2008 AMC 10B — Official Competition Problems (February 2008)
📅 2008 B 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
综合
A basketball player made 5 baskets during a game. Each basket was worth either 2 or 3 points. How many different numbers could represent the total points scored by the player? (A)\ 2 (B)\ 3 (C)\ 4 (D)\ 5 (E)\ 6
💡 解题思路
The number of points could have been 10, 11, 12, 13, 14, or 15. This is because the minimum is 2*5=10 and the maximum is 3*5=15. The numbers between 10 and 15 are possible as well. Thus, the answer is
2
第 2 题
规律与数列
A 4× 4 block of calendar dates is shown. First, the order of the numbers in the second and the fourth rows are reversed. Then, the numbers on each diagonal are added. What will be the positive difference between the two diagonal sums? \begin{tabular}[t]{|c|c|c|c|} \multicolumn{4}{c}{} ; \hline 1&2&3&4 ; \hline 8&9&10&11 ; \hline 15&16&17&18 ; \hline 22&23&24&25 ; \hline \end{tabular}
💡 解题思路
After reversing the numbers on the second and fourth rows, the block will look like this:
3
第 3 题
规律与数列
Assume that x is a positive real number . Which is equivalent to \sqrt[3]{x√(x)} ? (A)\ x^{1/6} (B)\ x^{1/4} (C)\ x^{3/8} (D)\ x^{1/2} (E)\ x
A semipro baseball league has teams with 21 players each. League rules state that a player must be paid at least 15,000 dollars, and that the total of all players' salaries for each team cannot exceed 700,000 dollars. What is the maximum possiblle salary, in dollars, for a single player?
💡 解题思路
We want to find the maximum any player could make, so assume that everyone else makes the minimum possible and that the combined salaries total the maximum of $700,000$
5
第 5 题
综合
For real numbers a and b , define a * b=(a-b)^2 . What is (x-y)^2*(y-x)^2 ? (A)\ 0 (B)\ x^2+y^2 (C)\ 2x^2 (D)\ 2y^2 (E)\ 4xy
💡 解题思路
Since $(-a)^2 = a^2$ , it follows that $(x-y)^2 = (y-x)^2$ , and \[(x-y)^2 * (y-x)^2 = [(x-y)^2 - (y-x)^2]^2 = [(x-y)^2 - (x-y)^2]^2 = 0\ \mathrm{(A)}.\]
6
第 6 题
分数与比例
Points B and C lie on \overline{AD} . The length of \overline{AB} is 4 times the length of \overline{BD} , and the length of \overline{AC} is 9 times the length of \overline{CD} . The length of \overline{BC} is what fraction of the length of \overline{AD} ?
💡 解题思路
Let $CD = 1$ . Then $AB = 4(BC + 1)$ and $AB + BC = 9\cdot1$ . From this system of equations, we obtain $BC = 1$ . Adding $CD$ to both sides of the second equation, we obtain $AD = AB + BC + CD = 9 +
7
第 7 题
几何·面积
An equilateral triangle of side length 10 is completely filled in by non-overlapping equilateral triangles of side length 1 . How many small triangles are required? (A)\ 10 (B)\ 25 (C)\ 100 (D)\ 250 (E)\ 1000
💡 解题思路
The area of the large triangle is $\frac{10^2\sqrt3}{4}$ , while the area of each small triangle is $\frac{1^2\sqrt3}{4}$ . Dividing these two quantities results in $100$ , therefore $\boxed{100} \mat
8
第 8 题
应用题
A class collects 50 dollars to buy flowers for a classmate who is in the hospital. Roses cost 3 dollars each, and carnations cost 2 dollars each. No other flowers are to be used. How many different bouquets could be purchased for exactly 50 dollars? (A)\ 1 (B)\ 7 (C)\ 9 (D)\ 16 (E)\ 17
💡 解题思路
The cost of a rose is odd, hence we need an even number of roses. Let there be $2r$ roses for some $r\geq 0$ . Then we have $50-3\cdot 2r = 50-6r$ dollars left. We can always reach the sum exactly $50
9
第 9 题
方程
A quadratic equation ax^2 - 2ax + b = 0 has two real solutions. What is the average of these two solutions? (A)\ 1 (B)\ 2 (C)\ \frac ba (D)\ \frac{2b}a (E)\ √(2b-a)
💡 解题思路
Dividing both sides by $a$ , we get $x^2 - 2x + b/a = 0$ . By Vieta's formulas, the sum of the roots is $2$ , therefore their average is $1\Rightarrow \boxed{A}$ .
