} is a function satisfying the conditions (i) f(x)2≤axrf(xa)f (x)^{2}\leq ax^{r}f (\frac{x}{a}) for all x>0x > 0, (ii) f(x)<22000f (x) < 2^{2000} for all x<122000x < \frac{1}{2^{2000}}, then f(x)≤xra1−rf (x) \leq x^{r}a^{1-r} for all x>0x > 0. (b) Construct a function f:R+→R+f : \mathbb{R}^{+}\to\mathbb{ R}^{+} satisfying condition (i) such that for all x>0,f(x)>xra1−rx > 0, f (x) > x^{r}a^{1-r} .
functionalgebra unsolvedalgebra