2003A年AMC10.doc

Problem 1 What is the difference between the sum of the first??even counting numbers and the sum of the first??odd counting numbers? Solution Problem 2 Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $4 per pair and each T-shirt costs $5 more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is $2366, how many members are in the League? Solution Problem 3 A solid box is??cm by??cm by??cm. A new solid is formed by removing a cube??cm on a side from each corner of this box. What percent of the original volume is removed? Solution Problem 4 It takes Mary??minutes to walk uphill??km from her home to school, but it takes her only??minutes to walk from school to her home along the same route. What is her average speed, in km/hr, for the round trip? Solution Problem 5 Let??and??denote the solutions of?. What is the value of?? Solution Problem 6 Define??to be??for all real numbers??and?. Which of the following statements is not true? ?for all??and? ?for all??and? ?for all? ?for all? ?if? Solution Problem 7 How many non-congruent triangles with perimeter??have integer side lengths? Solution Problem 8 What is the probability that a randomly drawn positive factor of??is less than? Solution Problem 9 Simplify . Solution Problem 10 The polygon enclosed by the solid lines in the figure consists of 4 congruent squares joined edge-to-edge. One more congruent square is attatched to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing? Solution Problem 11 The sum of the two 5-digit numbers??and??is?. What is?? Solution Problem 12 A point??is randomly picked from inside the rectangle with vertices?,?,?, and?. What is the probability that?? Solution Problem 13 The sum of three numbers is?. The first is four times the sum of the other two. The second is seven times the thir