美国数学竞赛AMC8 -- 2005年真题解析(英文解析+中文解析)

美国数学竞赛AMC8 – 2005年真题解析(英文解析+中文解析)

Problem 1

Answer: B

Solution:

If x is the number, then 2x=60 and x=30. Dividing the number by 2 yields 15.

中文解析：

按照Connie的计算，这个数乘以2是60，可知这个数是30. 应该做的计算是30除以2，因而正确答案应该是15. 答案是B。

Problem 2

Answer: C

Solution:

Karl paid 5*2.5=$12.5. 20% of this cost that he saved is 12.5*0.2=$2.5.

中文解析：

Karl按原价买了5个文件夹，支付的费用是：2.5*5=12.5. 折扣价是：1.25*0.8=10。如果Karl 等一天，可以省2.5元。答案是C.

Problem 3

Answer: D

Solution:

Rotating square ABCD counterclockwise 45° so that the line of symmetry BD is a vertical line makes it easier to see that 4 squares need to be colored to match its corresponding square.

中文解析：

如上图所示，以BD为对称轴，标蓝色的方块需要涂黑。共4块，答案是D。

Problem 4

Answer: C

Solution:

The perimeter of the triangle is 6.1+8.2+9.7=24cm. A square's perimeter is four times its side length, since all its side lengths are equal. If the square's perimeter is 24, the side length is

24/4=6, and the area is 6*6=36.

中文解析：

三角形的周长是：6.1+8.2+9.7=24. 正方形的周长和三角形相等，也是24，则其边长是24/4=6. 其面积是：6*6=36. 答案是C。

Problem 5

Answer: B

Solution:

Start by buying the largest packs first. After three 24-packs, 90-3*24=18 cans are left. After one 12-pack, 18-12=6 cans are left. Then buy one more 6-pack. The total number of packs is

3+1+1=5.

中文解析：

共90罐，先试试3包24罐的，是3*24=72，还剩下90-72=18罐，再用1包12罐的，剩下6罐，正好1包6罐的。共用了3+1+1=5包。答案是B。

Problem 6

Answer: C

Solution:

We see that 2.0055 works but 2.0045 does not. The digit d can be from 5 through 9, which is 5 values.

中文解析：

为了让2.00d5比2.005大，d至少要是5，即2.0055，比2.005大。d可以是5，6，7，8，9. 即有5个数字，答案是C。

Problem 7

Answer: B

Solution:

Draw a picture：（rectangle, length is 3/4, height is 1/2+1/2）. Find the length of the diagonal of the rectangle to find the length of the direct line to the starting time using Pythagorean Theorem. (1/2+1/2)^2+(3/4)^2=25/16=(5/4)^2. Thus the answer is 5/4.

中文解析：

如上图所示：AB=1/2，BC=3/4，CD=1/2. 做辅助线BE和ED，其中BE和CD平行且相等，ED 和BC平行且相等。在直角三角形AED中，AE=1，ED=3/4. 根据勾股定理，可得AD的平方是: 1*1+3/4 *3/4=25/16. 则AE=5/4. 答案是D。

Problem 8

Answer: E

Solution:

Assume that m =1 and n=3. Plugging into each of the choices, we get 10, 0, 30, 36, and 9 . The only odd integer is E.

中文解析：

假设m=1,n=3. 代入A选项得10，B选项得0；C选项得30；D选项得36；E选项得9. 只有E选项是奇数，因此答案是E。

Problem 9

Answer: D

Solution:

Because AD=CD and

AC=DC=17.

中文解析：

在三角形ACD中，AD=CD=17，且角ADC=60°，则三角形ACD是等边三角形。AC也是17. 答案是D。

Problem 10

Answer: D

Solution:

We know that it took Joe 6 minutes to walk halfway. If he ran 3 times as fast as that, he went

6/3=2 minutes for the other half. Therefore, we have 6+2=8 minutes.

中文解析：

Joe走路走一半路程需要6分钟，他跑步的速度是走路速度的3倍，因此跑步后一半路程的时间是6/3=2分钟。从家到学校的总时间需要6+2=8分钟。答案是D。

Problem 11

Answer: C

Solution:

The price Jacks rings up is 90*.106*0.8. The price Jill rings up is 90*0.8*1.06. By the commutative property of multiplication, these quantities are the same, and the difference is 0.

