MathDB

3

Part of 2003 Vietnam National Olympiad

Problems(2)

number of permutations

Source: 41-st Vietnamese Mathematical Olympiad 2003

3/10/2007
Let SnS_{n} be the number of permutations (a1,a2,...,an)(a_{1}, a_{2}, ... , a_{n}) of (1,2,...,n)(1, 2, ... , n) such that 1≤∣ak−k∣≤21 \leq |a_{k}-k | \leq 2 for all kk. Show that 74Sn−1<Sn<2Sn−1\frac{7}{4}S_{n-1}< S_{n}< 2 S_{n-1} for n>6.n > 6.
combinatorics proposedcombinatorics
$f(x) \geq A x$

Source: 41-st Vietnamese Mathematical Olympiad 2003

3/10/2007
Let F\mathcal{F} be the set of all functions f:(0,∞)→(0,∞)f : (0,\infty)\to (0,\infty) such that f(3x)≥f(f(2x))+xf(3x) \geq f( f(2x) )+x for all xx. Find the largest AA such that f(x)≥Axf(x) \geq A x for all f∈Ff\in\mathcal{F} and all xx.
functioninductionlimitalgebra proposedalgebra