2004 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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题目涉及图形的部分，原题以文字描述代替（图形题建议配合原版试卷使用）
1
第 1 题
行程问题
On a map, a 12 -centimeter length represents 72 kilometers. How many kilometers does a 17 -centimeter length represent?
💡 解题思路
We set up the proportion $\frac{12 \text{cm}}{72 \text{km}}=\frac{17 \text{cm}}{x \text{km}}$ . Thus $x=102 \Rightarrow \boxed{\textbf{(B)}\ 102}$
2
第 2 题
数字运算
How many different four-digit numbers can be formed by rearranging the four digits in 2004 ?
💡 解题思路
We can solve this problem easily, just by calculating how many choices there are for each of the four digits. First off, we know there are only $2$ choices for the first digit, because $0$ isn't a val
3
第 3 题
综合
Twelve friends met for dinner at Oscar's Overstuffed Oyster House, and each ordered one meal. The portions were so large, there was enough food for 18 people. If they shared, how many meals should they have ordered to have just enough food for the 12 of them?
💡 解题思路
Set up the proportion $\frac{12\ \text{meals}}{18\ \text{people}}=\frac{x\ \text{meals}}{12\ \text{people}}$ . Solving for $x$ gives us $x= \boxed{\textbf{(A)}\ 8}$ .
4
第 4 题
计数
Ms. Hamilton’s eighth-grade class wants to participate in the annual three-person-team basketball tournament. Lance, Sally, Joy, and Fred are chosen for the team. In how many ways can the three starters be chosen?
💡 解题思路
There are $\binom{4}{3}$ ways to choose three starters. Thus the answer is $\boxed{\textbf{(B)}\ 4}$ .
5
第 5 题
规律与数列
Ms. Hamilton's eighth-grade class wants to participate in the annual three-person-team basketball tournament. The losing team of each game is eliminated from the tournament. If sixteen teams compete, how many games will be played to determine the winner?
💡 解题思路
Note that the winning team will the be the only team that wins all of the games. Therefore, to find the total number of games to determine the winner has a 1:1 correspondence to the number of ways to
6
第 6 题
分数与比例
After Sally takes 20 shots, she has made 55\% of her shots. After she takes 5 more shots, she raises her percentage to 56\% . How many of the last 5 shots did she make?
💡 解题思路
Sally made $0.55*20=11$ shots originally. Letting $x$ be the number of shots she made, we have $\frac{11+x}{25}=0.56$ . Solving for $x$ gives us $x=\boxed{\textbf{(C)}\ 3}$
7
第 7 题
行程问题
An athlete's target heart rate, in beats per minute, is 80\% of the theoretical maximum heart rate. The maximum heart rate is found by subtracting the athlete's age, in years, from 220 . To the nearest whole number, what is the target heart rate of an athlete who is 26 years old?
💡 解题思路
The maximum heart rate is $220-26=194$ beats per minute. The target heart rate is then $0.8*194 \approx \boxed{\textbf{(B)}\ 155}$ beats per minute.
8
第 8 题
数字运算
Find the number of two-digit positive integers whose digits total 7 .
💡 解题思路
The numbers are $16, 25, 34, 43, 52, 61, 70$ which gives us a total of $\boxed{\textbf{(B)}\ 7}$ .
9
第 9 题
统计
The average of the five numbers in a list is 54 . The average of the first two numbers is 48 . What is the average of the last three numbers?
💡 解题思路
Let the $5$ numbers be $a, b, c, d$ , and $e$ . Thus $\frac{a+b+c+d+e}{5}=54$ and $a+b+c+d+e=270$ . Since $\frac{a+b}{2}=48$ , $a+b=96$ . Substituting back into our original equation, we have $96+c+d+
10
第 10 题
应用题
Handy Aaron helped a neighbor 1 \frac14 hours on Monday, 50 minutes on Tuesday, from 8:20 to 10:45 on Wednesday morning, and a half-hour on Friday. He is paid \textdollar 3 per hour. How much did he earn for the week?
💡 解题思路
Let's convert everything to minutes and add them together. On Monday he worked for $\frac54 \cdot 60 = 75$ minutes. On Tuesday he worked $50$ minutes. On Wednesday he worked for $2$ hours $25$ minutes
11
第 11 题
统计
The numbers -2, 4, 6, 9 and 12 are rearranged according to these rules: What is the average of the first and last numbers?
💡 解题思路
From rule 1, the largest number, $12$ , can be second or third. From rule 2, because there are five places, the smallest number $-2$ can either be third or fourth. The median, $6$ can be second, third
12
第 12 题
行程问题
Niki usually leaves her cell phone on. If her cell phone is on but she is not actually using it, the battery will last for 24 hours. If she is using it constantly, the battery will last for only 3 hours. Since the last recharge, her phone has been on 9 hours, and during that time she has used it for 60 minutes. If she doesn’t use it any more but leaves the phone on, how many more hours will the battery last?
💡 解题思路
When not being used, the cell phone uses up $\frac{1}{24}$ of its battery per hour. When being used, the cell phone uses up $\frac{1}{3}$ of its battery per hour. Since Niki's phone has been on for $9
13
第 13 题
逻辑推理
Amy, Bill and Celine are friends with different ages. Exactly one of the following statements is true. Rank the friends from the oldest to youngest.
