Subcontests
(3)Find all natural numbers n such that p_n > \frac{1}{3}
From the set of all permutations f of {1,2,...,n} that satisfy the condition:
f(i)≥i−1 i=1,...,n
one is chosen uniformly at random. Let pn be the probability that the chosen permutation f satisfies
f(i)≤i+1 i=1,...,n
Find all natural numbers n such that pn>31. Find all n such that x_n = 111111
Let p(k) be the smallest prime not dividing k. Put q(k)=1 if p(k)=2, or the product of all primes <p(k) if p(k)>2. Define the sequence x0,x1,x2,... by x0=1, xn+1=q(xn)xnp(xn). Find all n such that xn=111111