2004 AMC 10A — Official Competition Problems (January 2004)
📅 2004 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
应用题
You and five friends need to raise 1500 dollars in donations for a charity, dividing the fundraising equally. How many dollars will each of you need to raise? (A) \ 250 (B) \ 300 (C) \ 1500 (D) \ 7500 (E) \ 9000
💡 解题思路
There are $6$ people to split the $1500$ dollars among, so each person must raise $\frac{1500}6=250$ dollars. $\Rightarrow\boxed{\mathrm{(A)}\ 250}$
2
第 2 题
分数与比例
For any three real numbers a , b , and c , with b≠ c , the operation \otimes is defined by: \[\otimes(a,b,c)=\frac{a}{b-c}\] What is \otimes(\otimes(1,2,3),\otimes(2,3,1),\otimes(3,1,2)) ? (A) \ -\frac{1}{2} (B) \ -\frac{1}{4} (C) \ 0 (D) \ \frac{1}{4} (E) \ \frac{1}{2}
Alicia earns 20 dollars per hour, of which 1.45\% is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes? (A) \ 0.0029 (B) \ 0.029 (C) \ 0.29 (D) \ 2.9 (E) \ 29
💡 解题思路
$20$ dollars is the same as $2000$ cents, and $1.45\%$ of $2000$ is $0.0145\times2000=29$ cents. $\Rightarrow\boxed{\mathrm{(E)}\ 29}$ .
4
第 4 题
综合
What is the value of x if |x-1|=|x-2| ? (A) \ -\frac12 (B) \ \frac12 (C) \ 1 (D) \ \frac32 (E) \ 2
💡 解题思路
$|x-1|$ is the distance between $x$ and $1$ ; $|x-2|$ is the distance between $x$ and $2$ .
5
第 5 题
概率
A set of three points is randomly chosen from the grid shown. Each three point set has the same probability of being chosen. What is the probability that the points lie on the same straight line? (A) \ \frac{1}{21} (B) \ \frac{1}{14} (C) \ \frac{2}{21} (D) \ \frac{1}{7} (E) \ \frac{2}{7}
💡 解题思路
There are $\binom{9}{3}$ ways to choose three points out of the 9 there. There are 8 combinations of dots such that they lie in a straight line: three vertical, three horizontal, and the diagonals.
6
第 6 题
综合
Bertha has 6 daughters and no sons. Some of her daughters have 6 daughters, and the rest have none. Bertha has a total of 30 daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and grand-daughters have no daughters? (A) \ 22 (B) \ 23 (C) \ 24 (D) \ 25 (E) \ 26
💡 解题思路
Since Bertha has $6$ daughters, she has $30-6=24$ granddaughters, of which none have daughters. Of Bertha's daughters, $\frac{24}6=4$ have daughters, so $6-4=2$ do not have daughters. Therefore, of Be
7
第 7 题
统计
A grocer stacks oranges in a pyramid-like stack whose rectangular base is 5 oranges by 8 oranges. Each orange above the first level rests in a pocket formed by four oranges below. The stack is completed by a single row of oranges. How many oranges are in the stack? (A) \ 96 (B) \ 98 (C) \ 100 (D) \ 101 (E) \ 134
💡 解题思路
There are $5\times8=40$ oranges on the $1^{\text{st}}$ layer of the stack. The $2^{\text{nd}}$ layer that is added on top of the first will be a layer of $4\times7=28$ oranges. When the third layer is
8
第 8 题
综合
A game is played with tokens according to the following rule. In each round, the player with the most tokens gives one token to each of the other players and also places one token in the discard pile. The game ends when some player runs out of tokens. Players A , B , and C start with 15 , 14 , and 13 tokens, respectively. How many rounds will there be in the game? (A) \ 36 (B) \ 37 (C) \ 38 (D) \ 39 (E) \ 40
💡 解题思路
We look at a set of three rounds, where the players begin with $x+1$ , $x$ , and $x-1$ tokens. After three rounds, there will be a net loss of $1$ token per player (they receive two tokens and lose th
9
第 9 题
几何·面积
In the overlapping triangles \triangle{ABC} and \triangle{ABE} sharing common side AB , \angle{EAB} and \angle{ABC} are right angles , AB=4 , BC=6 , AE=8 , and \overline{AC} and \overline{BE} intersect at D . What is the difference between the areas of \triangle{ADE} and \triangle{BDC} ? [图] \mathrm {(A)}\ 2 \mathrm {(B)}\ 4 \mathrm {(C)}\ 5 \mathrm {(D)}\ 8 \mathrm {(E)}\ 9
💡 解题思路
Looking, we see that the area of $[\triangle EBA]$ is 16 and the area of $[\triangle ABC]$ is 12. Set the area of $[\triangle ADB]$ to be x. We want to find $[\triangle ADE]$ - $[\triangle CDB]$ . So,
10
第 10 题
概率
Coin A is flipped three times and coin B is flipped four times. What is the probability that the number of heads obtained from flipping the two fair coins is the same? (A) \ \frac{19}{128} (B) \ \frac{23}{128} (C) \ \frac14 (D) \ \frac{35}{128} (E) \ \frac12
💡 解题思路
There are $4$ ways that the same number of heads will be obtained; $0$ , $1$ , $2$ , or $3$ heads.
