Leo's Jump Rope

Leo’s jump rope is 8ft long. He uses his rope to wrap a rectangle ABCD with no remainder of the rope outside ABCD. Kyle draws a circle with center A and radius AC. Kyle’s circle and the extension of AB intersect at E; Kyle’s circle and the extension of AD intersect at F. If AE = 3. What is the area of  BDFE? (FE is an arc, not a chord.)

Making a Perfect Square

The sum n² + 2000n is a perfect square. Find the largest n.

Counting Trails

48 Charlotte Math Meetup students have a meetup together. Each holds a square tile of 1 m². They put the tiles together to form a big rectangle of 48 m². Leo tries to complete a trail from the bottom left tile to the top right tile. (The trails are not along the edges of the tiles, but from one tile to another tile.) If every time he can only go up or right, and if the students try to make the rectangle such that there are as many trails as possible. what’s the most number of trails?