AMC 10A 試題及答案解析

1、2006 AMC 10A1、Sandwiches at Joes Fast Food cost each and sodas cost each. How many dollars will it cost to purchase 5 sandwiches and 8 sodas?Solution Thesandwichescostdollars.Thesodascost dollars. In total, the purchase costsdollars. Theanswer is .2、Define . What is ?Solution Bythedefinitionof,wehav

2、e.Then. The answer is .3、The ratio of Marys age to Alices age is. Alice isyears old. How old is Mary?Solution Letbe Marys age. Then . Solving for, we obtain. The answer is .4 、 A digital watch displays hours and minutes withand. What is the largest possible sum of the digits in the display?Solution

3、From the greedy algorithm, we havein the hours section andin the minutes section. 5 、 Doug and Dave shared a pizza withequally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was , and there was an additional cost of for putting anchovi

4、es on one half. Dave ate all the slices of anchovy pizza and one plain slice. Doug ate the remainder. Each paid for what he had eaten. How many more dollars did Dave paythan Doug?Solution Dave and Doug paiddollars in total. Doug paid for threeslices of plain pizza, which cost . Dave paiddollars. Dav

5、e paidmore dollars than Doug. The answer is .6、What non-zero real value forsatisfies ?Solution 7 、 The rectangle is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is ?Solution Taking the seventh root of both s

6、ides, we get . Simplifying the LHS gives , which then simplifies to.Thus, , and the answer is.8、Aparabolawithequationpassesthroughthe points and . What is?Solution Substitute the points (2,3) and (4,3) into the given equation for (x,y).Then we get a system of two equations:Subtracting the first equa

7、tion from the second we have:Then usingin the first equation:is the answer.Alternatively, notice that since the equation is that of a monic parabola, the vertex is likely . Thus, the form of the equation ofthe parabola is . Expanding this out, we find that.9 、 How many sets of two or more consecutiv

8、e positive integers have a sum of 15?Solution Notice that if the consecutive positive integers have a sum of 15, then their average (which could be a fraction) must be a divisor of15. If the number of integers in the list is odd, then the average must be either 1, 3, or 5, and 1 is clearly not possi

9、ble. The other two possibilities both work:If the number of integers in the list is even, then the average will have a . The only possibility is , from which we get:Thus, the correct answer is 3, answer choice .10、For how many real values ofisan integer?Solution For to be an integer,must be a perfec

10、t square. Since cant be negative, .Theperfectsquaresthatarelessthanorequaltoare, so there arevalues for .Since every value of gives one and only one possible value for, the number of values ofis.11、Which of the following describes the graph of the equation?Solution Expanding the left side, we haveTh

11、us there are two lines described in this graph, the horizontal line and the vertical line . Thus, our answer is .12 、 Rolly wishes to secure his dog with an 8-foot rope to a square shed that is 16 feet on each side. His preliminary drawings are shown.Which of these arrangements gives the dog the gre

12、ater area to roam, and by how many square feet?Solution Let us first examine the area of both possible arrangements. The rope outlines a circular boundary that the dog may dwell in. Arrangement I allows the dog square feet of area. Arrangement II allows square feet plus a little more on the top part

13、 of the fence. So we already know that Arrangement II allows more freedom - only thing left is to find out how much. The extraarea can be represented by a quarter of a circle with radius 4. So the extra area is . Thus the answer is .13 、 A player pays to play a game. A die is rolled. If the number o

14、n the die is odd, the game is lost. If the number on the die is even, the die is rolled again. In this case the player wins if the second number matches the first and loses otherwise. How much should the player win if the game is fair? (In a fair game the probability of winning times the amount won

15、is what the player should pay.)Solution There arepossible combinations of 2 dice rolls. The only possible winning combinations are , and . Since there are winning combinations and possible combinations of dice rolls,the probability of winning is .Let be the amount won in a fair game. By the definiti

16、on of a fair game,.Therefore, .14 、 A number of linked rings, each 1 cm thick, are hanging on a peg. The top ring has an outside diameter of 20 cm. The outside diameter of each of the outer rings is 1 cm less than that of the ring above it. The bottom ring has an outside diameter of 3 cm. What is th

17、e distance, in cm, from the top of the top ring to the bottom of the bottom ring?Solution The inside diameters of the rings are the positive integers from 1 to18. The total distance needed is the sum of these values plus 2 for the top of the first ring and the bottom of the last ring. Using the form

18、ula for the sum of an arithmetic series, the answer is.Alternatively, the sum of the consecutive integers from 3 to 20 is. However, the 17 intersections between the rings must be subtracted, and we also get .15、Odell and Kershaw run for 30 minutes on a circular track. Odell runs clockwise at 250 m/m

19、in and uses the inner lane with a radius of 50 meters. Kershaw runs counterclockwise at 300 m/min and uses the outer lane with a radius of 60 meters, starting on the same radial line as Odell. How many times after the start do they pass each other?Solution SolutionSince , we note that Odell runs one

