# 8. Variant 1

At the base of the quadrilateral pyramid $S A B C D$ lies a square $A B C D, S A$ - the height of the pyramid. Let $M$ and $N$ be the midpoints of the edges $S C$ and $A D$. What is the maximum value that the area of triangle $B S A$ can have if $M N=3 ?$
Answer: 9.

Solution: Let $O$ be the center of the square $ABCD$. Then $MO$ is the midline of the triangle $SAC$, so $SA = 2MO$. Similarly, $ON$ is the midline of the triangle $BDA$, so $AB = 2ON$. Therefore, $SA^2 + AB^2 = 4(MO^2 + ON^2) = MN^2 = 36$. Let $SA = x, AB = y$. From the formula $S = 0.5 \cdot SA \cdot AB = 0.5xy$, the equality $x^2 + y^2 = SB^2 = 36$, and the inequality $x^2 + y^2 \geq 2xy$, it follows that the area of the triangle is maximized when $2xy = x^2 + y^2$, and it is equal to 9.

## Variant 2

In the base of the quadrilateral pyramid $SABCD$ lies a square $ABCD$, and $SA$ is the height of the pyramid. Let $M$ and $N$ be the midpoints of the edges $SC$ and $AD$. What is the maximum value that the area of the triangle $BSA$ can have if $MN = 4$?

Answer: 16.

## Variant 3

In the base of the quadrilateral pyramid $SABCD$ lies a square $ABCD$, and $SA$ is the height of the pyramid. Let $M$ and $N$ be the midpoints of the edges $SC$ and $AD$. What is the maximum value that the area of the triangle $BSA$ can have if $MN = 5$?

Answer: 25.

## Variant 4

In the base of the quadrilateral pyramid $SABCD$ lies a square $ABCD$, and $SA$ is the height of the pyramid. Let $M$ and $N$ be the midpoints of the edges $SC$ and $AD$. What is the maximum value that the area of the triangle $BSA$ can have if $MN = 6$?

Answer: 36.

## Variant 5

In the base of the quadrilateral pyramid $SABCD$ lies a square $ABCD$, and $SA$ is the height of the pyramid. Let $M$ and $N$ be the midpoints of the edges $SC$ and $AD$. What is the maximum value that the area of the triangle $BSA$ can have if $MN = 2$?

Answer: 4.