2003 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
计数
Jamie counted the number of edges of a cube, Jimmy counted the numbers of corners, and Judy counted the number of faces. They then added the three numbers. What was the resulting sum?
💡 解题思路
On a cube, there are $12$ edges, $8$ corners, and $6$ faces. Adding them up gets $12+8+6= \boxed{\mathrm{(E)}\ 26}$ .
2
第 2 题
数论
Which of the following numbers has the smallest prime factor? (A)\ 55 (B)\ 57 (C)\ 58 (D)\ 59 (E)\ 61
💡 解题思路
The smallest prime factor is $2$ , and since $58$ is the only multiple of $2$ , the answer is $\boxed{\mathrm{(C)}\ 58}$ .
3
第 3 题
分数与比例
A burger at Ricky C's weighs 120 grams, of which 30 grams are filler. What percent of the burger is not filler? (A)\ 60\% (B)\ 65\% (C)\ 70\% (D)\ 75\% (E)\ 90\%
💡 解题思路
There are $30$ grams of filler, so there are $120-30= 90$ grams that aren't filler. $\frac{90}{120}=\frac{3}{4}=\boxed{\mathrm{(D)}\ 75\%}$ .
4
第 4 题
计数
A group of children riding on bicycles and tricycles rode past Billy Bob's house. Billy Bob counted 7 children and 19 wheels. How many tricycles were there? (A)\ 2 (B)\ 4 (C)\ 5 (D)\ 6 (E)\ 7
💡 解题思路
If all the children were riding bicycles, there would be $2 \times 7=14$ wheels. Each tricycle adds an extra wheel and $19-14=5$ extra wheels are needed, so there are $\boxed{\mathrm{(C)}\ 5}$ tricycl
5
第 5 题
综合
If 20\% of a number is 12 , what is 30\% of the same number?
💡 解题思路
$20\%$ of a number is equal to $\frac{1}{5}$ of that number. Let $n$ =the number
6
第 6 题
几何·面积
Given the areas of the three squares in the figure, what is the area of the interior triangle? [图] (A)\ 13 (B)\ 30 (C)\ 60 (D)\ 300 (E)\ 1800
💡 解题思路
The sides of the squares are $5, 12$ and $13$ for the square with area $25, 144$ and $169$ , respectively. The legs of the interior triangle are $5$ and $12$ , so the area is $\frac{5 \times 12}{2}=\b
7
第 7 题
统计
Blake and Jenny each took four 100 -point tests. Blake averaged 78 on the four tests. Jenny scored 10 points higher than Blake on the first test, 10 points lower than him on the second test, and 20 points higher on both the third and fourth tests. What is the difference between Jenny's average and Blake's average on these four tests? (A)\ 10 (B)\ 15 (C)\ 20 (D)\ 25 (E)\ 40
💡 解题思路
Blake scored a total of $4 \times 78=312$ points. Jenny scored $10-10+20+20=40$ points higher than Blake, so her average is $\frac{312+40}{4}=88$ . the difference is $88-78=\boxed{\mathrm{(A)}\ 10}$ .
8
第 8 题
几何·面积
Problems 8, 9 and 10 use the data found in the accompanying paragraph and figures Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown. \circ Art's cookies are trapezoids. [图] \circ Roger's cookies are rectangles. [图] \circ Paul's cookies are parallelograms. [图] \circ Trisha's cookies are triangles. [图] Each friend uses the same amount of dough, and Art makes exactly 12 cookies. Who gets the fewest cookies from one batch of cookie dough?
💡 解题思路
Out of all the cookies, Art's has an area of $12 \text{ in}^2$ , which was the greatest area out of all the cookies' areas. Roger's cookie had an area of $8 \text{ in} ^2$ , and both Paul and Trisha's
9
第 9 题
几何·面积
Problems 8, 9 and 10 use the data found in the accompanying paragraph and figures Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown. \circ Art's cookies are trapezoids: [图] \circ Roger's cookies are rectangles: [图] \circ Paul's cookies are parallelograms: [图] \circ Trisha's cookies are triangles: [图] Each friend uses the same amount of dough, and Art makes exactly 12 cookies. Art's cookies sell for 60 cents each. To earn the same amount from a single batch, how much should one of Roger's cookies cost in cents?