10
第 10 题
几何·面积
Points A and B are on a circle of radius 5 and AB=6 . Point C is the midpoint of the minor arc AB . What is the length of the line segment AC ? (A)\ √(10) (B)\ \frac{7}{2} (C)\ √(14) (D)\ √(15) (E)\ 4
💡 解题思路
Let the center of the circle be $O$ , and let $D$ be the intersection of $\overline{AB}$ and $\overline{OC}$ (then $D$ is the midpoint of $\overline{AB}$ ). $OA=OB=5$ , since they are both radii of th
11
第 11 题
规律与数列
Suppose that (u_n) is a sequence of real numbers satifying u_{n+2}=2u_{n+1}+u_n , and that u_3=9 and u_6=128 . What is u_5 ? (A)\ 40 (B)\ 53 (C)\ 68 (D)\ 88 (E)\ 104
💡 解题思路
If we plug in $n=4$ , we get
12
第 12 题
计数
Postman Pete has a pedometer to count his steps. The pedometer records up to 99999 steps, then flips over to 00000 on the next step. Pete plans to determine his mileage for a year. On January 1 Pete sets the pedometer to 00000. During the year, the pedometer flips from 99999 to 00000 forty-four times. On December 31 the pedometer reads 50000. Pete takes 1800 steps per mile. Which of the following is closest to the number of miles Pete walked during the year? (A)\ 2500 (B)\ 3000 (C)\ 3500 (D)\ 4000 (E)\ 4500
💡 解题思路
Every time the pedometer flips from $99999$ to
13
第 13 题
统计
For each positive integer n , the mean of the first n terms of a sequence is n . What is the 2008^{th} term of the sequence? (A)\ {{{2008}}} (B)\ {{{4015}}} (C)\ {{{4016}}} (D)\ {{{4,030,056}}} (E)\ {{{4,032,064}}}
💡 解题思路
Since the mean of the first $n$ terms is $n$ , the sum of the first $n$ terms is $n^2$ . Thus, the sum of the first $2007$ terms is $2007^2$ and the sum of the first $2008$ terms is $2008^2$ . Hence,
14
第 14 题
几何·面积
Triangle [katex]OAB[/katex] has [katex]O=(0,0)[/katex], [katex]B=(5,0)[/katex], and [katex]A[/katex] in the first quadrant. In addition, [katex]\angle ABO=90^\circ[/katex] and [katex]\angle AOB=30^\circ[/katex]. Suppose that [katex]OA[/katex] is rotated [katex]90^\circ[/katex] counterclockwise about [katex]O[/katex]. What are the coordinates of the image of [katex]A[/katex]? (A)\ ( - \frac {10}{3}\sqrt {3},5) (B)\ ( - \frac {5}{3}\sqrt {3},5) (C)\ (\sqrt {3},5) (D)\ (\frac {5}{3}\sqrt {3},5) (E)\ (\frac {10}{3}\sqrt {3},5)
💡 解题思路
Since $\angle ABO=90^\circ$ , and $\angle AOB=30^\circ$ , we know that this triangle is one of the Special Right Triangles .
15
第 15 题
几何·面积
How many right triangles have integer leg lengths a and b and a hypotenuse of length b+1 , where b<100 ? (A)\ 6 (B)\ 7 (C)\ 8 (D)\ 9 (E)\ 10
💡 解题思路
By the Pythagorean theorem, $a^2+b^2=b^2+2b+1$
16
第 16 题
概率
Two fair coins are to be tossed once. For each head that results, one fair die is to be rolled. What is the probability that the sum of the die rolls is odd? (Note that if no die is rolled, the sum is 0.) (A)\ {{{\frac{3} {8}}}} (B)\ {{{\frac{1} {2}}}} (C)\ {{{\frac{43} {72}}}} (D)\ {{{\frac{5} {8}}}} (E)\ {{{\frac{2} {3}}}}
💡 解题思路
We consider 3 cases based on the outcome of the coin:
17
第 17 题
概率
A poll shows that 70\% of all voters approve of the mayor's work. On three separate occasions a pollster selects a voter at random. What is the probability that on exactly one of these three occasions the voter approves of the mayor's work? (A)\ {{{0.063}}} (B)\ {{{0.189}}} (C)\ {{{0.233}}} (D)\ {{{0.333}}} (E)\ {{{0.441}}}
💡 解题思路
Letting Y stand for a voter who approved of the work, and N stand for a person who didn't approve of the work, the pollster could select responses in $3$ different ways: $\text{YNN, NYN, and NNY}$ . T
18
第 18 题
时间问题
Bricklayer Brenda takes 9 hours to build a chimney alone, and bricklayer Brandon takes 10 hours to build it alone. When they work together, they talk a lot, and their combined output decreases by 10 bricks per hour. Working together, they build the chimney in 5 hours. How many bricks are in the chimney? (A)\ 500 (B)\ 900 (C)\ 950 (D)\ 1000 (E)\ 1900
💡 解题思路
Let $x$ be the number of bricks in the chimney. The work done is the rate multiplied by the time.
