21st AMC 8 2005
6
23. Isosceles right triangle ABC encloses a semicircle of are a 2π. Th e circle has its center O on hypotenuse AB and is tangent to sides AC and BC. What is the area of triangle ABC?
(A) 6 (B) 8
(C) 3π (D) 10
24. A certain calculator has only two keys [+1] and [×2]. When you press one of the keys, the calculator automatically displays the result. For instance, if the calculator originally displayed “9” and you pressed [+1], it would display “10.” If you then pressed [×2], it would display “20.” Starting with the display “1,” what is the fewest number of keystrokes you would need to reach “200”? (A) 8
(B) 9
(C) 10
(D) 11
25. A square with side length 2 and a circle share the same center. The total area of the regions that are inside the circle and outside the square is equal to the total area of the regions that are outside the circle and inside the square. What is the radius of the circle?
√ (A) √2 3
π (B) 1+ 2 2
(C) 2
(D) √
3
(E) 4π
(E) 12
(E) √
π
