美国数学竞赛AMC8 -- 2008年真题解析(英文解析+中文解析)

美国数学竞赛AMC8 – 2008年真题解析(英文解析+中文解析)

Problem 1

Answer: B

Solution:

50-12-24=14

中文解析：

总共花的钱是：12+12*2=36元。剩余50-36=14元。答案是B

Problem 2

Answer: A

Solution:

We can derive that c=8,L=6, U=7,and E=1. Therefore, the answer is 8671.

中文解析：

这10个字母的对应关系是： B -0；E-1; S-2; ......K -9. 按照这个对应关系：C-8，L-6，U-7，E-1. 即8671. 答案是A。

Problem 3

Answer: A

Solution:

We can go backwards by days, but we can also backwards by weeks. If we go backwards by weeks, we see that February 6 is a Friday. If we now go backwards by days, February 1 is a Sunday.

中文解析：

13日是周五，则13-7=6，即6日也是周五，则倒推2月1日是周日。答案是A。

Problem 4

Answer: C

Solution:

The area outside the small triangle but inside the large triangle is 16-1=15. This is equally distributed between the three trapezoids. Each trapezoid has an area of 15/3=5.

中文解析：

大三角形的面积等于小的等边三角形的面积加上3个梯形的面积。据此，三个梯形的面积是16-1=15. 每个梯形的面积是15/3=5. 答案是C。

Problem 5

Answer: E

Solution:

Barney travels 1661-1441=220 miles in 4+6=10 hours for an average of 220/10=22 miles per hour.

中文解析：

行驶的里程是：1661-1441=220. 时间是4+6=10小时。平均速度是220/10=22. 答案是E。Problem 6

Answer: D

Solution:

Dividing the gray square into four smaller squares, there are 6 gray tiles and 10 white tiles, giving a ratio of 3:5.

中文解析：

灰色的方块是6个，白色的方块是（4*4）-6=10个。因此比值是：6：10=3：5. 答案是D。Problem 7

Answer: E

Solution:

Separate into two equations 3/5=M/45 , 3/5=60/N，and solve for the unknowns. M=27 , N=100, therefore M+N=127.

中文解析：

M/45=3/5 ,因此M=27. 3/5=60/N；因此N=100. M+N=27+100=127. 答案是E。

Problem 8

Answer: D

Solution:

There are a total of 100+60+40+120=320 dollars of sales spread through 4 months, for an average of 320/4=80.

中文解析：

这四个月的数据分别是：100，60，40，120. 平均值是（100+60+40+120）/4 =80. 答案是D。

Problem 9

Answer: D

Solution:

After the 15% loss, Tammy has 100*0.85=85 dollars. After the 20% gain, she has 85*1.2=102 dollars. This is an increase in 2 dollars from her original 100 dollars, a 2% gain.

中文解析：

第一年后的金额是：100*0.85，即85. 第二年后的金额是：85*1.2=102。这两年收益2%。答案是D。

Problem 10

Answer: D

Solution:

The total of all their ages over the number of people is (6*40+4*25)/(6+4)=34.

中文解析：

A房间的总年龄数是：40*6；B房间的总年龄数是25*4. 这两个房间合一起后，总年龄数是

40*6+25*4=340. 总人数是10人。因此平均年龄是340/10=34. 答案是D。

Problem 11

Answer: A

Solution:

The union of two sets is equal to the sum of each set minus their intersection. The number of students that have both a dog and a cat is 20+26-39=7.

中文解析：

画Venn Diagram。20个学生有狗，26个学生有猫，总人数是39人。既有猫又有狗的人数是：20+26-39=7人。答案是A .

Problem 12

Answer: C

Solution:

Each bounce is 2/3 times the height of the previous bounce. The first bounce reaches 2 meters, the second 4/3, the third 8/9, the fourth 16/27, and the fifth 32/81. Half of 81 is 40.5, so the ball does not reach the required height on bounce 5.

中文解析：

这是一个等比数列（Geometric Sequence）：2，4/3，8/9，16/27，32/81. 其中32/81已经比0.5小。是第5次。答案是C。

Problem 13

Answer: C

Solution:

Each box is weighed twice during this, so the combined weight of the three boxes is half the weight of these separate measures.

中文解析：

两个箱子组合一起的重量分别是：122，125，127. 则这三个数字的和是这三个箱子的总重量的2倍。因此这三个箱子的总重量是187.

Problem 14

Answer: C

Solution:

There are 2 ways to place the B in first row. Accordingly, there are 2 ways to place the B in second row each. The remaining A, B or C no more choice. Totally 2*2=4.

