Problem 1 What is the maximum number of coins can be arranged in cells of the table n×nn \times nn×n (each cell is not more of the one coin) so that any coin was not simultaneously below and to the right than any other?
Problem 3 a,b,c≥0, a3+b3+c3+abc=4a,b,c\ge 0,\ \ \ a^3+b^3+c^3+abc=4a,b,c≥0, a3+b3+c3+abc=4 Prove that a3b+b3c+c3b≤3a^3b+b^3c+c^3b \le 3a3b+b3c+c3b≤3
Problem 4 Find all f:N→Nf:N\rightarrow Nf:N→N , such that ∀m,n∈N\forall m,n\in N∀m,n∈N 2f(mn)≥f(m2+n2)−f(m)2−f(n)2≥2f(m)f(n)2f(mn) \geq f(m^2+n^2)-f(m)^2-f(n)^2 \geq 2f(m)f(n)2f(mn)≥f(m2+n2)−f(m)2−f(n)2≥2f(m)f(n)