2014AIME.docVIP

2007 AIME 1．How many positive perfect squares less than are multiples of 24? 2．A 100 foot long moving walkway moves at a constant rate of 6 feet per second. Al steps onto the start of the walkway and stands. Bob steps onto the start of the walkway two seconds later and strolls forward along the walkway at a constant rate of 4 feet per second. Two seconds after that, Cy reaches the start of the walkway and walks briskly forward beside the walkway at a constant rate of 8 feet per second. At a certain time, one of these three persons is exactly halfway between the other two. At that time, find the distance in feet between the start of the walkway and the middle person. 3．The complex number is equal to, where is a positive real number and . Given that the imaginary parts of and are the same, what is equal to? 4．Three planets orbit a star circularly in the same plane. Each moves in the same direction and moves at constant speed. Their periods are, 84, and 140. The three planets and the star are currently collinear. What is the fewest number of years from now that they will all be collinear again? 5．The formula for converting a Fahrenheit temperature to the corresponding Celsius temperature is An integer Fahrenheit temperature is converted to Celsius, rounded to the nearest integer, converted back to Fahrenheit, and again rounded to the nearest integer. For how many integer Fahrenheit temperatures between 32 and 1000 inclusive does the original temperature equal the final temperature? 6．A frog is placed at the origin on the number line, and moves according to the following rule: in a given move, the frog advances to either the closest point with a greater integer coordinate that is a multiple of 3, or to the closest point with a greater integer coordinate that is a multiple of 13. A move sequence is a sequence of coordinates which correspond to valid moves, beginning with 0, and ending with 39. For example, is a move sequence. How many move sequen