中文解析：

第一行的B有个位置可选择，分别对应的第二行的B有两个位置可选择，共4种选择。当第二行的B也选好位置后，剩下的位置就没有别的选择机会了。

Problem 15

Answer: B

Solution:

The total number of points from the first 8 games is 37. We have to make this a multiple of 9 by scoring less than 10 points. The closest multiple of 9 is 45. 45-37=8, now we have to add a number to get a multiple of 10. The next multiple is 50 we added 5, multiplying these together you get 8*5 is 40. The answer is 40.

中文解析：

前八次比赛的分数综合是37. 假设第九次的分数是x, (x<10)，且前9次的平均值是个整数,则37+x 是9的倍数，x只能是8. 这样前9次的总分是45分。同样道理，假设第10次的分数是y, 则45+y 必须是10的倍数，且y<10, y只能是5. 因此第9次和第10次的分数乘积是8*5=40. 答案是B。

Problem 16

Answer: D

Solution:

The volume is of seven unit cubes which is 7. The surface area is the same for each of the protruding cubes which is 5*6=30. The ratio of the volume to the surface area is 7:30.

中文解析：

这个形状是由7个单位体积的小正方体组成，因此体积是7. 每个小正方体的表面积都是6，7个小正方体的表面积和是6*7=42. 其中看不到的面有12个。因此这个形状的表面积是42-12=30个。体积与表面积的比值是：7：30. 答案是D。

Problem 17

Answer: D

Solution:

A rectangle's area is maximized when its length and width are equivalent, or the two side lengths are closest together in this case with integer lengths. This occurs with the sides 12*13=156. Likewise, the area is smallest when the side lengths have the greatest difference, which is

1*24=24. The difference in area is 156-24=132.

中文解析：

长方形的面积最大是当长和宽的长度尽量相近时，由于长方形的周长是50，相邻两条边的和是25，则当一条边是12，另一条边是13时，面积最大，是12*13=156. 长方形的面积最小是当长方形的长和宽相差最大时，即一条边是24，另一条边是1时，此时的面积24*1=24. 两种情况下的面积差是156-24=132. 答案是D。

Problem 18

Answer: E

Solution:

We will deal with this part by part: Part 1: 1/4 circumference of big circle=1/4*(2*Pi*20)=10Pi; Part 2: Big radius minus small radius=20-10=10; Part 3: 1/4 circumference of small

circle=1/4*(2*Pi*10)=5Pi; Part 4: Diameter of small circle:2*10=20; Part 5: Same as part 3: 5Pi; Part 6: Same as part 2: 10; Total:10Pi +10+5Pi+20+5Pi+10=20Pi+40.

中文解析：

大圆的1/4弧长是：Pi*2*20/4=10Pi. 小圆的1/4弧长是Pi*2*10/4=5Pi. 按照题目箭头所示路径的长度从A到K，分别是: 10Pi, 10, 5Pi, 20, 5Pi, 10. 总和是40+20Pi. 答案是E。

Problem 19

Answer: B

Solution:

Arbitrarily pick a point in the grid. Clearly, we see two options for the other point to be placed, so the answer is 2/7.

中文解析：

任意一个点，有7个点可以连接成一个线段。其中，在1个单位距离内的，只有相邻的有2个点。因此本题所求的概率是2/7. 答案是B。

Problem 20

Answer: B

Solution:

Let b be the number of boys and g be the number of girls. 2/3*b=3/4 * g; =>b=9/8*g; For g and b to be integers, he smallest possible value of g is 8. This yields 9 boys. The minimum number of students is 8+9=17.

中文解析：

2/3的男生的人数等于3/4的女生人数。假设男生有b人，女生有g人，得：2/3 *b=3/4 * g. 上式两边等乘以12，得到：8*b=9*g。由于男生和女生的人数都必须是正整数，因此最小b=9,g=8. 班上总人数是9+8=17人。答案是B。

Problem 21

Answer: C

Solution:

The slice is cutting the cylinder into two equal wedges with equal area. The cylinder's volume is Pi*4*4*6=96PI,. The area of the wedge is half this which is 48Pi, 151 is the closest.

中文解析：

这个圆柱的体积是Pi*4*4*6=96Pi. 分成的这两个wedge体积相等，因此一个的体积是96Pi/2=48Pi,即151，答案是C。

Problem 22

Answer: A

Solution:

If n/3 is a three digit whole number, n must be divisible by 3 and be>=100*3=300. . If 3n is three digits, n must be<=999/3=333. So it must be divisible by three and between 300 and 333. There are 12 such numbers, which you can find by direct counting.