20、 lap in minutes, while Kershaw also runs one lap in minutes. They take the same amount of time to run a lap, and since they are running in opposite directions they will meet exactly twice per lap (once at the starting point, the other at the half-way point). Thus, there arelaps run by both, or meeti

21、ng points.16 、 A circle of radius 1 is tangent to a circle of radius 2. The sides ofare tangent to the circles as shown, and the sides andare congruent. What is the area of?Solution Note that . Using the first pair of similar triangles, we write the proportion:By the Pythagorean Theorem we have that

22、 . Now using ,Theareaofthetriangleis.17、In rectangle, pointsandtrisect , and pointsand trisect.Inaddition,.Whatistheareaofquadrilateralshown in the figure?Solution Solution 1It is not difficult to see by symmetry thatis a square.Draw. Clearly . Thenis isosceles, and is a. Hence , and .There are many

23、 different similar ways to come to the same conclusion using different 45-45-90 triangles.Solution 2Draw the lines as shown above, and count the squares. There are 12, so we have .18、A license plate in a certain state consists of 4 digits, not necessarily distinct, and 2 letters, also not necessaril

24、y distinct. These six characters may appear in any order, except that the two letters must appear next to each other. How many distinct license plates are possible?Solution There are ways to choose 4 digits. There are ways to choose the 2 letters.For the letters to be next to each other, they can be

25、 the 1st and 2nd, 2nd and 3rd, 3rd and 4th, 4th and 5th, or the 5th and 6th characters. So, there arechoices for the position of the letters.Therefore, the number of distinct license plates is .19、How many non-similar triangles have angles whose degree measures are distinct positive integers in arit

26、hmetic progression?Solution The sum of the angles of a triangle is degrees. For an arithmetic progression with an odd number of terms, the middle term is equal to the average of the sum of all of the terms, making itdegrees. The minimum possibly value for the smallest angle is and the highest possib

27、le is (since the numbers are distinct), so there are possibilities .20 、 Six distinct positive integers are randomly chosen between 1 and 2006, inclusive. What is the probability that some pair of these integers has a difference that is a multiple of 5?Solution For two numbers to have a difference t

28、hat is a multiple of 5, the numbers must be congruent (their remainders after division by 5 must be the same).are the possible values of numbers in . Since there are only 5 possible values in and we are picking numbers, by the Pigeonhole Principle, two of the numbers must be congruent .Therefore the

29、 probability that some pair of the 6 integers has a difference that is a multiple of 5 is.21 、 How many four-digit positive integers have at least one digit that is a 2 or a 3?Solution Since we are asked for the number of positive 4-digit integers with at least 2 or 3 in it, we can find this by find

30、ing the total number of 4-digit integers and subtracting off those which do not have any 2s or 3s as digits.The total number of 4-digit integers is, since we have 10 choices for each digit except the first (which cant be 0).Similarly, the total number of 4-digit integers without any 2 or 3 is.Theref

31、ore, the total number of positive 4-digit integers that have at least one 2 or 3 in their decimal representation is22 、 Two farmers agree that pigs are worth dollars and that goats are worth dollars. When one farmer owes the other money, he pays the debt in pigs or goats, with change received in the

32、 form of goats or pigs as necessary. (For example, a dollar debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way?Solution The problem can be restated as an equation of the form, where is the number of p

33、igs, is the number of goats, and is the positive debt. The problem asks us to find the lowest x possible. p and g must be integers, which makes the equation a Diophantine equation. The Euclidean algorithm tells us that there are integer solutions to the Diophantine equation , where is the greatest c

34、ommon divisor of and , and no solutions for any smaller . Therefore, the answer is the greatestcommon divisor of 300 and 210, which is 30, Alternatively, note that is divisible by 30 nomatter what and are, so our answer must be divisible by 30. In addition, three goats minus two pigs give usexactly.

35、 Since our theoretical best can be achived, it must really be the best,and the answer is . debt that can be resolved.23 、 Circles with centers and have radii 3 and 8, respectively. A common internal tangent intersects the circles at and , respectively. Lines and intersect at , and. What is?Solution

36、andare vertical angles so they are congruent, as are angles and(both are right angles because the radius and tangent line at a point on a circle are always perpendicular). Thus,.By the Pythagorean Theorem, line segment. The sides are proportional,so.Thismakesand.24 、 Centers of adjacent faces of a u

37、nit cube are joined to form a regular octahedron. What is the volume of this octahedron?Solution We can break the octahedron into two square pyramids by cutting it along a plane perpendicular to one of its internal diagonals. The cube has edges of length one so the edge of the octahedron is of lengt

38、h . Then the square base of the pyramid has area. We also know that the height of the pyramid is half theheight of the cube, so it is . The volume of a pyramid with base areaand heightis so each of the pyramids has volume. The whole octahedron is twice this volume, so.25 、 A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose