2009 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
综合
Bridget bought a bag of apples at the grocery store. She gave half of the apples to Ann. Then she gave Cassie 3 apples, keeping 4 apples for herself. How many apples did Bridget buy?
💡 解题思路
If we set up an equation, we find out $x=(3+4)\cdot 2$ because 3 apples were left after giving half, then four away. We can simplify the equations to $x=7\cdot 2=14.$ The answer is $\text{(E) } 14.$
2
第 2 题
统计
On average, for every 4 sports cars sold at the local dealership, 7 sedans are sold. The dealership predicts that it will sell 28 sports cars next month. How many sedans does it expect to sell?
💡 解题思路
This means the ratio is $4:7$ . If the ratio now is $28:x$ , then that means $x= \boxed{\textbf{(D) }49}$ .
3
第 3 题
坐标几何
The graph shows the constant rate at which Suzanna rides her bike. If she rides a total of a half an hour at the same speed, how many miles would she have ridden? [图]
💡 解题思路
Suzanna's speed is $\frac{1}{5}$ . This means she runs $\frac{1}{5} \cdot 30 = \boxed{ \textbf{(C) }6 }$
4
第 4 题
统计
The five pieces shown below can be arranged to form four of the five figures shown in the choices. Which figure cannot be formed? [图]
💡 解题思路
The answer is $\boxed{\textbf{(B)}}$ because the longest piece cannot fit into the figure.
5
第 5 题
规律与数列
A sequence of numbers starts with 1 , 2 , and 3 . The fourth number of the sequence is the sum of the previous three numbers in the sequence: 1+2+3=6 . In the same way, every number after the fourth is the sum of the previous three numbers. What is the eighth number in the sequence?
💡 解题思路
List them out, adding the three previous numbers to get the next number,
6
第 6 题
时间问题
Steve's empty swimming pool will hold 24,000 gallons of water when full. It will be filled by 4 hoses, each of which supplies 2.5 gallons of water per minute. How many hours will it take to fill Steve's pool?
💡 解题思路
Each of the four hoses hose fills $24,000/4 = 6,000$ gallons of water. At the rate it goes at it will take $6,000/2.5 = 2400$ minutes, or $\boxed{\textbf{(A)}\ 40}$ hours.
7
第 7 题
几何·面积
The triangular plot of ACD lies between Aspen Road, Brown Road and a railroad. Main Street runs east and west, and the railroad runs north and south. The numbers in the diagram indicate distances in miles. The width of the railroad track can be ignored. How many square miles are in the plot of land ACD? [图]
💡 解题思路
The area of a triangle is $\frac12 bh$ . If we let $CD$ be the base of the triangle, then the height is $AB$ , and the area is $\frac12 \cdot 3 \cdot 3 = \boxed{\textbf{(C)}\ 4.5}$ .
8
第 8 题
几何·面积
The length of a rectangle is increased by 10\% percent and the width is decreased by 10\% percent. What percent of the old area is the new area?
💡 解题思路
In a rectangle with dimensions $10 \times 10$ , the new rectangle would have dimensions $11 \times 9$ . The ratio of the new area to the old area is $99/100 = \boxed{\textbf{(B)}\ 99}$ .
9
第 9 题
几何·面积
Construct a square on one side of an equilateral triangle. On one non-adjacent side of the square, construct a regular pentagon, as shown. On a non-adjacent side of the pentagon, construct a hexagon. Continue to construct regular polygons in the same way, until you construct an octagon. How many sides does the resulting polygon have? [图]
💡 解题思路
Of the six shapes used to create the polygon, the triangle and octagon are adjacent to the others on one side, and the others are adjacent on two sides. In the triangle and octagon $3+8-2(1)=9$ sides
10
第 10 题
几何·面积
On a checkerboard composed of 64 unit squares, what is the probability that a randomly chosen unit square does not touch the outer edge of the board? [图]
💡 解题思路
There are $8^2=64$ total squares. There are $(8-1)(4)=28$ unit squares on the perimeter and therefore $64-28=36$ NOT on the perimeter. The probability of choosing one of those squares is $\frac{36}{64
11
第 11 题
应用题
The Amaco Middle School bookstore sells pencils costing a whole number of cents. Some seventh graders each bought a pencil, paying a total of 1.43 dollars. Some of the 30 sixth graders each bought a pencil, and they paid a total of 1.95 dollars. How many more sixth graders than seventh graders bought a pencil?
💡 解题思路
Because the pencil costs a whole number of cents, the cost must be a factor of both $143$ and $195$ . They can be factored into $11\cdot13$ and $3\cdot5\cdot13$ . The common factor cannot be $1$ or th
12
第 12 题
数论
The two spinners shown are spun once and each lands on one of the numbered sectors. What is the probability that the sum of the numbers in the two sectors is prime? [图] [图]
A three-digit integer contains one of each of the digits 1 , 3 , and 5 . What is the probability that the integer is divisible by 5 ?
💡 解题思路
The three digit numbers are $135,153,351,315,513,531$ . The numbers that end in $5$ are divisible are $5$ , and the probability of choosing those numbers is $\boxed{\textbf{(B)}\ \frac13}$ .
14
第 14 题
统计
Austin and Temple are 50 miles apart along Interstate 35. Bonnie drove from Austin to her daughter's house in Temple, averaging 60 miles per hour. Leaving the car with her daughter, Bonnie rode a bus back to Austin along the same route and averaged 40 miles per hour on the return trip. What was the average speed for the round trip, in miles per hour?