💡 解题思路
If Bill is the oldest, then Amy is not the oldest, and both statements I and II are true, so statement I is not the true one.
14
第 14 题
几何·面积
What is the area enclosed by the geoboard quadrilateral below? [图]
Thirteen black and six white hexagonal tiles were used to create the figure below. If a new figure is created by attaching a border of white tiles with the same size and shape as the others, what will be the difference between the total number of white tiles and the total number of black tiles in the new figure? [图]
💡 解题思路
The first ring around the middle tile has $6$ tiles, and the second has $12$ . From this pattern, the third ring has $18$ tiles. Of these, $6+18=24$ are white and $1+12=13$ are black, with a differenc
16
第 16 题
分数与比例
Two 600 mL pitchers contain orange juice. One pitcher is 1/3 full and the other pitcher is 2/5 full. Water is added to fill each pitcher completely, then both pitchers are poured into one large container. What fraction of the mixture in the large container is orange juice?
💡 解题思路
The first pitcher contains $600 \cdot \frac13 = 200$ mL of orange juice. The second pitcher contains $600 \cdot \frac25 = 240$ mL of orange juice. In the large pitcher, there is a total of $200+240=44
17
第 17 题
计数
Three friends have a total of 6 identical pencils, and each one has at least one pencil. In how many ways can this happen?
💡 解题思路
For each person to have at least one pencil, assign one pencil to each of the three friends so that you have $3$ left. In partitioning the remaining $3$ pencils into $3$ distinct groups, use Ball-and-
18
第 18 题
规律与数列
Five friends compete in a dart-throwing contest. Each one has two darts to throw at the same circular target, and each individual's score is the sum of the scores in the target regions that are hit. The scores for the target regions are the whole numbers 1 through 10 . Each throw hits the target in a region with a different value. The scores are: Alice 16 points, Ben 4 points, Cindy 7 points, Dave 11 points, and Ellen 17 points. Who hits the region worth 6 points?
💡 解题思路
The only way to get Ben's score is with $1+3=4$ . Cindy's score can be made of $3+4$ or $2+5$ , but since Ben already hit the $3$ , Cindy hit $2+5=7$ . Similarly, Dave's darts were in the region $4+7=
19
第 19 题
数论
A whole number larger than 2 leaves a remainder of 2 when divided by each of the numbers 3, 4, 5, and 6 . The smallest such number lies between which two numbers?
💡 解题思路
The smallest number divisible by $3,4,5,$ and $6$ , or their least common multiple, can be found to be $60$ . When $2$ is added to a multiple of number, its remainder when divided by that number is $2
20
第 20 题
综合
Two-thirds of the people in a room are seated in three-fourths of the chairs. The rest of the people are standing. If there are 6 empty chairs, how many people are in the room?
💡 解题思路
Working backwards, if $3/4$ of the chairs are taken and $6$ are empty, then there are three times as many taken chairs as empty chairs, or $3 \cdot 6 = 18$ . If $x$ is the number of people in the room
21
第 21 题
概率
Spinners A and B are spun. On each spinner, the arrow is equally likely to land on each number. What is the probability that the product of the two spinners' numbers is even? [图]
💡 解题思路
An even number comes from multiplying an even and even, even and odd, or odd and even. Since an odd number only comes from multiplying an odd and odd, there are less cases and it would be easier to fi
22
第 22 题
分数与比例
At a party there are only single women and married men with their wives. The probability that a randomly selected woman is single is \frac25 . What fraction of the people in the room are married men?
💡 解题思路
Assume arbitrarily (and WLOG) there are $5$ women in the room, of which $5 \cdot \frac25 = 2$ are single and $5-2=3$ are married. Each married woman came with her husband, so there are $3$ married men
23
第 23 题
坐标几何
Tess runs counterclockwise around rectangular block JKLM . She lives at corner J . Which graph could represent her straight-line distance from home? [图] (A) [图] (B) [图] (C) [图] (D) [图] (E) [图]
💡 解题思路
For her distance to be represented as a constant horizontal line, Tess would have to be running in a circular shape with her home as the center. Since she is running around a rectangle, this is not po
24
第 24 题
几何·面积
In the figure, ABCD is a rectangle and EFGH is a parallelogram. Using the measurements given in the figure, what is the length d of the segment that is perpendicular to \overline{HE} and \overline{FG} ? [图]
💡 解题思路
The area of the parallelogram can be found in two ways. The first is by taking rectangle $ABCD$ and subtracting the areas of the triangles cut out to create parallelogram $EFGH$ . This is \[(4+6)(5+3)
25
第 25 题
几何·面积
Two 4 × 4 squares intersect at right angles, bisecting their intersecting sides, as shown. The circle's diameter is the segment between the two points of intersection. What is the area of the shaded region created by removing the circle from the squares? [图] (A)\ 16-4π (B)\ 16-2π (C)\ 28-4π (D)\ 28-2π (E)\ 32-2π
💡 解题思路
If the circle was shaded in, the intersection of the two squares would be a smaller square with half the sidelength, $2$ . The area of this region would be the two larger squares minus the area of the