2009 Math Olympics Level I

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1 Saginaw Valle State Universit 009 Math Olmpics Level I. A man and his wife take a trip that usuall takes three hours if the drive at an average speed of 60 mi/h. After an hour and a half of driving at a stead rate of 60 mi/h the stop for a 5 minute lunch break. How fast do the need to drive on the rest of the trip to arrive three hours after the started? (a) 90 mi/h 0 mi/h (c) 80 mi/h (d) 05 mi/h (e) None of the above. Ali and Fred are driving to Bing s house. Ali confuses left with right 50 0 / 0 of the time (This means half the time, when he means to sa left he sas right instead, and half the time when he means to sa left he reall sas left. ). Fred gets left and right confused 5 0 / 0 of the time. (This means when someone tells him to turn left, 75 0 / 0 of the time he will turn left, and 5 0 / 0 he will turn right instead, and when someone tells him to turn right, 75 0 / 0 of the time he will turn right, and 5 0 / 0 he will turn left instead.) If Fred is driving and Ali is giving directions, and there are two left turns on the wa to Bing s house (and no other turns) what is the probabilit the will get there? (a) 9 6 (c) (d) (e) None of the above. If W XY Z is a parallelogram, the t equals: (a) 8 9 (c) 0 (d) (e) W (, ) X(5, ). The smbol R k stands for a positive integer whose baseten representation is a sequence of k ones, that is R =, R =, R =, etc. The quotient R R is an integer whose base-ten representation contains onl digits 0 and. The number of digits 0 in this representation is: Z(, 8) Y (t, 8) (a) 6 (c) 5 (d) 6 (e) 0 5. A square with side a has a semicircle constructed inside it such that the diameter of the semicircle is one of the sides of the square. A circle with maimal radius is then constructed inside the square, but outside of the semicircle (see illustration). Epress the radius r of the circle in terms of a. r a (a) r = a( ) r = a/ a (c) r = a( ) (d) r = a/ (e) None of the above

2 009 SVSU Math Olmpics Level I page of 5 6. In the trapezoid ABCD (the picture is not drawn to scale), the sides AB and CD are parallel, and AB = and CD =. The point E is the intersection point of the two diagonals, and the segment EF is parallel to AB and CD. The length of EF is D E C F (a) A B 6 (c) /7 (d) / (e) Not enough information given 7. A compan offers three tpes of benefits, A, B, and C. As part of this plan, the individual emploees ma choose either eactl two benefit tpes, or no benefits at all. The proportion of the compan s emploees that choose benefit tpes A, B and C are /, / and 5/, respectivel. Determine the probabilit that a randoml chosen emploee will choose no benefits at all. (a) 0 7/ (c) / (d) 97/ (e) 7/9 8. The (possibl out-of-scale) picture shows a window whose shape consists of a semicircle on top of a square. If the perimeter of the window is 0 ft, what is the eact length of the bottom side of the window? (a) 0 π+ ft 0 π+6 ft (c) 0 π+ ft (d) 0 π+ ft (e) None of the above 9. A rectangular parallelepiped is cut into 8 smaller rectangular parallelepipeds b three cuts parallel to the sides. The 8 smaller solids are then separated, as shown in the picture. What is the percent increase in the total surface area? (a) 5 0 / 0 0 / 0 (c) 50 0 / 0 (d) 75 0 / 0 (e) 00 0 / 0

3 009 SVSU Math Olmpics Level I page of 5 0. Find (g f )(). (a) = f () (c) (d) (e) Not enough information given = g(). The picture shows the graph of a third degree polnomial f (). Which of the following gives f ()? (a) f () = ( )( ) f () = ( ) ( ) (c) f () = ( ) ( ) (d) f () = ( )( ) (e) f () = ( + )( ). On three sides of a cube, diagonals are drawn as shown in the picture. If we unfold the cube, which of the following could we obtain? (a) (c) (d) (e) None of the above. The ten points on the right form a square grid. How man isosceles triangles can be drawn with vertices at the points? (a) 8 6 (c) (d) (e) 0

4 009 SVSU Math Olmpics Level I page of 5. The numbers from to 00 are written onto slips of paper and placed into a hat. How man numbers do we have to pull out of the hat to make sure that their product is divisible b 0? (a) 5 80 (c) 8 (d) 8 (e) 9 5. Simplif (a) 0 (c) 8 (d) 9 (e) None of the above 6. In the drawing on the right, the small white circle is the largest circle that fits inside the semicircle. What is the ratio of the area of the small circle to the shaded area? (a) : : (c) : (d) : (e) : π 7. What is 5 % of 5 % of 000? (a) 50 (c) 5 (d) 00 (e) If a b is the greatest common factor of two positive integers a and b, how man of the following equations are not alwas true? a = a a (b c) = (a b) c a b = b a a (b + c) = a b + a c (a) 0 (c) (d) (e) 9. Suppose we arrange the numbers,,, and 5 in the five squares so that the horizontal and the vertical line both add up to 8. Which number has to go in the middle square? (a) (c) (d) (e) 5 0. For integers a, b and c, define (a b c) to mean a b b c + c a. Then ( ) equals (a) (c) 0 (d) (e)

5 009 SVSU Math Olmpics Level I page 5 of 5. A frog starts out b hopping one inch. Each successive hop is inches longer than the previous hop. How far has he gone all together at the end of his th hop? (a) 5 in 69 in (c) 9 in (d) 57 in (e) None of the above. Four points are on a line segment, as shown. If AB : BC = : and BC : CD = 8 : 5, then AB : BD equals A B C D (a) : : (c) : 7 (d) : (e) : 7. A square is cut into three congruent rectangles along two lines parallel to a side, as shown. If the perimeter of each of the three rectangles is, then the area of the original square is (a) 6 (c) 6 (d) 8 (e) 96. Which of the following triangles cannot eist? (a) An acute isosceles triangle An isosceles right triangle (c) An obtuse right triangle (d) A scalene right triangle (e) A scalene obtuse triangle 5. Frida the th occurred two months in a row this ear. What is the net ear that this will happen? (a) 0 06 (c) 05 (d) 0 (e) None of the above

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