19
第 19 题
立体几何
A cylindrical tank with radius 4 feet and height 9 feet is lying on its side. The tank is filled with water to a depth of 2 feet. What is the volume of water, in cubic feet? (A)\ 24π - 36 \sqrt {2} (B)\ 24π - 24 \sqrt {3} (C)\ 36π - 36 \sqrt {3} (D)\ 36π - 24 \sqrt {2} (E)\ 48π - 36 \sqrt {3} https://www.youtube.com/watch?v=I9RRrumPRK4
💡 解题思路
Any vertical cross-section of the tank parallel with its base looks as follows: [asy] unitsize(0.8cm); defaultpen(0.8); pair s=(0,0), bottom=(0,-4), mid=(0,-2); pair x[]=intersectionpoints( (-10,-2)--
20
第 20 题
概率
The faces of a cubical die are marked with the numbers 1 , 2 , 2 , 3 , 3 , and 4 . The faces of another die are marked with the numbers 1 , 3 , 4 , 5 , 6 , and 8 . What is the probability that the sum of the top two numbers will be 5 , 7 , or 9 ? (A)\ 5/18 (B)\ 7/18 (C)\ 11/18 (D)\ 3/4 (E)\ 8/9
💡 解题思路
One approach is to write a table of all $36$ possible outcomes, do the sums, and count good outcomes.
21
第 21 题
统计
Ten chairs are evenly spaced around a round table and numbered clockwise from 1 through 10 . Five married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or across from his/her spouse. How many seating arrangements are possible? (A)\ 240 (B)\ 360 (C)\ 480 (D)\ 540 (E)\ 720
💡 解题思路
For the first man, there are $10$ possible seats. For each subsequent man, there are $4$ , $3$ , $2$ , or $1$ possible seats. After the men are seated, there are only two possible arrangements for the
22
第 22 题
概率
Three red beads, two white beads, and one blue bead are placed in line in random order. What is the probability that no two neighboring beads are the same color? (A)\ 1/12 (B)\ 1/10 (C)\ 1/6 (D)\ 1/3 (E)\ 1/2
💡 解题思路
There are two ways to arrange the red beads, where $R$ represents a red bead and $-$ represents a blank space.
23
第 23 题
几何·面积
A rectangular floor measures a by b feet, where a and b are positive integers and b > a . An artist paints a rectangle on the floor with the sides of the rectangle parallel to the floor. The unpainted part of the floor forms a border of width 1 foot around the painted rectangle and occupies half the area of the whole floor. How many possibilities are there for the ordered pair (a,b) ?
💡 解题思路
Because the unpainted part of the floor covers half the area, then the painted rectangle covers half the area as well. Since the border width is 1 foot, the dimensions of the rectangle are $a-2$ by $b
24
第 24 题
几何·角度
Quadrilateral ABCD has AB = BC = CD , m\angle ABC = 70^\circ and m\angle BCD = 170^\circ . What is the degree measure of \angle BAD ? (A)\ 75 (B)\ 80 (C)\ 85 (D)\ 90 (E)\ 95
💡 解题思路
To start off, draw a diagram like in solution two and label the points. Create lines $\overline{AC}$ and $\overline{BD}$ . We can call their intersection point $Y$ . Note that triangle $BCD$ is an iso
25
第 25 题
行程问题
Michael walks at the rate of 5 feet per second on a long straight path. Trash pails are located every 200 feet along the path. A garbage truck traveling at 10 feet per second in the same direction as Michael stops for 30 seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leaving the next pail. How many times will Michael and the truck meet? (A)\ 4 (B)\ 5 (C)\ 6 (D)\ 7 (E)\ 8
💡 解题思路
Pick a coordinate system where Michael's starting pail is $0$ and the one where the truck starts is $200$ . Let $M(t)$ and $T(t)$ be the coordinates of Michael and the truck after $t$ seconds. Let $D(