💡 解题思路
The area of one of Art's cookies is $3 \cdot 3 + \frac{2 \cdot 3}{2}=9+3=12$ . As he has $12$ cookies in a batch, the amount of dough each person used is $12 \cdot 12=144$ . Roger's cookies have an ar
10
第 10 题
几何·面积
Problems 8, 9 and 10 use the data found in the accompanying paragraph and figures Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown. \circ Art's cookies are trapezoids: [图] \circ Roger's cookies are rectangles: [图] \circ Paul's cookies are parallelograms: [图] \circ Trisha's cookies are triangles: [图] How many cookies will be in one batch of Trisha's cookies?
💡 解题思路
Art's cookies have areas of $3 \cdot 3 + \frac{2 \cdot 3}{2}=9+3=12$ . There are 12 cookies in one of Art's batches so everyone used $12 \cdot 12=144 \text{ in}^2$ of dough. Trisha's cookies have an a
11
第 11 题
分数与比例
Business is a little slow at Lou's Fine Shoes, so Lou decides to have a sale. On Friday, Lou increases all of Thursday's prices by 10 percent. Over the weekend, Lou advertises the sale: "Ten percent off the listed price. Sale starts Monday." How much does a pair of shoes cost on Monday that cost 40 dollars on Thursday?
💡 解题思路
On Friday, the shoes would cost $40 \cdot 1.1= 44$ dollars. Then on Monday, the shoes would cost $44- \frac{44}{10}=44-4.4=\boxed{\textbf{(B)}\ 39.60}$ .
12
第 12 题
数论
When a fair six-sided dice is tossed on a table top, the bottom face cannot be seen. What is the probability that the product of the faces that can be seen is divisible by 6 ?
💡 解题思路
We have six cases: each different case, every one where a different number cannot be seen. The rolls that omit numbers one through five are all something times six: an example would be where the numbe
13
第 13 题
行程问题
Fourteen white cubes are put together to form the figure on the right. The complete surface of the figure, including the bottom, is painted red. The figure is then separated into individual cubes. How many of the individual cubes have exactly four red faces [图]
💡 解题思路
This is the number cubes that are adjacent to another cube on exactly two sides. The bottom corner cubes are connected on three sides, and the top corner cubes are connected on one. The number we are
14
第 14 题
数字运算
In this addition problem, each letter stands for a different digit. \setlength{\tabcolsep}{0.5mm}\begin{array}{cccc}&T & W & O ; +&T & W & O ; \hline F& O & U & R\end{array} If T = 7 and the letter O represents an even number, what is the only possible value for W?
💡 解题思路
Since both $T$ 's are $7$ , then $O$ has to equal $4$ , because $7 + 7 = 14$ . Then, $F$ has to equal $1$ . To get $R$ , we do $4 + 4$ (since $O = 4$ ) to get $R = 8$ . The value for $W$ then has to b
15
第 15 题
立体几何
A figure is constructed from unit cubes. Each cube shares at least one face with another cube. What is the minimum number of cubes needed to build a figure with the front and side views shown? [图]
💡 解题思路
In order to minimize the amount of cubes needed, we must match up as many squares of our given figures with each other to make different sides of the same cube. One example of the solution with $\boxe
16
第 16 题
统计
Ali, Bonnie, Carlo, and Dianna are going to drive together to a nearby theme park. The car they are using has 4 seats: 1 Driver seat, 1 front passenger seat, and 2 back passenger seats. Bonnie and Carlo are the only ones who know how to drive the car. How many possible seating arrangements are there?
