2008 AMC 10A — Official Competition Problems (February 2008)
📅 2008 A 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
行程问题
A bakery owner turns on his doughnut machine at 8:30\ {\smallAM} . At 11:10\ {\smallAM} the machine has completed one third of the day's job. At what time will the doughnut machine complete the job? (A)\ 1:50\ {\smallPM} (B)\ 3:00\ {\smallPM} (C)\ 3:30\ {\smallPM} (D)\ 4:30\ {\smallPM} (E)\ 5:50\ {\smallPM}
💡 解题思路
The machine completes one-third of the job in $\text{11:10}-\text{8:30}=\text{2:40}$ hours. Thus, the entire job is completed in $3\cdot(\text{2:40})=\text{8:00}$ hours.
2
第 2 题
几何·面积
A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is 2:1 . The ratio of the rectangle's length to its width is 2:1 . What percent of the rectangle's area is inside the square? (A)\ 12.5 (B)\ 25 (C)\ 50 (D)\ 75 (E)\ 87.5
💡 解题思路
Since they are asking for the "ratio" of two things, we can say that the side of the square is anything that we want. So if we say that it is $1$ , then the width of the rectangle is $2$ , and the len
3
第 3 题
规律与数列
For the positive integer n , let \langle n\rangle denote the sum of all the positive divisors of n with the exception of n itself. For example, \langle 4\rangle=1+2=3 and \langle 12 \rangle =1+2+3+4+6=16 . What is \langle\langle\langle 6\rangle\rangle\rangle ? (A)\ 6 (B)\ 12 (C)\ 24 (D)\ 32 (E)\ 36
Suppose that \tfrac{2}{3} of 10 bananas are worth as much as 8 oranges. How many oranges are worth as much as \tfrac{1}{2} of 5 bananas? (A)\ 2 (B)\ \frac{5}{2} (C)\ 3 (D)\ \frac{7}{2} (E)\ 4
💡 解题思路
If $\frac{2}{3}\cdot10\ \text{bananas}=8\ \text{oranges}$ , then $\frac{1}{2}\cdot5\ \text{bananas}=\left(\frac{1}{2}\cdot 5\ \text{bananas}\right)\cdot\left(\frac{8\ \text{oranges}}{\frac{2}{3}\cdot1
5
第 5 题
综合
Which of the following is equal to the product \[\frac{8}{4}·\frac{12}{8}·\frac{16}{12}··s·\frac{4n+4}{4n}··s·\frac{2008}{2004}?\]
A triathlete competes in a triathlon in which the swimming, biking, and running segments are all of the same length. The triathlete swims at a rate of 3 kilometers per hour, bikes at a rate of 20 kilometers per hour, and runs at a rate of 10 kilometers per hour. Which of the following is closest to the triathlete's average speed, in kilometers per hour, for the entire race? (A)\ 3 (B)\ 4 (C)\ 5 (D)\ 6 (E)\ 7
💡 解题思路
Let $d$ be the length of one segment of the race.
7
第 7 题
分数与比例
The fraction \[\frac{(3^{2008})^2-(3^{2006})^2}{(3^{2007})^2-(3^{2005})^2}\] simplifies to which of the following? (A)\ 1 (B)\ \frac{9}{4} (C)\ 3 (D)\ \frac{9}{2} (E)\ 9
💡 解题思路
Simplifying, we get \[\frac{3^{4016}-3^{4012}}{3^{4014}-3^{4010}}.\] Factoring out $3^{4012}$ in the numerator and factoring out $3^{4010}$ in the denominator gives us \[\frac{(3^4-1)(3^{4012})}{(3^4-
8
第 8 题
应用题
Heather compares the price of a new computer at two different stores. Store A offers 15\% off the sticker price followed by a \90 rebate, and store B offers 25\% off the same sticker price with no rebate. Heather saves \15 by buying the computer at store A instead of store B . What is the sticker price of the computer, in dollars? (A)\ 750 (B)\ 900 (C)\ 1000 (D)\ 1050 (E)\ 1500
💡 解题思路
Let the sticker price be $x$ .
9
第 9 题
数论
Suppose that \[\frac{2x}{3}-\frac{x}{6}\] is an integer. Which of the following statements must be true about x ? (A)\ It is negative. ; (B)\ It is even, but not necessarily a multiple of 3. ; (C)\ It is a multiple of 3, but not necessarily even. ; (D)\ It is a multiple of 6, but not necessarily a multiple of 12. ; (E)\ It is a multiple of 12.
