Make A Square

Statement You are give a set of 2-d vectors (x,y) [ -10^8 ≤ x,y ≤ 10^8, x ≠ y ] of size n (1 ≤ n ≤ 50000 ). Lets denote the ith vector as vi. You are to determine if the following system of equations has a solution : a1*v1 + a2*v2 + a3*v3 + ... ai*v1 + ... an*vn = (k,k) for some arbitrary k. All ai's are non-negative integers [ 0 ≤ ai ≤ 10^16 ] and ALL ai's CANNOT BE ZERO ;). In other words,