I have a question on the Infinity Grid and integers
Imagine that there is an infinite grid, within which each box contains an integer. Prove that the only way each box has a value that is less than or equal to the mean of the values in the 4 surrounding it, is if every box is equal.
The problem is not true. If one assigns box $(x,y)$ the number $x^2+y^2$, then we have
$$\frac{(x+1)^2+y^2+x^2+(y+1)^2+(x-1)^2+y^2+x^2+(y-1)^2}{4}=\frac{4x^2+4y^2+4}{4}=x^2+y^2+1>x^2+y^2,$$
so each box has a value $\leq$ the average of the values in its neighboring boxes.
However, if the problem were asking you to show that each number must be the same if each box has a value $\geq$ the average of the values in its neighboring boxes, AND the boxes must each have positive integers, then it would be true (I suspect this is what the original problem asked). To show this, think about a box that contains the smallest positive integer contained by any box, and go outwards.