2002 AMC 8 — Official Competition Problems (November 2019)
📅 2011 年11月📝 25题选择题⏱ 40分钟🎯 满分25分✅ 含解题思路👥 612 人已练习
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1
第 1 题
几何·面积
A circle and two distinct lines are drawn on a sheet of paper. What is the largest possible number of points of intersection of these figures? \text {(A)}\ 2 \text {(B)}\ 3 {(C)}\ 4 {(D)}\ 5 {(E)}\ 6
💡 解题思路
The two lines can both intersect the circle twice, and can intersect each other once, so $2+2+1= \boxed {\text {(D)}\ 5}.$
2
第 2 题
计数
How many different combinations of \5 bills and \2 bills can be used to make a total of \17? Order does not matter in this problem. \text {(A)}\ 2 \text {(B)}\ 3 \text {(C)}\ 4 \text {(D)}\ 5 \text {(E)}\ 6$
💡 解题思路
https://youtu.be/Zhsb5lv6jCI?t=701
3
第 3 题
统计
What is the smallest possible average of four distinct positive even integers? (A)\ 3 (B)\ 4 (C)\ 5 (D)\ 6 (E)\ 7
💡 解题思路
In order to get the smallest possible average, we want the 4 even numbers to be as small as possible. The first 4 positive even numbers are 2, 4, 6, and 8. Their average is $\frac{2+4+6+8}{4}=\frac{20
4
第 4 题
数字运算
The year 2002 is a palindrome (a number that reads the same from left to right as it does from right to left). What is the product of the digits of the next year after 2002 that is a palindrome? (A)\ 0 (B)\ 4 (C)\ 9 (D)\ 16 (E)\ 25
💡 解题思路
The palindrome right after 2002 is 2112. The product of the digits of 2112 is $\boxed{\text{(B)}\ 4}$ .
5
第 5 题
综合
Carlos Montado was born on Saturday, November 9, 2002. On what day of the week will Carlos be 706 days old? (A)\ Monday (B)\ Wednesday (C)\ Friday (D)\ Saturday (E)\ Sunday
💡 解题思路
Days of the week have a cycle that repeats every $7$ days. Thus, after $100$ cycles, or $700$ days, it will be Saturday again. Six more days will make it $\text{Friday} \rightarrow \boxed{C}$
6
第 6 题
坐标几何
A birdbath is designed to overflow so that it will be self-cleaning. Water flows in at the rate of 20 milliliters per minute and drains at the rate of 18 milliliters per minute. Which one of these graphs shows the volume of water in the birdbath during the filling time and continuing into the overflow time? [图] (A)\ A (B)\ B (C)\ C (D)\ D (E)\ E
💡 解题思路
The change in the water volume has a net gain of $20-18=2$ millimeters per minute. The birdbath's volume increases at a constant rate until it reaches its maximum and starts overflowing to keep a cons
7
第 7 题
坐标几何
The students in Mrs. Sawyer's class were asked to do a taste test of five kinds of candy. Each student chose one kind of candy. A bar graph of their preferences is shown. What percent of her class chose candy E? [图] (A)\ 5 (B)\ 12 (C)\ 15 (D)\ 16 (E)\ 20
💡 解题思路
From the bar graph, we can see that $5$ students chose candy E. There are $6+8+4+2+5=25$ total students in Mrs. Sawyers class. The percent that chose E is $\frac{5}{25} \cdot 100 = \boxed{\text{(E)}\
8
第 8 题
坐标几何
Problems 8,9 and 10 use the data found in the accompanying paragraph and table: [图] How many of his European stamps were issued in the '80s? (A)\ 9 (B)\ 15 (C)\ 18 (D)\ 24 (E)\ 42
💡 解题思路
France and Spain are European countries. The number of '80s stamps from France is $15$ and the number of '80s stamps from Spain is $9$ . The total number of stamps is $15+9=\boxed{\text{(D)}\ 24}$ .