中文解析：

按照Jack的算法：90*1.06*0.8；按照Jill 的算法：90*0.8*1.06. 这两个计算的结果是相同的，仅顺序不同。因此总价的差别是0. 答案是C。

Problem 12

Answer: D

Solution:

There are 5 days from May 1 to May 5. The number of bananas he eats each day is an arithmetic sequence. He eats n bananas on May 5, and n-4*6=n-24 bananas on May 1. The sum of this arithmetic sequence is equal to 100.((n-24)+n)/2 *5=100, thus n=32.

中文解析：

假设第三天吃x个，则这5天分别吃：x-12, x-6, x, x+6,x+12. 这5个数的和是100，平均数是最中间的那个数，即x. 因此x=100/5=20. 第5天吃的是20+12=32个。答案是D。

Problem 13

Answer: C

Solution:

Notice that AF+DE=BC, so DE=4. Let O be the intersection of the extensions of AF and DC, which makes rectangle ABCO. The area of the polygon is the area of FEDO subtracted from the area of ABCO. Area=52=8*9-EF*4. Solving for the unknown, EF=5. Therefore, DE+EF=4+5=9. 中文解析：

这个图形可以看成上面一个小长方形，下面一个小长方形。上面长方形的面积是8*5=40. 则下面的小长方形的面积是52-40=12. 下面的下长方形中，ED=9-5=4，因此DC=12/4=3.

EF=AB-CD=8-3=5. 题目所求DE+EF=4+5=9. 答案是C。

Problem 14

Answer: B

Solution:

Within each division, there are C(6,2)=15 pairings, and each of these games happens twice. The same goes for the other division so that there are 2*15*2=60 games within their own divisions. The number of games between the two divisions is 6*6=36. Together there are 6-+36=96 conference games.

中文解析：

组内6个队伍，如果相互之间仅比赛一场，则需C（6，2）场，即15场，组内每队伍之间比赛2场，则需要比15*2=30场。两个分组，则组内共进行30*2=60场。组间的比赛场次：6*6=36场。因此合计共需要组织比赛60+36=96场。答案是B。

Problem 15

Answer: C

Solution:

Let b be the base of the isosceles triangles, and let a be the lengths of the other legs. From this, 2a+b=23 and =23-2a. From triangle inequality, 2a>b, then plug in the value from the previous equation to get 2a>23-2a or a>5.75. The maximum value of a occurs when b=1, in which from

the first equation a=11. Thus, a can have integer side lengths from 6 to 11 and there are 6 triangles.

中文解析：

假设等腰三角形的腰是a, 底是b. 则：2a+b=23. 即b=23-2a。另外根据三角形边长的不等式关系，两边之和大于第三边，则2a>b。由于b=23-2a，所以2a>23-2a, 可以得到a>=6. a的最大值是当b=1时，a=（23-b)/2=11. 综合起来，a最小是6，最大是11，共6个值。因此答案是C。

Problem 16

Answer: D

Solution:

The Martian can pull out 12 socks, 4 of each color, without having 5 of the same kind yet. However, the next one he pulls out must be the fifth of one of the colors so he must remove 13 socks.

中文解析：

最坏情况下，Martian 取出了红，白，蓝各4个，即共去取出了12个袜子。在这种情况下，再取一个袜子，不论是什么颜色，则一定有一种颜色的袜子达到了5双。即要取12+1次。答案是D。

Problem 17

Answer: E

Solution:

Average speed is distance over time, E runs the greatest distance for the shortest amount of time.

中文解析：

从图中可以看出，Evelyn在最短的时间内，距离最长。因此Evelyn 的平均速度最快。答案是E。

Problem 18

Answer: C

Solution:

Let k be any positive integer so that 13k is a multiple of 13. For the smallest three-digit number, 3k>100 and k>7.7. For the greatest three-digit number, 13,<999 and k<76.8. The number k can range from 8 to 76 so there are 69 three-digit numbers.

中文解析：

这道题可以理解为：在100到999之间有多少个13的倍数。13 * 8=104；13*77=1001. 从8 到76共有69个数字。因此答案是C。

Problem 19

Answer: A

Solution:

Draw altitudes from B and C to base AD to create a rectangle and two right triangles. The side opposite BC is equal to 50. The bases of the right triangles can be found using Pythagorean or special triangles to be 18 and 7. Add it together to get AD=18+50+7=75. The perimeter is

75+30+50+25=180.

中文解析：

在直角三角形ABE中，根据勾股定理，AE是18. 在直角三角形CFD中，CF=24，CD=25，根据勾股定理，FD=7. 因此AD的长度是18+50+7=75. 这个梯形的周长是：50+30+75+25=180. 答案是A。

Problem 20

Answer: A

Solution:

Alice moves 5k steps and Bob moves 9k steps, where k is the turn they are on. Alice and Bob coincide when the number of steps they move collectively, 14k, is a multiple of 12. Since 14 has a factor 2, k must have a factor of 6. The smallest number of turns that is a multiple of 6 is 6.

中文解析：

Alice顺时针走，每次5点，其位置依次分别是：5，10，3，8，1，6；Bob 逆时针走，每次9点，其位置依次分别是：3，6，9，12，3，6，两个人在6点处相遇，当时各够了6次。答案是A。

Problem 21

Answer: C

Solution:

The number of ways to choose three points to make a triangle is C(6,3)=20. However, two of these are a straight line so we subtract 2 to get 18.

中文解析：

从6个点中，任意选出3个点，有C（6，3）=20. 其中当3个点是一条线时无法组成三角形，这种情况有2次。因此能组成的三角形有20-2=18个。答案是C。

Problem 22

Answer: E

Solution:

Suppose the small size costs $1 and the large size has 10 oz. The medium size then costs $1.5 and has 8 oz. The small size has 5 oz and the large size costs 1.5*1.3=$1.95. The small, medium, and large size cost respectively,0.2, 0.188, 0.195 dollars per oz. The sizes from best to worst buy are MLS.

中文解析：

假设S号包含的洗涤剂成分是1 oz, 价格是$1 . 则L号包含的洗涤剂成分是2 oz. M号包含的洗涤剂成分是0.8*2=1.6 oz. M号的价格则是$1.5，L号的价格是1.5*1.3=$1.95. 综合起来每1 oz 的洗涤剂，价格是：S号是：1/1=1；M号是1.5/1.6=0.94；L号是： 1.95/2=0.975；因此从高到低是：LMS. 答案是E。

Problem 23

Answer: B

Solution:

The semi circle has an area of Pi*r*r/2=2*Pi and a radius of 2.

Because this is an isosceles right triangle, the center is the midpoint of the hypotenuse. Radii drawn to the tangent points of the semicircle and the radii also divide the legs into two equal segments. They also create a square in the top left corner. From this, we can conclude the legs of the triangle are twice the length of the radii, 4. The area of the triangle is 4*4/2=8.

中文解析：

假设这个半圆的半径是r, 半圆的面积是Pi*r*r/2=2Pi, 因此半径是2. 如上图所示, OD 垂直与AC 与点D。在直角三角形ADO中，角A是45°，因此AD=OD=2，另外，CD=EO=2. 因此AC=2+2=4. 三角形ABC的面积是：4*4/2=8. 答案是B。

Problem 24

Answer: B

Solution:

It is easier to work backwards from 200, and have keys that display [-1] and [x2]. Use the second key when the number is even, and the first key when the number is odd until you get one. We get: 200, 100, 50, 25, 24, 12, 6, 3, 2, 1, This takes 9 keystrokes.

中文解析：

倒推法，从200倒推，依次是200，100，50，25，24，12，6，3，2， 1. 因此按了9次。答案是B。

Problem 25

Answer: A

Solution:

We realize that since the areas of the regions outside of the circle and the square are equal to each other, the area of the circle must be equal to the area of the square. => r=2/sqrt(Pi).

中文解析：

从题目中说的两部分相等，可以推出正方形的面积和圆形的面积相等。即Pi*r*r=4. pi=2/sqrt(Pi) . 答案是A。