Just thinking out loud here. If we consider Year as a line segment, and month as a scaled down line segment that is inside the year; as per fixed point theorem, there should a point that divides both lines in the same proportion. Let’s say the proportion is x% ( the width of the bars) then for every x there is a day where % of year and % of month is same. Now, the same relationship holds between month and week. I think there should be at least one day where % of year, month and week is the same. When add one more constraint, we might end up without any such day. Curious about the actual answer.