9
第 9 题
坐标几何
Problems 8,9 and 10 use the data found in the accompanying paragraph and table: [图] In dollars and cents, how much did his South American stamps issued before the ’70s cost him? (A)\ \0.40 (B)\ \1.06 (C)\ \1.80 (D)\ \2.38 (E)\ \2.64$
💡 解题思路
Brazil 50s and 60s total 11 stamps with each 6 cents, Peru 50s and 60s total 10 stamps with each 4 cents. So total $11*0.06+10*0.04 = \boxed{\text{(B)}\ \$1.06}$
10
第 10 题
坐标几何
Problems 8,9 and 10 use the data found in the accompanying paragraph and table: [图] The average price of his '70s stamps is closest to (A)\ 3.5 cents (B)\ 4 cents (C)\ 4.5 cents (D)\ 5 cents (E)\ 5.5 cents
💡 解题思路
The price of all the stamps in the '70s together over the total number of stamps is equal to the average price.
11
第 11 题
几何·面积
A sequence of squares is made of identical square tiles. The edge of each square is one tile length longer than the edge of the previous square. The first three squares are shown. How many more tiles does the seventh square require than the sixth? [图] (A)\ 11 (B)\ 12 (C)\ 13 (D)\ 14 (E)\ 15
💡 解题思路
The first square has a sidelength of $1$ , the second square $2$ , and so on. The seventh square has $7$ and is made of $7^2=49$ unit tiles. The sixth square has $6$ and is made of $6^2=36$ unit tiles
12
第 12 题
概率
A board game spinner is divided into three regions labeled A , B and C . The probability of the arrow stopping on region A is \frac{1}{3} and on region B is \frac{1}{2} . The probability of the arrow stopping on region C is: (A)\ \frac{1}{12} (B)\ \frac{1}{6} (C)\ \frac{1}{5} (D)\ \frac{1}{3} (E)\ \frac{2}{5}
💡 解题思路
Since the arrow must land in one of the three regions, the sum of the probabilities must be 1. Thus the answer is $1-\frac{1}{2}-\frac{1}{3}=\boxed{\text{(B)}\ \frac16}$ .
13
第 13 题
工程问题
For his birthday, Bert gets a box that holds 125 jellybeans when filled to capacity. A few weeks later, Carrie gets a larger box full of jellybeans. Her box is twice as high, twice as wide and twice as long as Bert's. Approximately, how many jellybeans did Carrie get? (A)\ 250 (B)\ 500 (C)\ 625 (D)\ 750 (E)\ 1000
💡 解题思路
Since the volume ratio is equal to the sides ratio cubed, then the ratio of the larger box's volume to the smaller one is 2 cubed.
14
第 14 题
计数
A merchant offers a large group of items at 30\% off. Later, the merchant takes 20\% off these sale prices. The total discount is (A)\ 35\% (B)\ 44\% (C)\ 50\% (D)\ 56\% (E)\ 60\%
💡 解题思路
Let's assume that each item is $100$ dollars. First we take off $30\%$ off of $100$ dollars. $100\cdot0.7=70$
15
第 15 题
几何·面积
Which of the following polygons has the largest area? [图]
💡 解题思路
Each polygon can be partitioned into unit squares and right triangles with sidelength $1$ . Count the number of boxes enclosed by each polygon, with the unit square being $1$ , and the triangle being
16
第 16 题
几何·面积
Right isosceles triangles are constructed on the sides of a 3-4-5 right triangle, as shown. A capital letter represents the area of each triangle. Which one of the following is true? [图]
💡 解题思路
The area of a right triangle can be found by using the legs of triangle as the base and height. In the three isosceles triangles, the length of their second leg is the same as the side that is connect
17
第 17 题
综合
In a mathematics contest with ten problems, a student gains 5 points for a correct answer and loses 2 points for an incorrect answer. If Olivia answered every problem and her score was 29, how many correct answers did she have? (A)\ 5 (B)\ 6 (C)\ 7 (D)\ 8 (E)\ 9
💡 解题思路
We can try to guess and check to find the answer. If she got five right, her score would be $(5*5)-(5*2)=15$ . If she got six right her score would be $(6*5)-(2*4)=22$ . That's close, but it's still n
