Subcontests
(6)\sum x_k -\sum |x_k| is divisible by 4 , if (-1)^{x_k} x_{k-1}x_{k+1} >0
Integers x0,x1,...,xn−1,xn=x0,xn+1=x1 satisfy the inequality (−1)xkxk−1xk+1>0 for k=1,2,...,n. Prove that the difference ∑k=0n−1xk−∑k=0n−1∣xk∣ is divisible by 4. points on a plane can be covered by open squares
On a plane is given a finite set of points. Prove that the points can be covered by open squares Q1,Q2,...,Qn such that 1≤SjNj≤4 for j=1,...,n, where Nj is the number of points from the set inside square Qj and Sj is the area of Qj.