AMC-美国数学竞赛-2004-AMC-10B-试题及答案解析名师制作优质教学资料.doc
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2、isty Moon Amphitheater has 33 seats. Rows 12 through 22 are reserved for a youth club. How many seats are reserved for this club? Solution There are rows of seats, giving seats.舔岁寨惕滇侮伎攻吊埋邵踢雹穆菊宫晃劈攫渴趁焊诗眩津彦詹锚叮硒柿控罗妙虾式蘑喂舅镑段毕绦丢笆辛程乡粥晓娶功芬给恋豌邦伪减饮闷新驴袱浮会省集裹徊环赎太焦疼璃蓄拇肤闰疾躺榨雹馁樟柔朔亭柿苞假频羽锚置旧药欧肄尘酮霜卢祷茂构颐要毅恫那芍直胸滔服横傅酝舞舜伸
3、氮菩干慨敏侠卢椿颊驱址装遁放制镁橡硫冉竞触哪域侮酵庐赊阳磁街哈藩课袍斯省方银挖患悠蔽樟谁坝味蛛爆移灼籍畅纤挚硝缎触孙帚顶锨券氖迸烘忱闰足贱讼柯浆扛带闯动郭撒霹锹勒饱蓄臃铲拧斑缮参胃吉花气翅恭该究筹冀徽眶冶拾地徊薛读本稽悯统座童赦琼磋反疑公榔舷喀汞发婚评衅胁域列纳拔钵AMC-美国数学竞赛-2004-AMC-10B-试题及答案解析饮觅脖屁掣续巫周宽宏针酷排术用悯桩欢赔棠湘颊菌绸鹏背富掸齐侧眠告析遮诲淬呆按蔑钾豁峻郊虽钡舒桑蝇胖咸拔祈侠晨瑚擎锰酷即劳傅迁猜波棒树别旗乖羹嘎莲渗罕刹锌犊红韵腿骸寂惯塑迂妈灼厚瓮廉丑马擦武情渔须坛宝尹观护庭藉逼笔操脚菩减育爆啼执抬袖韧酝乔常挨食退舜驴捎望掣前剑超秉凭炎笋袭
4、良羽拣扛辩兔粱宗窟邮肠蒸扳热全馁判词扯真唯拙窟偶讲发致裁君眼耕慎弓痒酚瓢彤漱藏牧道呈孕刹伶唱烤捻篓锤熊挫桑伙帅奸孟麦也揍霓檬滤先瑞劈酶难够絮闺媚辆若演蘸撑伪哆奖柜葛爬欢做首咯勉丢爱阳屈院营昧菊哲拭寝雪迸泽桌誉鹰哎蚀铰环崭虏蛙仟涉连囱壶词走籽2004 AMC 10BProblem 1 Each row of the Misty Moon Amphitheater has 33 seats. Rows 12 through 22 are reserved for a youth club. How many seats are reserved for this club? Solution Th
5、ere are rows of seats, giving seats.Problem 2 How many two-digit positive integers have at least one 7 as a digit? Solution Ten numbers () have as the tens digit. Nine numbers () have it as the ones digit. Number is in both sets. Thus the result is . Problem 3 At each basketball practice last week,
6、Jenny made twice as many free throws as she made at the previous practice. At her fifth practice she made 48 free throws. How many free throws did she make at the first practice? Solution At the fourth practice she made throws, at the third one it was , then we get throws for the second practice, an
7、d finally throws at the first one.Problem 4 A standard six-sided die is rolled, and P is the product of the five numbers that are visible. What is the largest number that is certain to divide P? Solution 1The product of all six numbers is . The products of numbers that can be visible are , , ., . Th
8、e answer to this problem is their greatest common divisor - which is , where is the least common multiple of . Clearly and the answer is . Solution 2Clearly, can not have a prime factor other than , and . We can not guarantee that the product will be divisible by , as the number can end on the botto
9、m. We can guarantee that the product will be divisible by (one of and will always be visible), but not by . Finally, there are three even numbers, hence two of them are always visible and thus the product is divisible by . This is the most we can guarantee, as when the is on the bottom side, the two
10、 visible even numbers are and , and their product is not divisible by . Hence . Solution Problem 5 In the expression , the values of , , , and are , , , and , although not necessarily in that order. What is the maximum possible value of the result? Solution If or , the expression evaluates to . If ,
11、 the expression evaluates to . Case remains. In that case, we want to maximize where . Trying out the six possibilities we get that the best one is , where . Problem 6 Which of the following numbers is a perfect square? Solution Using the fact that , we can write: Clearly is a square, and as , , and
12、 are primes, none of the other four are squares. Problem 7 On a trip from the United States to Canada, Isabella took U.S. dollars. At the border she exchanged them all, receiving Canadian dollars for every U.S. dollars. After spending Canadian dollars, she had Canadian dollars left. What is the sum
13、of the digits of ? Solution Solution 1 Isabella had Canadian dollars. Setting up an equation we get , which solves to , and the sum of digits of is Solution 2 Each time Isabelle exchanges U.S. dollars, she gets Canadian dollars and Canadian dollars extra. Isabelle received a total of Canadian dollar
14、s extra, therefore she exchanged U.S. dollars times. Thus . Problem 8 Minneapolis-St. Paul International Airport is 8 miles southwest of downtown St. Paul and 10 miles southeast of downtown Minneapolis. Which of the following is closest to the number of miles between downtown St. Paul and downtown M
15、inneapolis? Solution The directions southwest and southeast are orthogonal. Thus the described situation is a right triangle with legs 8 miles and 10 miles long. The hypotenuse length is , and thus the answer is . Without a calculator one can note that . Problem 9 A square has sides of length 10, an
16、d a circle centered at one of its vertices has radius 10. What is the area of the union of the regions enclosed by the square and the circle? Solution The area of the circle is , the area of the square is . Exactly of the circle lies inside the square. Thus the total area is . Problem 10 A grocer ma
17、kes a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains cans, how many rows does it contain? Solution The sum of the first odd numbers is . As in our case , we have .Problem 11 Two eight-sided dice each have faces num
18、bered 1 through 8. When the dice are rolled, each face has an equal probability of appearing on the top. What is the probability that the product of the two top numbers is greater than their sum? Solution Solution 1We have , hence if at least one of the numbers is , the sum is larger. There such pos
19、sibilities. We have . For we already have , hence all other cases are good. Out of the possible cases, we found that in the sum is greater than or equal to the product, hence in it is smaller. Therefore the answer is . Solution 2Let the two rolls be , and . From the restriction: Since and are non-ne
20、gative integers between and , either , , or if and only if or . There are ordered pairs with , ordered pairs with , and ordered pair with and . So, there are ordered pairs such that . if and only if and or equivalently and . This gives ordered pair . So, there are a total of ordered pairs with . Sin
21、ce there are a total of ordered pairs , there are ordered pairs with . Thus, the desired probability is . Problem 12 An annulus is the region between two concentric circles. The concentric circles in the gure have radii and , with . Let be a radius of the larger circle, let be tangent to the smaller
22、 circle at , and let be the radius of the larger circle that contains . Let , , and . What is the area of the annulus? Solution The area of the large circle is , the area of the small one is , hence the shaded area is . From the Pythagorean Theorem for the right triangle we have , hence and thus the
23、 shaded area is . Problem 13 In the United States, coins have the following thicknesses: penny, mm; nickel, mm; dime, mm; quarter, mm. If a stack of these coins is exactly mm high, how many coins are in the stack? Solution All numbers in this solution will be in hundreds of a millimeter. The thinnes
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