💡 解题思路
There are only $2$ people who can go in the driver's seat--Bonnie and Carlo. Any of the $3$ remaining people can go in the front passenger seat. There are $2$ people who can go in the first back passe
17
第 17 题
综合
The six children listed below are from two families of three siblings each. Each child has blue or brown eyes and black or blond hair. Children from the same family have at least one of these characteristics in common. Which two children are Jim's siblings? \[\begin{array}{c|c|c}Child&Eye Color&Hair Color ; \hlineBenjamin&Blue&Black ; \hlineJim&Brown&Blonde ; \hlineNadeen&Brown&Black ; \hlineAustin&Blue&Blonde ; \hlineTevyn&Blue&Black ; \hlineSue&Blue&Blonde ; \hline\end{array}\] (E)\ Austin and Sue
💡 解题思路
Jim has brown eyes and blonde hair. If you look for anybody who has brown eyes or blonde hair, you find that Nadeen, Austin, and Sue are Jim's possible siblings. However, the children have at least on
18
第 18 题
坐标几何
Each of the twenty dots on the graph below represents one of Sarah's classmates. Classmates who are friends are connected with a line segment. For her birthday party, Sarah is inviting only the following: all of her friends and all of those classmates who are friends with at least one of her friends. How many classmates will not be invited to Sarah's party? [图]
💡 解题思路
There are $3$ people who are friends with only each other who won't be invited, plus $1$ person who has no friends, and $2$ people who are friends of friends of friends who won’t be invited. So the an
19
第 19 题
数论
How many integers between 1000 and 2000 have all three of the numbers 15, 20, and 25 as factors?
💡 解题思路
Find the least common multiple of $15, 20, 25$ by turning the numbers into their prime factorization. \[15 = 3 \cdot 5, 20 = 2^2 \cdot 5, 25 = 5^2\] Gather all necessary multiples $3, 2^2, 5^2$ when m
20
第 20 题
几何·角度
What is the measure of the acute angle formed by the hands of the clock at 4:20 PM?
💡 解题思路
Imagine the clock as a circle. The minute hand will be at the 4 at 20 minutes past the hour. The central angle formed between $4$ and $5$ is $30$ degrees (since it is 1/12 of a full circle, 360). By $
21
第 21 题
几何·面积
The area of trapezoid ABCD is 164 cm^2 . The altitude is 8 cm, AB is 10 cm, and CD is 17 cm. What is BC , in centimeters? [图]
💡 解题思路
Using the formula for the area of a trapezoid, we have $164=8(\frac{BC+AD}{2})$ . Thus $BC+AD=41$ . Drop perpendiculars from $B$ to $AD$ and from $C$ to $AD$ and let them hit $AD$ at $E$ and $F$ respe
22
第 22 题
几何·面积
The following figures are composed of squares and circles. Which figure has a shaded region with largest area? [图]
💡 解题思路
First we have to find the area of the shaded region in each of the figures. In figure $\textbf{A}$ the area of the shaded region is the area of the circle subtracted from the area of the square. That
23
第 23 题
几何·面积
In the pattern below, the cat moves clockwise through the four squares, and the mouse moves counterclockwise through the eight exterior segments of the four squares. If the pattern is continued, where would the cat and mouse be after the 247th move?
💡 解题思路
Break this problem into two parts: where the cat will be after the $247^{\text{th}}$ move, and where the mouse will be.
24
第 24 题
坐标几何
A ship travels from point A to point B along a semicircular path, centered at Island X . Then it travels along a straight path from B to C . Which of these graphs best shows the ship's distance from Island X as it moves along its course? [图]
💡 解题思路
The distance from Island $\text{X}$ to any point on the semicircle will always be constant. On the graph, this will represent a straight line. The distance between Island $\text{X}$ and line $\text{BC
25
第 25 题
几何·面积
In the figure, the area of square WXYZ is 25 cm^2 . The four smaller squares have sides 1 cm long, either parallel to or coinciding with the sides of the large square. In \triangle ABC , AB = AC , and when \triangle ABC is folded over side \overline{BC} , point A coincides with O , the center of square WXYZ . What is the area of \triangle ABC , in square centimeters? [图]
💡 解题思路
We see that $XY = 5$ , the vertical distance between $B$ and $X$ is $1$ , and the vertical distance between $C$ and $Y$ is $1$ . Therefore, $BC = 5 - 1 - 1 = 3$ . We are given that the length of the a