2005 AMC 10A Problems.doc
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1、2005 AMC 10A2005 AMC 10A ProblemsProblem 1While eating out, Mike and Joe each tipped their server2dollars. Mike tipped 10%of his bill and Joe tipped 20%of his bill. What was the difference, in dollars between their bills?(A)2 (B)4 (C)5 (D)10 (E)20Problem 2For each pair of real numbers , define the o
2、perationas .What is the value of? (A) (B) (C)0 (D) (E)This value is not defined.Problem 3The equations andhave the same solution. What is the value of?(A)-8 (B)-4 (C)2 (D)4 (E)8Problem 4A rectangle with a diagonal of lengthis twice as long as it is wide. What is the area of the rectangle?(A) (B) (C)
3、 (D) (E)Problem 5A store normally sells windows at 100 each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately?(A)100 (B)200 (C
4、)300 (D)400 (E)500Problem 6The average (mean) of20numbers is30, and the average of30other numbers is20. What is the average of all50numbers?(A)23 (B)24 (C)25 (D)26 (E)27Problem 7Josh and Mike live13miles apart. Yesterday Josh started to ride his bicycle toward Mikes house. A little later Mike starte
5、d to ride his bicycle toward Joshs house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mikes rate. How many miles had Mike ridden when they met?(A)4 (B)5 (C)6 (D)7 (E)8Problem 8In the figure, the length of side ABof square ABCDis and BE=1. What is the are
6、a of the inner square EFGH?(A)25 (B)32 (C)36 (D)40 (E)42Problem 9Three tiles are markedXand two other tiles are markedO. The five tiles are randomly arranged in a row. What is the probability that the arrangement reads XOXOX?(A) (B) (C) (D) (E)Problem 10There are two values offor which the equation
7、has only one solution for. What is the sum of those values of?(A)-16 (B)-8 (C)0 (D)8 (E)20Problem 11A wooden cubeunits on a side is painted red on all six faces and then cut into unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is?(A)3 (B)4 (C)5 (D)6 (E)7Pr
8、oblem 12The figure shown is called atrefoiland is constructed by drawing circular sectors about the sides of the congruent equilateral triangles. What is the area of a trefoil whose horizontal base has length2?(A) (B) (C) (D) (E)Problem 13How many positive integerssatisfy the following condition: ?(
9、A)0 (B)7 (C)12 (D)65 (E)125Problem 14How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?(A)41 (B)42 (C)43 (D)44 (E)45Problem 15How many positive cubes divide3!5!7! ?(A)2 (B)3 (C)4 (D)5 (E)6Problem 16The sum of the digits of a two-d
10、igit number is subtracted from the number. The units digit of the result is6. How many two-digit numbers have this property?(A)5 (B)7 (C)9 (D)10 (E)19Problem 17In the five-sided star shown, the letters A, B, C, D,andEare replaced by the numbers 3, 5, 6, 7 and9, although not necessarily in this order
11、. The sums of the numbers at the ends of the line segments AB, BC, CD, DE, and EAform an arithmetic sequence, although not necessarily in this order. What is the middle term of the sequence?(A)9 (B)10 (C)11 (D)12 (E)13Problem 18Team A and team B play a series. The first team to win three games wins
12、the series. Each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If team B wins the second game and team A wins the series, what is the probability that team B wins the first game?(A) (B) (C) (D) (E)Problem 19Three one-inch square
13、s are placed with their bases on a line. The center square is lifted out and rotated 45 degrees, as shown. Then it is centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the pointfrom the line on which the bases of the original squares w
14、ere placed?(A)1 (B) (C) (D) (E)2Problem 20An equiangular octagon has four sides of length 1 and four sides of length , arranged so that no two consecutive sides have the same length. What is the area of the octagon?(A) (B) (C) (D) (E)7Problem 21For how many positive integersdoesevenly divide?(A)3 (B
15、)5 (C)7 (D)9 (E)11Problem 22LetSbe the set of the2005smallest positive multiples of4, and letTbe the set of the 2005smallest positive multiples of6. How many elements are common toSandT?(A)166 (B)333 (C)500 (D)668 (E)1001Problem 23Let ABbe a diameter of a circle and letCbe a point on ABwith 2AC=BC.
16、LetDandEbe points on the circle such that DCABand DEis a second diameter. What is the ratio of the area of DCEto the area of ABD?(A) (B) (C) (D) (E)Problem 24For each positive integer , let denote the greatest prime factor of. For how many positive integersis it true that both and ?(A)0 (B)1 (C)3 (D
17、)4 (E)5Problem 25In ABCwe have AB=25,BC=39, and AC=42. PointsDand Eare on ABandACrespectively, withAD=19and AE=14. What is the ratio of the area of triangleADEto the area of the quadrilateral BCED?(A) (B) (C) (D) (E)12005 AMC 10A SolutionsProblem 1Letbe Mikes bill andbe Joes bill. , so m=20; ,so j=1
18、0So the desired difference is D Problem 2Problem 3 , , B Problem 4Let the width of the rectangle be.Then the length is 2. Using thePythagorean Theorem:, So theareaof therectangleis B Problem 5The stores offer means that every 5th window is free.Dave would getfree window. Doug would getfree window.Th
19、is is a total of2free windows. Together, they would getfree windows.So they get 3-2=1additional window if they purchase the windows together.Therefore they save 1100=100 A Problem 6Since theaverageof the first20numbers is30, their sum is 20*30=600.Since the average of30other numbers is20, their sum
20、is 30*20=600.So the sum of all50numbers is 600+600=1200Therefore, the average of all50numbers is 1200/50=24 B Problem 7Letbe the distance in miles that Mike rode.Since Josh rode for twice the length of time as Mike and at four-fifths of Mikes rate, he rode miles.Since their combined distance was13mi
21、les, B Problem 8(C)We see that side BE, which we know is 1, is also the shorter leg of one of the four right triangles (which are congruent, Ill not prove this). So,AH=1. Then HB=HE+BE=HE+1, andHEis one of the sides of the square whose area we want to find. So:, So, the area of the square is .Proble
22、m 9There are distinct arrangements of threes and twos.There is only1distinct arrangement that readsxoxoxTherefore the desiredprobabilityis 1/10 B Problem 10Aquadratic equationhas exactly onerootif and only if it is aperfect square. So set, Twopolynomialsare equal only if theircoefficientsare equal,
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