EXAM AB HIGH SCHOOL MATH CONTEST FALL 2004 1. The decimal representation of 4/7 is 0.571428 . What is the 2003 rd digit after the decimal point? 2. The earth rotates about its axis. How many degrees does the earth rotate in 12 minutes? 3. Starting at home, a man traveled 9 miles east, then 7 miles north, then 3 miles east, then x miles south. At that point, he was 13 miles from home. Find all possible values for x 4. ABCD is a rectangle whose dimensions are 6 × 10 inches. Squares are drawn on each side of the exterior of the rectangle, and their centers are joined to form a convex quadrilateral. Find the area of the quadrilateral. B C A D 5. A figure skater is facing north when she begins to spin to her right. She spins 2250 degrees. Which direction (north, south, east or west) is she facing when she finishes her spin? 6. By using the answers to the following questions, Patrick determines Sam’s secret whole number. • Is it a factor of 30? Yes • Is it a prime number? No • Is it a multiple of 3? No • Is it less than 3? No What is Sam’s secret number? 7. When this net of six squares is cut out and folded to form a cube, what is the product of the numbers on the four faces adjacent to the one labeled with a 1? 1 3 4 5 6 2 8. How many square units are in the area of the quadrilateral with vertices (0, 0), (3, 0), (2, 2) and (0, 3)? 9. Find the sum of the numerical coefficients in the expansion of (x + y)10 . 10. One solution of a(b − c)x2 + b(c − a)x + c(b − a) = 0 is x = 1 for some fixed real numbers a, b, c. What is the other solution for x? 11. How many of the numbers 21 , 22 , 23 , . . . , 2100 , written in base-ten notation, have a ones digit of 6? 12. Find the numerical value of 1 + 3 4 1+ 3 5 1+ 3 6 1+ 3 7 × ··· × 1 + 3 21 1+ 3 22 . 13. Three tennis balls are stacked in a cylinder that touches the stack on all sides, on the top and on the bottom. Find the ratio of the volume of the balls to the volume inside the can. 14. Find the number of digits in 416 525 . 15. The 120 permutations of AHSME are arranged in dictionary order, as if each were an ordinary five-letter word. Find the 85 th word in the list. 16. Mary typed a six-digit number, but the two 1’s she typed did not show. What appeared instead was 2002. How many different six-digit numbers could she have typed? 17. The coordinates of the vertices of a parallelogram are (10, 1), (7, −2), (4, 1) and (x, y). What is the sum of the distinct possible values for x? 18. Katrina and Abbie start a game with one pile of 40 pennies. They take turns. On each turn, a player must take 1,2,3,4 or 5 pennies from the pile. The player who takes the last penny from the pile of 40 pennies wins the game. If Abbie plays first, what number of pennies must she take from the pile on her first turn in order to guarantee that she can win the game? 19. In this array of numbers, the first number in each row is 2 and the last number in each row is 5. Each of the other entries is the sum of the two numbers nearest it in the row directly above it. What is the sum of all of the numbers in row 10? Row 1 Row 2 Row 3 2 5 2 7 2 5 9 12 5 20. Isosceles triangle ABE of area 100 square inches is cut by CD into an isosceles trapezoid and a smaller isosceles triangle. The area of the trapezoid is 75 square inches. If the altitude of triangle ABE from A is 20 inches, what is the number of inches in the length of CD ? A D E C B

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