IMC_2012_web_solutions

Clockwise around a third semicircle with centre 3

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clockwise around a third semicircle with centre ) 0 , 3 ( . Where does Peri end this expedition? A ) 0 , 0 ( B ) 0 , 1 ( C ) 0 , 2 ( D ) 0 , 4 ( E ) 0 , 6 ( Solution: C As may be seen from the diagram, Peri first moves along the semicircle with centre ) 0 , 4 ( from the point ) 0 , 0 ( to the point ) 0 , 8 ( , then along the semicircle with centre ) 0 , 6 ( to the point ) 0 , 4 ( , and finally along the semicircle with centre ) 0 , 3 ( to end up at the point ) 0 , 2 ( . 19. The shaded region shown in the diagram is bounded by four arcs, each of the same radius as that of the surrounding circle. What fraction of the surrounding circle is shaded? A 1 4 B 4 1 C 2 1 D 3 1 E it depends on the radius of the circle Solution: A Suppose that the surrounding circle has radius r . In the diagram we have drawn the square with side length 2 r which touches the circle at the points where it meets the arcs. The square has area 2 2 4 ) 2 ( r r . The unshaded area inside the square is made up of four quarter circles with radius r , and thus has area 2 r . Hence the shaded area is 2 2 2 ) 4 ( 4 r r r . The circle has area 2 r . So the fraction of the circle that is shaded is 1 4 4 ) 4 ( 2 2 r r . 20. A rectangle with area 125 cm 2 has sides in the ratio 4:5. What is the perimeter of the rectangle? A 18 cm B 22.5 cm C 36 cm D 45 cm E 54 cm Solution: D Since the side lengths of the rectangle are in the ratio 4:5, they are 4 a cm and 5 a cm, for some positive number a . This means that the rectangle has area 2 20 5 4 a a a cm 2 . Hence 125 20 2 a . So 2 a 4 25 20 125 , and hence 2 5 a . Hence the rectangle has perimeter 2 5 18 18 ) 5 4 ( 2 a a a 45 cm. 3
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11 21. The parallelogram PQRS is formed by joining together four equilateral triangles of side 1 unit, as shown. What is the length of the diagonal SQ ? A 7 B 8 C 3 D 6 E 5 Solution: A Let T be the foot of the perpendicular from Q to the line SR extended. Now RQT is half of an equilateral triangle with side length 1. Hence the length of RT is 2 1 and hence ST has length 2 5 2 1 1 1 . By Pythagoras’ Theorem applied to the right angled triangle RQT ,   2 2 2 1 2 ) ( 1 QT . Therefore 4 3 4 1 2 2 1 2 2 1 ) ( ) 1 ( QT . Hence, by Pythagoras’ Theorem applied to the right angled triangle SQT , 7 ) ( 4 3 4 25 4 3 2 2 5 2 2 2 QT ST SQ . Therefore, 7 SQ . 22. What is the maximum possible value of the median number of cups of coffee bought per customer on a day when Sundollars Coffee Shop sells 477 cups of coffee to 190 customers, and every customer buys at least one cup of coffee? A 1.5 B 2 C 2.5 D 3 E 3.5 Solution: E Put the set of numbers of cups of coffee drunk by the individual customers into numerical order with the smallest first. This gives an increasing sequence of positive integers with sum 477. Because 190 is even, the median of these numbers is the mean of the 95th and 96th numbers in this list. Suppose these are a and b , respectively. Then the median is ) ( 2 1 b a .
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