💡 解题思路
\[\frac{2x}{3}-\frac{x}{6}\quad\Longrightarrow\quad\frac{4x}{6}-\frac{x}{6}\quad\Longrightarrow\quad\frac{3x}{6}\quad\Longrightarrow\quad\frac{x}{2}\] For $\frac{x}{2}$ to be an integer, $x$ must be e
10
第 10 题
几何·面积
Each of the sides of a square S_1 with area 16 is bisected, and a smaller square S_2 is constructed using the bisection points as vertices. The same process is carried out on S_2 to construct an even smaller square S_3 . What is the area of S_3 ? (A)\ \frac{1}{2} (B)\ 1 (C)\ 2 (D)\ 3 (E)\ 4
💡 解题思路
Since the area of the large square is $16$ , the side length equals $4$ . If all sides are bisected, the resulting square has side length $2\sqrt{2}$ , thus making the area $8$ . If we repeat this pro
11
第 11 题
行程问题
While Steve and LeRoy are fishing 1 mile from shore, their boat springs a leak, and water comes in at a constant rate of 10 gallons per minute. The boat will sink if it takes in more than 30 gallons of water. Steve starts rowing towards the shore at a constant rate of 4 miles per hour while LeRoy bails water out of the boat. What is the slowest rate, in gallons per minute, at which LeRoy can bail if they are to reach the shore without sinking? (A)\ 2 (B)\ 4 (C)\ 6 (D)\ 8 (E)\ 10
💡 解题思路
It will take $\frac{1}{4}$ of an hour or $15$ minutes to get to shore.
12
第 12 题
综合
In a collection of red, blue, and green marbles, there are 25\% more red marbles than blue marbles, and there are 60\% more green marbles than red marbles. Suppose that there are r red marbles. What is the total number of marbles in the collection? (A)\ 2.85r (B)\ 3r (C)\ 3.4r (D)\ 3.85r (E)\ 4.25r
💡 解题思路
The number of blue marbles is $\frac{4}{5}r$ , the number of green marbles is $\frac{8}{5}r$ , and the number of red marbles is $r$ .
13
第 13 题
方程
Doug can paint a room in 5 hours. Dave can paint the same room in 7 hours. Doug and Dave paint the room together and take a one-hour break for lunch. Let t be the total time, in hours, required for them to complete the job working together, including lunch. Which of the following equations is satisfied by t ? (A)\ (\frac{1}{5}+\frac{1}{7})(t+1)=1 (B)\ (\frac{1}{5}+\frac{1}{7})t+1=1 (C)\ (\frac{1}{5}+\frac{1}{7})t=1 ; (D)\ (\frac{1}{5}+\frac{1}{7})(t-1)=1 (E)\ (5+7)t=1
💡 解题思路
Doug can paint $\frac{1}{5}$ of a room per hour, Dave can paint $\frac{1}{7}$ of a room per hour, and the time they spend working together is $t-1$ .
14
第 14 题
分数与比例
Older television screens have an aspect ratio of 4: 3 . That is, the ratio of the width to the height is 4: 3 . The aspect ratio of many movies is not 4: 3 , so they are sometimes shown on a television screen by "letterboxing" - darkening strips of equal height at the top and bottom of the screen, as shown. Suppose a movie has an aspect ratio of 2: 1 and is shown on an older television screen with a 27 -inch diagonal. What is the height, in inches, of each darkened strip? [图] (A)\ 2 (B)\ 2.25 (C)\ 2.5 (D)\ 2.7 (E)\ 3
💡 解题思路
Let the width and height of the screen be $4x$ and $3x$ respectively, and let the width and height of the movie be $2y$ and $y$ respectively.
15
第 15 题
统计
Yesterday Han drove 1 hour longer than Ian at an average speed 5 miles per hour faster than Ian. Jan drove 2 hours longer than Ian at an average speed 10 miles per hour faster than Ian. Han drove 70 miles more than Ian. How many more miles did Jan drive than Ian? (A)\ 120 (B)\ 130 (C)\ 140 (D)\ 150 (E)\ 160
💡 解题思路
Set the time Ian traveled as $I$ , and set Han's speed as $H$ . Therefore, Jan's speed is $H+5.$
16
第 16 题
几何·面积
Points A and B lie on a circle centered at O , and \angle AOB = 60^\circ . A second circle is internally tangent to the first and tangent to both \overline{OA} and \overline{OB} . What is the ratio of the area of the smaller circle to that of the larger circle? (A)\ \frac{1}{16} (B)\ \frac{1}{9} (C)\ \frac{1}{8} (D)\ \frac{1}{6} (E)\ \frac{1}{4}
💡 解题思路
[asy]size(200); defaultpen(fontsize(10)); pair O=(0,0), A=(3,0), B=(3/2,3/2*3^.5), C=(3^.5,1), D=(3^.5,0), F=(1.5*3^.5,1.5), G=(2*3^.5,2); picture p = new picture; draw(p,Circle(O,0.2)); clip(p,O--C--
17
第 17 题
几何·面积