中文解析：

n/3 是3位的正整数，则n 可以是：100*3, 101*3,102*3,......999*3. 这些书都是3的倍数。3n 是3位的正整数，则n可以是：34，35，36, ...333. 两者都满足要求的n 是300到333之间的3的倍数，即：300，303，309，312，。。。333. 即：100*3，101*3，102*3，103*3，......111*3. 共计：111-100+1=12个。答案是A。

Problem 23

Answer: C

Solution:

The area of triangle BFD is the area of square ABCD subtracted by the area of the three triangles around it. Arbitrarily assign the side length of the square to be 3. The ratio of the area of triangle BFD to the area of ABCD is (3*3-1/2*3*2-1/2*3*2-1/2*1*1)/3*3=2.5/9=5/18.

中文解析：

假设这个正方形的边长是3，则AF=2，EF=1，CD=2，DE=1. 三角形ABF的面积是1/2*3*2=3. 三角形BCD的面积和ABF相同。三角形EDF的面积是：1/2 *1*1=0.5. 三角形BFD的面积是正方形的面积减去三角形ABF，减去三角形BCD，减去三角形EDF的面积，即：3*3-3-3-0.5=2.5. 其和正方形的面积的比值是： 2.5/9 .即5/18. 答案是C。

Problem 24

Answer: C

Solution:

The numbers can at most multiply to be 60. The squares less than 60 are 1,4,9,16,25,36,and49. The possible pairs are(1,1)，（1，4），（2，2），（4，1），（3，3，）（9，1），（4，4），（8，2），（5，5）, (6,6), and (9,4). There are 11 choices and 60 possibilities giving a probability of 11/60.

中文解析：

Tile的编号是1，2，...10. 色子的编号是：1，2，...6. Tile和Die共可以组成10*6=60组。其乘积最大值是60. 小于等于60的平方数有：1，4，9，16，25，36，49. Tile和die的数字相乘，乘积是平方数的数对分别是：（1，1），（1，4），（4，1），（2，2），（3，3），（9，1），（8，2），（4，4），（5，5），（9，4），（6，6）. 即有11对。占比：11/60. 答案是C。

Problem 25

Answer: A

Solution:

Circle 1(black): radius: 2, area: 4Pi;

Circle 2(white)：radius:4，area：16Pi；

Circle 3(black): radius: 6, area: 36Pi;

Circle 4(white)：radius: 8，area：64Pi；

Circle 5(black): radius: 10, area: 100Pi;

Circle 6(white)：radius:12，area：144Pi；

The entire circle's area is 144Pi. The area of the black regions is (100-64)PI+(36-16)Pi+4Pi=60Pi. The percentage of the design that is black is 60Pi/144Pi=5/12=42.

中文解析：

从小到大的圆形（包括黑色和白色）的半径分别是：2，4，6，8，10，12. 从小到大的圆形（包括黑色和白色）的面积分别是: 4Pi, 16Pi, 36Pi, 64Pi, 100Pi, 144Pi. 最小的黑色圆形的面积是4Pi，其次的圆形部分的面积是36Pi-16Pi=20Pi, 最大的黑色部分的面积是100Pi-64Pi=36Pi. 3个黑色部分的面积之和是：4Pi+20Pi+36Pi=60Pi. 整个图形的面积是144Pi。黑色部分所占的比例是60/144=42%. 答案是A。

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AMC12美国数学竞赛 2012-2014

AMC12 2014A Problem 1 What is Solution At the theater children get in for half price. The price for adult tickets and child tickets is . How much would adult tickets and child tickets cost? Solution Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible? Solution Suppose that cows give gallons of milk in days. At this rate, how many gallons of milk will cows give in days? Solution

On an algebra quiz, of the students scored points, scored points, scored points, and the rest scored points. What is the difference between the mean and median score of the students' scores on this quiz? Solution The difference between a two-digit number and the number obtained by reversing its digits is times the sum of the digits of either number. What is the sum of the two digit number and its reverse? Solution The first three terms of a geometric progression are , , and . What is the fourth term? Solution A customer who intends to purchase an appliance has three coupons, only one of which may be used: Coupon 1: off the listed price if the listed price is at least Coupon 2: dollars off the listed price if the listed price is at least Coupon 3: off the amount by which the listed price exceeds For which of the following listed prices will coupon offer a greater price reduction than either coupon or coupon ?