18
第 18 题
统计
Gage skated 1 hr 15 min each day for 5 days and 1 hr 30 min each day for 3 days. How long would he have to skate the ninth day in order to average 85 minutes of skating each day for the entire time? (A)\ 1 hr (B)\ 1 hr 10 min (C)\ 1 hr 20 min (D)\ 1 hr 40 min (E)\ 2 hr
💡 解题思路
Converting into minutes and adding, we get that she skated $75*5+90*3+x = 375+270+x = 645+x$ minutes total, where $x$ is the amount she skated on day $9$ . Dividing by $9$ to get the average, we get $
19
第 19 题
综合
How many whole numbers between 99 and 999 contain exactly one 0? (A)\ 72 (B)\ 90 (C)\ 144 (D)\ 162 (E)\ 180
💡 解题思路
Numbers with exactly one zero have the form $\overline{a0b}$ or $\overline{ab0}$ , where the $a,b \neq 0$ . There are $(9\cdot1\cdot9)+(9\cdot9\cdot1) = 81+81 = \boxed{162}$ such numbers, hence our an
20
第 20 题
几何·面积
The area of triangle XYZ is 8 square inches. Points A and B are midpoints of congruent segments \overline{XY} and \overline{XZ} . Altitude \overline{XC} bisects \overline{YZ} . The area (in square inches) of the shaded region is [图]
💡 解题思路
We know the area of triangle $XYZ$ is $8$ square inches. The area of a triangle can also be represented as $\frac{bh}{2}$ or in this problem $\frac{XC\cdot YZ}{2}$ . By solving, we have \[\frac{XC\cdo
21
第 21 题
概率
Harold tosses a coin four times. The probability that he gets at least as many heads as tails is (A)\ \frac{5}{16} (B)\ \frac{3}{8} (C)\ \frac{1}{2} (D)\ \frac{5}{8} (E)\ \frac{11}{16}
💡 解题思路
Case 1: There are two heads and two tails. There are $\binom{4}{2} = 6$ ways to choose which two tosses are heads, and the other two must be tails.
22
第 22 题
几何·面积
Six cubes, each an inch on an edge, are fastened together, as shown. Find the total surface area in square inches. Include the top, bottom, and sides. [图]
💡 解题思路
Count the number of sides that are not exposed, where a cube is connected to another cube and subtract it from the total number of faces. There are $5$ places with two adjacent cubes, covering $10$ si
23
第 23 题
分数与比例
A corner of a tiled floor is shown. If the entire floor is tiled in this way and each of the four corners looks like this one, then what fraction of the tiled floor is made of darker tiles? [图]
💡 解题思路
The same pattern is repeated for every $6 \times 6$ tile. Looking closer, there is also symmetry of the top $3 \times 3$ square, so the fraction of the entire floor in dark tiles is the same as the fr
24
第 24 题
分数与比例
Miki has a dozen oranges of the same size and a dozen pears of the same size. Miki uses her juicer to extract 8 ounces of pear juice from 3 pears and 8 ounces of orange juice from 2 oranges. She makes a pear-orange juice blend from an equal number of pears and oranges. What percent of the blend is pear juice? (A)\ 30 (B)\ 40 (C)\ 50 (D)\ 60 (E)\ 70
💡 解题思路
A pear gives $8/3$ ounces of juice per pear. An orange gives $8/2=4$ ounces of juice per orange. If the pear-orange juice blend used one pear and one orange each, the percentage of pear juice would be
25
第 25 题
分数与比例
Loki, Moe, Nick and Ott are good friends. Ott had no money, but the others did. Moe gave Ott one-fifth of his money, Loki gave Ott one-fourth of his money and Nick gave Ott one-third of his money. Each gave Ott the same amount of money. What fractional part of the group's money does Ott now have? (A)\ \frac{1}{10} (B)\ \frac{1}{4} (C)\ \frac{1}{3} (D)\ \frac{2}{5} (E)\ \frac{1}{2}
💡 解题思路
Since Ott gets equal amounts of money from each friend, we can say that he gets $x$ dollars from each friend. This means that Moe has $5x$ dollars, Loki has $4x$ dollars, and Nick has $3x$ dollars. Th