Subcontests
(5)primes in a sequence
Let p∈N∖{0,1} be a fixed positive integer. Prove that for every K>0, there exist infinitely many n and N such that there are atleast log(N)KN primes among the following N numbers given by
n+1,n+2p,n+3p,⋯,n+Np.Proposed by Bimit Mandal Minimal number of polynomials to write as SOS
Let n be a fixed positive integer.
- Show that there exist real polynomials p1,p2,p3,⋯,pk∈R[x1,⋯,xn] such that
(x1+x2+⋯+xn)2+p1(x1,⋯,xn)2+p2(x1,⋯,xn)2+⋯+pk(x1,⋯,xn)2=n(x12+x22+⋯+xn2)
- Find the least natural number k, depending on n, such that the above polynomials p1,p2,⋯,pk exist.