美国数学竞赛AMC12词汇

A abbreviation 简写符号；简写 absolute error 绝对误差 absolute value 绝对值 accuracy 准确度 acute angle 锐角 acute-angled triangle 锐角三角形 add 加 addition 加法 addition formula 加法公式 addition law 加法定律 addition law(of probability)（概率）加法定律additive property 可加性 adjacent angle 邻角 adjacent side 邻边 algebra 代数 algebraic 代数的 algebraic equation 代数方程 algebraic expression 代数式 algebraic fraction 代数分式；代数分数式algebraic inequality 代数不等式 algebraic operation 代数运算 alternate angle （交）错角 alternate segment 交错弓形 altitude 高；高度；顶垂线；高线 ambiguous case 两义情况；二义情况 amount 本利和；总数 analysis 分析；解析 analytic geometry 解析几何 angle 角 angle at the centre 圆心角 angle at the circumference 圆周角 angle between a line and a plane 直与平面的交角 angle between two planes 两平面的交角 angle bisection 角平分 angle bisector 角平分线 ；分角线 angle in the alternate segment 交错弓形的圆周角angle in the same segment 同弓形内的圆周角angle of depression 俯角 angle of elevation 仰角 angle of greatest slope 最大斜率的角 angle of inclination 倾斜角angle of intersection 相交角；交角 angle of rotation 旋转角 angle of the sector 扇形角 angle sum of a triangle 三角形内角和 angles at a point 同顶角 annum(X% per annum) 年（年利率X%） anti-clockwise direction 逆时针方向；返时针方向anti-logarithm 逆对数；反对数 anti-symmetric 反对称 apex 顶点 approach 接近；趋近 approximate value 近似值 approximation 近似；略计；逼近 Arabic system 阿刺伯数字系统 arbitrary 任意 arbitrary constant 任意常数 arc 弧 arc length 弧长 arc-cosine function 反余弦函数 arc-sin function 反正弦函数 arc-tangent function 反正切函数 area 面积 arithmetic 算术 arithmetic mean 算术平均；等差中顶；算术中顶arithmetic progression 算术级数；等差级数arithmetic sequence 等差序列 arithmetic series 等差级数 arm 边 arrow 前号 ascending order 递升序 ascending powers of X X 的升幂 associative law 结合律 assumed mean 假定平均数 assumption 假定；假设 average 平均；平均数；平均值 average speed 平均速率 axiom 公理 axis 轴 axis of parabola 拋物线的轴 axis of symmetry 对称轴

AMC 美国数学竞赛试题 详解 英文版

2013 AMC8 Problems 1. Danica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars. What is the smallest number of additional cars she must buy in order to be able to arrange all her cars this way? 2. A sign at the fish market says, "50% off, today only: half-pound packages for just $3 per package." What is the regular price for a full pound of fish, in dollars? What is the value of ? 3. 4. Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven friends paid an extra $2.50 to cover her portion of the total bill. What was the total bill? 5. Hammie is in the grade and weighs 106 pounds. His quadruplet sisters are tiny babies and weigh 5, 5, 6, and 8 pounds. Which is greater, the average (mean) weight of these five children or the median weight, and by how many pounds?

AMC美国数学竞赛AMCB试题及答案解析

A M C美国数学竞赛 A M C B试题及答案解析 The latest revision on November 22, 2020

2003 AMC 10B 1、Which of the following is the same as 2、Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs more than a pink pill, and Al’s pills cost a total of for the two weeks. How much does one green pill cost 3、The sum of 5 consecutive even integers is less than the sum of the rst consecutive odd counting numbers. What is the smallest of the even integers 4、Rose fills each of the rectangular regions of her rectangular flower bed with a different type of flower. The lengths, in feet, of the rectangular regions in her flower bed are as shown in the gure. She plants one flower per square foot in each region. Asters cost 1 each, begonias each, cannas 2 each, dahlias each, and Easter lilies 3 each. What is the least possible cost, in dollars, for her garden 5、Moe uses a mower to cut his rectangular -foot by -foot lawn. The swath he cuts is inches wide, but he overlaps each cut by inches to make sure that no grass is missed. He walks at the rate of feet per hour while pushing the mower. Which of the following is closest to the number of hours it will take Moe to mow his lawn

2004 AMC12A(美国数学竞赛)

Alicia earns dollars per hour, of which is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes? Solution On the AMC 12, each correct answer is worth points, each incorrect answer is worth points, and each problem left unanswered is worth points. If Charlyn leaves of the problems unanswered, how many of the remaining problems must she answer correctly in order to score at least ? Solution For how many ordered pairs of positive integers is ? Solution Bertha has daughters and no sons. Some of her daughters have daughters, and the rest have none. Bertha has a total of daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and grand-daughters have no children? Solution