💡 解题思路
The way to Temple took $\frac{50}{60}=\frac56$ hours, and the way back took $\frac{50}{40}=\frac54$ for a total of $\frac56 + \frac54 = \frac{25}{12}$ hours. The trip is $50\cdot2=100$ miles. The aver
15
第 15 题
几何·面积
A recipe that makes 5 servings of hot chocolate requires 2 squares of chocolate, \frac{1}{4} cup sugar, 1 cup water and 4 cups milk. Jordan has 5 squares of chocolate, 2 cups of sugar, lots of water, and 7 cups of milk. If he maintains the same ratio of ingredients, what is the greatest number of servings of hot chocolate he can make?
💡 解题思路
Assuming excesses of the other ingredients, the chocolate can make $\frac52 \cdot 5=12.5$ servings, the sugar can make $\frac{2}{1/4} \cdot 5 = 40$ servings, the water can make unlimited servings, and
16
第 16 题
数字运算
How many 3 -digit positive integers have digits whose product equals 24 ?
💡 解题思路
With the digits listed from least to greatest, the $3$ -digit integers are $138,146,226,234$ . $226$ can be arranged in $\frac{3!}{2!} = 3$ ways, and the other three can be arranged in $3!=6$ ways. Th
17
第 17 题
几何·面积
The positive integers x and y are the two smallest positive integers for which the product of 360 and x is a square and the product of 360 and y is a cube. What is the sum of x and y ?
💡 解题思路
Take the prime factorization of $360$ . $360=2^3*3^2*5$ . We want $x$ to be as small as possible. And you want $x*360$ to be a square. So $x=2*5=10$ . $y$ is similar. $y=3*5^2=3*25=75$ So, $x+y=75+10=
18
第 18 题
几何·面积
The diagram represents a 7 -foot-by- 7 -foot floor that is tiled with 1 -square-foot black tiles and white tiles. Notice that the corners have white tiles. If a 15 -foot-by- 15 -foot floor is to be tiled in the same manner, how many white tiles will be needed? [图]
💡 解题思路
In a $1$ -foot-by- $1$ -foot floor, there is $1$ white tile. In a $3$ -by- $3$ , there are $4$ . Continuing on, you can deduce the $n^{th}$ positive odd integer floor has $n^2$ white tiles. $15$ is th
19
第 19 题
几何·面积
Two angles of an isosceles triangle measure 70^\circ and x^\circ . What is the sum of the three possible values of x ?
💡 解题思路
There are 3 cases: where $x^\circ$ is a base angle with the $70^\circ$ as the other angle, where $x^\circ$ is a base angle with $70^\circ$ as the vertex angle, and where $x^\circ$ is the vertex angle
20
第 20 题
几何·面积
How many non-congruent triangles have vertices at three of the eight points in the array shown below? [图]
💡 解题思路
Assume the base of the triangle is on the bottom four points because a congruent triangle can be made by reflecting the base on the top four points. For a triangle with a base of length $1$ , there ar
21
第 21 题
统计
Andy and Bethany have a rectangular array of numbers with 40 rows and 75 columns. Andy adds the numbers in each row. The average of his 40 sums is A . Bethany adds the numbers in each column. The average of her 75 sums is B . What is the value of \frac{A}{B} ?
💡 解题思路
Note that $40A=75B=\text{sum of the numbers in the array}$ . This means that $\frac{A}{B}=\boxed{\text{(D) } \frac{15}{8}}$ using basic algebraic manipulation.
22
第 22 题
数字运算
How many whole numbers between 1 and 1000 do not contain the digit 1?
💡 解题思路
Note that this is the same as finding how many numbers with up to three digits do not contain 1.
23
第 23 题
综合
On the last day of school, Mrs. Wonderful gave jelly beans to her class. She gave each boy as many jelly beans as there were boys in the class. She gave each girl as many jelly beans as there were girls in the class. She brought 400 jelly beans, and when she finished, she had six jelly beans left. There were two more boys than girls in her class. How many students were in her class?
💡 解题思路
If there are $x$ girls, then there are $x+2$ boys. She gave each girl $x$ jellybeans and each boy $x+2$ jellybeans, for a total of $x^2 + (x+2)^2$ jellybeans. She gave away $400-6=394$ jellybeans.
24
第 24 题
数字运算
The letters A , B , C and D represent digits. If \begin{tabular}{ccc}&A&B ; +&C&A ; \hline &D&A\end{tabular} and \begin{tabular}{ccc}&A&B ; -&C&A ; \hline &&A\end{tabular} ,what digit does D represent?
💡 解题思路
Because $B+A=A$ , $B$ must be $0$ . Next, because $B-A=A\implies0-A=A,$ we get $A=5$ as the "0" mentioned above is actually 10 in this case.
25
第 25 题
几何·面积
A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cut is \frac{1}{2} foot from the top face. The second cut is \frac{1}{3} foot below the first cut, and the third cut is \frac{1}{17} foot below the second cut. From the top to the bottom the pieces are labeled A, B, C, and D. The pieces are then glued together end to end as shown in the second diagram. What is the total surface area of this solid in square feet? [图] [图]
💡 解题思路
The areas of the tops of $A$ , $B$ , $C$ , and $D$ in the figure formed has sum $1+1+1+1 = 4$ as do the bottoms. Thus, the total so far is $8$ . Now, one of the sides has an area of one, since it comb