Subcontests
(5)a_n = A_n + B, M^2 smallest square, M < A +\sqrt{B}
Let A and B be positive integers. Define the arithmetic sequence a0,a1,a2,... by an=An+B. Suppose that there exists an n≥0 such that an is a square. Let M be a positive integer such that M2 is the smallest square in the sequence. Prove that M<A+B. M(a, b) = a -1/b +b(b + 3/a) is integer => square also
(a) Let a and b be positive integers such that M(a,b)=a−b1+b(b+a3) is an integer.
Prove that M(a,b) is a square.
(b) Find nonzero integers a and b such that M(a,b) is a positive integer, but not a square. min a_{2010}, a_n < a_{n+1}, a_i + a_l > a_j + a_k
Consider sequences a1,a2,a3,... of positive integers. Determine the smallest possible value of a2010 if
(i) an<an+1 for all n≥1,
(ii) ai+al>aj+ak for all quadruples (i,j,k,l) which satisfy 1≤i<j≤k<l.