2012 Math Day Competition

Transcription

1 2012 Math Day Competition 1. Two cars are on a collision course, heading straight toward each other. One car is traveling at 45 miles per hour and the other at 75 miles per hour. How far apart will the two cars be one minute before they collide? 2. The mass of a $20 gold piece is exactly twice the mass of a $10 gold piece. Which is worth more, one kilogram of $10 gold pieces or one-half kilogram of $20 gold pieces (or are they worth the same)? 3. Bobby has 5-pound weights and 10-pound weights. The total weight of the 5-pound set is the same as the total weight of the 10-pound set. If the whole set has 12 weights in all, what is their total weight? 4. It is said that Diophantus passed 1/12 of his life in infancy and 1/6 more in youth. Then he married and spent 1/7 of his life and 5 more years. Then he had a son whom he survived by 4 years. The son reached only 1/2 the father's age. How old was Diophantus when he died?

2 5. Five geese in a gaggle produce 55 eggs in 5555 days. What is the average number of days it takes a single goose to lay an egg? 6. If the five-digit number 5DDDD is divisible by 6, then determine the digit D. 7. A train, traveling at constant speed, takes 20 seconds from the time it first enters a tunnel that is 300 meters long until it completely emerges from the tunnel. One of the stationary ceiling lights in the tunnel is directly above the train for 10 seconds. Find the length of the train. 8. In 1988, the population of Abra increased by 20 percent while the population of Cadabra decreased by 10 percent, after which the two populations were equal. What percent of the original population of Cadabra was the original population of Abra?

3 9. Two numbers have a sum of 8 and a product of 11. Find the sum of their reciprocals. 10. A rectangular field is twice as long as it is wide. Express the area of the rectangle as a function of its perimeter P. 11. A point (x, y) is called integral if both x and y are integers. How many points on the graph of + = are integral points? x y A cube is formed by gluing together 27 standard cubical dice. (On a standard die, the numbers are 1 through 6, and the sum of the numbers of any pair of opposite faces is 7.) Find the smallest possible sum of all the numbers showing on the surface of the cube.

4 13. The 5 x 5 grid below contains squares with dimensions of 1 1 to 5 5. How many of these squares include the black square? 14. What is the greatest possible number of points of intersection for eight distinct lines in a plane? 15. How many rectangles can be drawn using four of these twelve evenly spaced points as corners?

5 16. Five points on a circle are numbered 1, 2, 3, 4, and 5 in clockwise order. A bug jumps in a counterclockwise direction from one point to another around the circle. If it is on an odd-numbered point, it moves two points, and if it is on an even-numbered point, it moves one point. If the bug begins on point 1, what point will it be on after 2007 jumps? 17. How many ways can you read OSLO in the diagram, using adjacent letters (horizontal, vertical, or diagonal)? Any O can be the first and the last letter of the same OSLO. f ( 1 x) 3 f ( x) = x 18. Suppose a function f is such that for every x 0. Find f (2). 19. Two trees of heights 20 m and 30 m have ropes running from the top of one to the bottom of the other. How high above the ground do the ropes intersect?

6 20. A function f is defined on the positive integers by: f (1) = 0 and f (n) = 3 f (n-1) + 1 for all n > 1. Compute f (1003) - f (1001).