11
第 11 题
分数与比例
A company sells peanut butter in cylindrical jars. Marketing research suggests that using wider jars will increase sales. If the diameter of the jars is increased by 25\% without altering the volume , by what percent must the height be decreased? (A) \ 10 (B) \ 25 (C) \ 36 (D) \ 50 (E) \ 60
💡 解题思路
When the diameter is increased by $25\%$ , it is increased by $\dfrac{5}{4}$ , so the area of the base is increased by $\left(\dfrac54\right)^2=\dfrac{25}{16}$ .
12
第 12 题
计数
Henry's Hamburger Haven offers its hamburgers with the following condiments: ketchup, mustard, mayonnaise, tomato, lettuce, pickles, cheese, and onions. A customer can choose one, two,or three meat patties and any collection of condiments. How many different kinds of hamburgers can be ordered? (A) \ 24 (B) \ 256 (C) \ 768 (D) \ 40,320 (E) \ 120,960
💡 解题思路
Think of the condiments as a set with 8 elements. There are $8$ total condiments to choose from. Therefore, there are $2^8=256$ ways to order the condiments. (You have two choices for each condiment-
13
第 13 题
综合
At a party, each man danced with exactly three women and each woman danced with exactly two men. Twelve men attended the party. How many women attended the party? (A) \ 8 (B) \ 12 (C) \ 16 (D) \ 18 (E) \ 24
💡 解题思路
If each man danced with $3$ women, then there will be a total of $3\times12=36$ pairs of men and women. However, each woman only danced with $2$ men, so there must have been $\frac{36}2 \Longrightarro
14
第 14 题
统计
The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is 20 cents. If she had one more quarter, the average value would be 21 cents. How many dimes does she have in her purse? \text {(A)}\ 0 \text {(B)}\ 1 \text {(C)}\ 2 \text {(D)}\ 3 \text {(E)}\ 4
💡 解题思路
Let the total value, in cents, of the coins Paula has originally be $v$ , and the number of coins she has be $n$ . Then $\frac{v}{n}=20\Longrightarrow v=20n$ and $\frac{v+25}{n+1}=21$ . Substituting y
15
第 15 题
综合
Given that -4≤ x≤-2 and 2≤ y≤4 , what is the largest possible value of \frac{x+y}{x} ? (A) \ -1 (B) \ -\frac12 (C) \ 0 (D) \ \frac12 (E) \ 1
💡 解题思路
Rewrite $\frac{(x+y)}x$ as $\frac{x}x+\frac{y}x=1+\frac{y}x$ .
16
第 16 题
几何·面积
The 5× 5 grid shown contains a collection of squares with sizes from 1× 1 to 5× 5 . How many of these squares contain the black center square? [图] (A) \ 12 (B) \ 15 (C) \ 17 (D) \ 19 (E) \ 20
💡 解题思路
Since there are five types of squares: $1 \times 1, 2 \times 2, 3 \times 3, 4 \times 4,$ and $5 \times 5.$ We must find how many of each square contain the black shaded square in the center.
