Math Olympiad 5

1. A square has an area with a value 5 times its perimeter. How long is each side of the square?

2. Two circles A and B are enclosed inside a larger circle C. There is no overlap between A and B, and they stretch the entire diameter of the larger circle C. The area of circle C is 64π. What is the largest possible non-shaded area inside C? Note that the diagram might not be drawn to scale.
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3. ABCD is a rectangle. E, and F are at the same height. If the area of the rectangle ABCD is 49, what is the area sum of the shaded triangles?
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4. The sides of a triangle are all integer lengths. Two sides have the lengths of 19 and 5. What is the smallest possible perimeter of the triangle?

5. What is the length of AE in the following diagram?

In the figure above, point R lies on segment OP. The area of the circle with center O is 400π, and the length of the line segment OP is 18. What is the area of the circle with center Q?

If the area of the square in the figure above is 9 and the perimeter of each of the 4 triangles is 7, what is the perimeter of the figure (a star) outlined by the solid line? Note that the figure is not drawn to scale.

8. In a parallelogram ABCD, if AC > BD, which of the following statement(s) must be true? Note that the diagram is not drawn to scale.

I. ∠B > ∠A
II.AB + BC > BC + CD
III.∠DBC > ∠DAC

9. By connecting the four midpoints of each of the sides of a square, a new square is formed enclosed in the original square. If the most external square is the 1st square, the one enclosed by the first one is the 2nd square, the one enclosed by the 2nd square is the 3rd square, ... and so on. If the 6th square has a perimeter of 4, what is the area of the 1st square?
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10. Given two squares, if the ratio of the area of the larger square to the area of the smaller square is 17, what is the ratio of the length of the diagonal of the larger square to the length of the diagonal of the smaller?