2010 AMC 12-A Problem 1 What is ? Solution . Problem 2 A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip, which starts at 3 PM. One day the boat captain notes that on the 10 AM trip there were 100 tourists on the ferry boat, and that on each successive trip, the number of tourists was 1 fewer than on the previous trip. How many tourists did the ferry take to the island that day? Solution It is easy to see that the ferry boat takes trips total. The total number of people taken to the island is Problem 3 Rectangle , pictured below, shares of its area with square . Square shares of its area with rectangle . What is ? Solution If we shift to coincide with , and add new horizontal lines to divide into five equal parts: This helps us to see that and , where . Hence . Problem 4 If , then which of the following must be positive? Solution is negative, so we can just place a negative value into each expression and find the one that is positive. Suppose we use . Obviously only is positive. Problem 5 Halfway through a 100-shot archery tournament, Chelsea leads by 50 points. For each shot a bullseye scores 10 points, with other possible scores being 8, 4, 2, and 0 points. Chelsea always scores at least 4 points on each shot. If Chelseas next shots are bullseyes she will be guaranteed victory. What is the minimum value for ? Solution Let be the number of points Chelsea currently has. In order to guarantee victory, we must consider the possibility that the opponent scores the maximum amount of points by getting only bullseyes. The lowest integer value that satisfies the inequality is . Problem 6 A , such as 83438, is a number that remains the same when its digits are reversed. The numbers and are three-digit and four-digit palindromes, respectively. What is the sum of the digits of ? Solution is at most , so is at most . The minimum value of is . However, the only palindrome between and is , which means that must be . It follows that is , so the sum of the digits is . Problem 7 Logan is constructing a scaled model of his town. The citys water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logans miniature water tower holds 0.1 liters. How tall, in meters, should Logan make his tower? Solution The water tower holds times more water than Logans miniature. Therefore, Logan should make his tower times shorter than the actual tower. This is meters high, or choice . Also, the fact that doesnt matter since only the ratios are important. Problem 8 Triangle has . Let and be on and , respectively, such that . Let be the intersection of segments and , and suppose that is equilateral. What is ? Solution Let . Since , triangle is a triangle, so Problem 9 A solid cube has side length inches. A -inch by -inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid? Solution Solution 1 Imagine making the cuts one at a time. The first cut removes a box . The second cut removes two boxes, each of dimensions , and the third cut does the same as the second cut, on the last two faces. Hence the total volume of all cuts is . Therefore the volume of the rest of the cube is . Solution 2 We can use Principle of Inclusion-Exclusion to find the final volume of the cube. There are 3 “cuts“ through the cube that go from one end to the other. Each of these “cuts“ has cubic inches. However, we can not just sum their volumes, as the central cube is included in each of these three cuts. To get the correct result, we can take the sum of the volumes of the three cuts, and subtract the volume of the central cube twice. Hence the total volume of the cuts is . Therefore the volume of the rest of the cube is . Solution 3 We can visualize the final figure and see a cubic frame. We can find the volume of the figure by adding up the volumes of the edges and corners. Each edge can be seen as a box, and each corner can be seen as a box. . Solution 4 First, you can find the volume, which is 27. Now, imagine there are three prisms of dimensions 2 x 2 x 3. Now subtract the prism volumes from 27. We have -9. From here we add two times 23, because we over-removed. This is 16 - 9 = 7 (A). Problem 10 The first four terms of an arithmetic sequence are , , , and . What is the term of this sequence? Solution and are consecutive terms, so the common difference is . The common difference is . The first term is and the term is Problem 11 The solution of the equation can be expressed in the form . What is ? Solution This problem is quickly solved with knowledge of the laws of exponents and logarithms. Since we are looking for the base of the logarithm, our answer is . Problem 12 In a magical swamp there are two species of talking amphibians: toads, whose stat