An equilateral triangle has side length 6 . What is the area of the region containing all points that are outside the triangle but not more than 3 units from a point of the triangle? (A)\ 36+24√(3) (B)\ 54+9π (C)\ 54+18√(3)+6π (D)\ (2√(3)+3)^2π ; (E)\ 9(√(3)+1)^2π
💡 解题思路
The region described contains three rectangles of dimensions $3 \times 6$ , and three $120^{\circ}$ degree arcs of circles of radius $3$ . Thus the answer is \[3(3 \times 6) + 3 \left( \frac{120^{\cir
18
第 18 题
几何·面积
A right triangle has perimeter 32 and area 20 . What is the length of its hypotenuse ? (A)\ \frac{57}{4} (B)\ \frac{59}{4} (C)\ \frac{61}{4} (D)\ \frac{63}{4} (E)\ \frac{65}{4}
💡 解题思路
Let the legs of the triangle have lengths $a,b$ . Then, by the Pythagorean Theorem , the length of the hypotenuse is $\sqrt{a^2+b^2}$ , and the area of the triangle is $\frac 12 ab$ . So we have the t
19
第 19 题
几何·面积
Rectangle PQRS lies in a plane with PQ=RS=2 and QR=SP=6 . The rectangle is rotated 90^\circ clockwise about R , then rotated 90^\circ clockwise about the point S moved to after the first rotation. What is the length of the path traveled by point P ? (A)\ (2√(3)+√(5))π (B)\ 6π (C)\ (3+√(10))π (D)\ (√(3)+2√(5))π ; (E)\ 2√(10)π
💡 解题思路
We let $P'Q'R'S'$ be the rectangle after the first rotation, and $P''Q''R''S''$ be the rectangle after the second rotation. Point $P$ pivots about $R$ in an arc of a circle of radius $\sqrt{2^2+6^2} =
20
第 20 题
几何·面积
Trapezoid ABCD has bases \overline{AB} and \overline{CD} and diagonals intersecting at K. Suppose that AB = 9 , DC = 12 , and the area of \triangle AKD is 24. What is the area of trapezoid ABCD ? (A)\ 92 (B)\ 94 (C)\ 96 (D)\ 98 (E)\ 100
💡 解题思路
Since $\overline{AB} \parallel \overline{DC}$ it follows that $\triangle ABK \sim \triangle CDK$ . Thus $\frac{KA}{KC} = \frac{KB}{KD} = \frac{AB}{DC} = \frac{3}{4}$ .
21
第 21 题
几何·面积
A cube with side length 1 is sliced by a plane that passes through two diagonally opposite vertices A and C and the midpoints B and D of two opposite edges not containing A or C , as shown. What is the area of quadrilateral ABCD ? [图] (A)\ \frac{√(6)}{2} (B)\ \frac{5}{4} (C)\ √(2) (D)\ \frac{5}{8} (E)\ \frac{3}{4}
💡 解题思路
Since $AB = AD = CB = CD = \sqrt{\left(\frac{1}{2}\right)^2+1^2}$ , it follows that $ABCD$ is a rhombus . The area of the rhombus can be computed by the formula $A = \frac 12 d_1d_2$ , where $d_1,\,d_
22
第 22 题
概率
Jacob uses the following procedure to write down a sequence of numbers. First he chooses the first term to be 6. To generate each succeeding term, he flips a fair coin. If it comes up heads, he doubles the previous term and subtracts 1. If it comes up tails, he takes half of the previous term and subtracts 1. What is the probability that the fourth term in Jacob's sequence is an integer ? (A)\ \frac{1}{6} (B)\ \frac{1}{3} (C)\ \frac{1}{2} (D)\ \frac{5}{8} (E)\ \frac{3}{4}
💡 解题思路
We construct a tree showing all possible outcomes that Jacob may get after $3$ flips; we can do this because there are only $8$ possibilities: \[6\quad\begin{cases} \ \text{H}: 11 &\quad \begin{cases}
23
第 23 题
计数
Two subsets of the set S=\lbrace a,b,c,d,e\rbrace are to be chosen so that their union is S and their intersection contains exactly two elements. In how many ways can this be done, assuming that the order in which the subsets are chosen does not matter? (A)\ 20 (B)\ 40 (C)\ 60 (D)\ 160 (E)\ 320
💡 解题思路
First, choose the two letters to be repeated in each set. $\dbinom{5}{2}=10$ . Now we have three remaining elements that we wish to place into two separate subsets. There are $2^3 = 8$ ways to do so b
24
第 24 题
数字运算
Let k={2008}^{2}+{2}^{2008} . What is the units digit of k^2+2^k ? (A)\ 0 (B)\ 2 (C)\ 4 (D)\ 6 (E)\ 8
A round table has radius 4 . Six rectangular place mats are placed on the table. Each place mat has width 1 and length x as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length x . Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is x ? [图] (A)\ 2√(5)-√(3) (B)\ 3 (C)\ \frac{3√(7)-√(3)}{2} (D)\ 2√(3) (E)\ \frac{5+2√(3)}{2}
💡 解题思路
Let one of the mats be $ABCD$ , and the center be $O$ as shown: