Question 112
Consider a square consisting of 9 little squares of side length $1$ arranged in $3$ rows and $3$ columns as in the figure below. Let us number the little squares as indicated in the figure.
| ① | ② | ③ |
|----|----|----|
| ⑧ | | ④ |
| ⑦ | ⑥ | ⑤ |
Let $A$, $B$, $C$, $D$ be points chosen from the interior (not on the boundary) of the squares $1$, $3$, $5$, $7$, respectively. Denote by $X$ the sum of the areas of the intersections of the quadrilateral $ABCD$ with the squares $1$, $3$, $5$ and $7$. Denote also by $Y$ the sum of the areas of the intersections of the quadrilateral $ABCD$ with the squares $2$, $4$, $6$ and $8$. Prove that the inequality $X \lt Y$ holds.
| ① | ② | ③ |
|----|----|----|
| ⑧ | | ④ |
| ⑦ | ⑥ | ⑤ |
Let $A$, $B$, $C$, $D$ be points chosen from the interior (not on the boundary) of the squares $1$, $3$, $5$, $7$, respectively. Denote by $X$ the sum of the areas of the intersections of the quadrilateral $ABCD$ with the squares $1$, $3$, $5$ and $7$. Denote also by $Y$ the sum of the areas of the intersections of the quadrilateral $ABCD$ with the squares $2$, $4$, $6$ and $8$. Prove that the inequality $X \lt Y$ holds.