17
第 17 题
行程问题
Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run 100 meters. They next meet after Sally has run 150 meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters? (A) \ 250 (B) \ 300 (C) \ 350 (D) \ 400 (E) \ 500
💡 解题思路
Call the length of the race track $x$ . When they meet at the first meeting point, Brenda has run $100$ meters, while Sally has run $\frac{x}{2} - 100$ meters. By the second meeting point, Sally has r
18
第 18 题
规律与数列
A sequence of three real numbers forms an arithmetic progression with a first term of 9 . If 2 is added to the second term and 20 is added to the third term, the three resulting numbers form a geometric progression . What is the smallest possible value for the third term in the geometric progression? \text {(A)}\ 1 \text {(B)}\ 4 \text {(C)}\ 36 \text {(D)}\ 49 \text {(E)}\ 81
💡 解题思路
Let $d$ be the common difference. Then $9$ , $9+d+2=11+d$ , $9+2d+20=29+2d$ are the terms of the geometric progression. Since the middle term is the geometric mean of the other two terms, $(11+d)^2 =
19
第 19 题
几何·面积
A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet? [图] (A) \ 120 (B) \ 180 (C) \ 240 (D) \ 360 (E) \ 480
💡 解题思路
The cylinder can be "unwrapped" into a rectangle, and we see that there are two stripes which is a parallelogram with base $3$ and height $40$ , each. Thus, we get $3\times40\times2=240\Rightarrow\box
20
第 20 题
几何·面积
Points E and F are located on square ABCD so that \triangle BEF is equilateral. What is the ratio of the area of \triangle DEF to that of \triangle ABE ? (A) \ \frac{4}{3} (B) \ \frac{3}{2} (C) \ √(3) (D) \ 2 (E) \ 1+√(3)
💡 解题思路
Since triangle $BEF$ is equilateral, $EA=FC$ , and $EAB$ and $FCB$ are $HL$ congruent. Thus, triangle $DEF$ is an isosceles right triangle. So we let $DE=x$ . Thus $EF=EB=FB=x\sqrt{2}$ . If we go angl
21
第 21 题
几何·面积
Two distinct lines pass through the center of three concentric circles of radii 3, 2, and 1. The area of the shaded region in the diagram is \frac{8}{13} of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: π radian is 180 degree.) (A) \ \frac{π}{8} (B) \ \frac{π}{7} (C) \ \frac{π}{6} (D) \ \frac{π}{5} (E) \ \frac{π}{4}
💡 解题思路
Let the area of the shaded region be $S$ , the area of the unshaded region be $U$ , and the acute angle that is formed by the two lines be $\theta$ . We can set up two equations between $S$ and $U$ :
22
第 22 题
几何·面积
Square ABCD has side length 2 . A semicircle with diameter \overline{AB} is constructed inside the square, and the tangent to the semicircle from C intersects side \overline{AD} at E . What is the length of \overline{CE} ? [图] (A) \ \frac{2+√(5)}{2} (B) \ √(5) (C) \ √(6) (D) \ \frac{5}{2} (E) \ 5-√(5)
Circles A, B and C are externally tangent to each other, and internally tangent to circle D . Circles B and C are congruent. Circle A has radius 1 and passes through the center of D . What is the radius of circle B ? (A) \frac23 (B) \frac {\sqrt3}{2} (C) \frac78 (D) \frac89 (E) \frac {1 + \sqrt3}{3} https://www.youtube.com/watch?v=4-lbEZkFJdc
💡 解题思路
[asy] import graph; size(400); defaultpen(fontsize(10)); pair OA=(-1,0),OB=(2/3,8/9),OC=(2/3,-8/9),OD=(0,0),E=(2/3,0); real t = 2.5; pair OA1=(-2+2*t,0),OB1=(4/3+2*t,16/9),OC1=(4/3+2*t,-16/9),OD1=(0+2
24
第 24 题
函数
Let f be a function with the following properties: (i) f(1) = 1 , and (ii) f(2n) = n · f(n) for any positive integer n . What is the value of f(2^{100}) ? \textbf {(A)}\ 1 \textbf {(B)}\ 2^{99} \textbf {(C)}\ 2^{100} \textbf {(D)}\ 2^{4950} \textbf {(E)}\ 2^{9999}
Three mutually tangent spheres of radius 1 rest on a horizontal plane. A sphere of radius 2 rests on them. What is the distance from the plane to the top of the larger sphere? \text {(A)}\ 3 + \frac {\sqrt {30}}{2} \text {(B)}\ 3 + \frac {\sqrt {69}}{3} \text {(C)}\ 3 + \frac {\sqrt {123}}{4} \text {(D)}\ \frac {52}{9} \text {(E)}\ 3 + 2\sqrt2
💡 解题思路
The height from the center of the bottom sphere to the plane is $1$ , and from the center of the top sphere to the tip is $2$ .