MathDB

3

Part of 1996 Polish MO Finals

Problems(2)

two n-tuples and inequality

Source:

10/30/2005
ai,xia_i, x_i are positive reals such that a1+a2+...+an=x1+x2+...+xn=1a_1 + a_2 + ... + a_n = x_1 + x_2 + ... + x_n = 1. Show that 2∑i<jxixj≤n−2n−1+∑aixi21−ai 2 \sum_{i<j} x_ix_j \leq \frac{n-2}{n-1} + \sum \frac{a_ix_i ^2}{1-a_i} When do we have equality?
inequalitiesLaTeXinequalities unsolved
Find all natural numbers n such that p_n &gt; \frac{1}{3}

Source:

10/30/2005
From the set of all permutations ff of {1,2,...,n}\{1, 2, ... , n\} that satisfy the condition: f(i)≥i−1f(i) \geq i-1 i=1,...,ni=1,...,n one is chosen uniformly at random. Let pnp_n be the probability that the chosen permutation ff satisfies f(i)≤i+1f(i) \leq i+1 i=1,...,ni=1,...,n Find all natural numbers nn such that pn>13p_n > \frac{1}{3}.
probabilityinequalitiesfunctioncombinatorics unsolvedcombinatorics