2004A AMC 12 — Official Competition Problems (February 2024)
📅 2024 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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第 1 题
应用题
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. $\Rightarrow\boxed{\mathrm{(E)}\ 29}$ .
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第 2 题
综合
On the AMC 12, each correct answer is worth 6 points, each incorrect answer is worth 0 points, and each problem left unanswered is worth 2.5 points. If Charlyn leaves 8 of the 25 problems unanswered, how many of the remaining problems must she answer correctly in order to score at least 100 ? \text {(A)} 11 \text {(B)} 13 \text {(C)} 14 \text {(D)} 16 \text {(E)}17
💡 解题思路
She gets $8*2.5=20$ points for the problems she didn't answer. She must get $\left\lceil \frac{100-20}{6} \right\rceil =14 \Rightarrow \text {(C)}$ problems right to score at least 100.
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第 3 题
整数运算
For how many ordered pairs of positive integers (x,y) is x + 2y = 100 ? \text {(A)} 33 \text {(B)} 49 \text {(C)} 50 \text {(D)} 99 \text {(E)}100
💡 解题思路
Every integer value of $y$ leads to an integer solution for $x$ Since $y$ must be positive, $y\geq 1$
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第 4 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
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第 5 题
坐标几何
The graph of the line y=mx+b is shown. Which of the following is true? \mathrm {(A)} mb<-1 \mathrm {(B)} -1<mb<0 \mathrm {(C)} mb=0 \mathrm {(D)}0<mb<1 \mathrm {(E)} mb>1
💡 解题思路
The line appears to have a slope of $-\dfrac{1}{2}$ and y-intercept of $\dfrac{4}{5}$ up.
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第 6 题
方程
Let U=2· 2004^{2005} , V=2004^{2005} , W=2003· 2004^{2004} , X=2· 2004^{2004} , Y=2004^{2004} and Z=2004^{2003} . Which of the following is the largest? \mathrm {(A)} U-V \mathrm {(B)} V-W \mathrm {(C)} W-X \mathrm {(D)} X-Y \mathrm {(E)} Y-Z
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
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第 8 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,
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第 9 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 $\left(\dfrac54\right)^2=\dfrac{25}{16}$ .
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第 10 题
统计
The sum of 49 consecutive integers is 7^5 . What is their median ? \text {(A)}\ 7 \text {(B)}\ 7^2 \text {(C)}\ 7^3 \text {(D)}\ 7^4 \text {(E)}\ 7^5
💡 解题思路
The median of a sequence is the middle number of the sequence when the sequence is arranged in order. Since the integers are consecutive, the median is also the mean , so the median is $\frac{7^5}{49}
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第 11 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
12
第 12 题
综合
Let A = (0,9) and B = (0,12) . Points A' and B' are on the line y = x , and \overline{AA'} and \overline{BB'} intersect at C = (2,8) . What is the length of \overline{A'B'} ? \text {(A)} 2 \text {(B)} 2\sqrt2 \text {(C)} 3 \text {(D)} 2 + \sqrt 2 \text {(E)}3\sqrt 2
💡 解题思路
The equation of $\overline{AA'}$ can be found using points $A, C$ to be $y - 9 = \left(\frac{9-8}{0-2}\right)(x - 0) \Longrightarrow y = -\frac{1}{2}x + 9$ . Similarily, $\overline{BB'}$ has the equat
13
第 13 题
坐标几何
Let S be the set of points (a,b) in the coordinate plane , where each of a and b may be - 1 , 0 , or 1 . How many distinct lines pass through at least two members of S ? \text {(A)}\ 8 \text {(B)}\ 20 \text {(C)}\ 24 \text {(D)}\ 27 \text {(E)}\ 36
💡 解题思路
Let's count them by cases:
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第 14 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 =
15
第 15 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
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第 16 题
综合
The set of all real numbers x for which \[\log_{2004}(\log_{2003}(\log_{2002}(\log_{2001}{x})))\] is defined is \{x\mid x > c\} . What is the value of c ? \textbf {(A) } 0 \textbf {(B) }2001^{2002} \textbf {(C) }2002^{2003} \textbf {(D) }2003^{2004} \textbf {(E) }2001^{2002^{2003}}
💡 解题思路
For all real numbers $a,b,$ and $c$ such that $b>1,$ note that:
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第 17 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}
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
20
第 20 题
概率
Select numbers a and b between 0 and 1 independently and at random, and let c be their sum. Let A, B and C be the results when a, b and c , respectively, are rounded to the nearest integer. What is the probability that A + B = C ? \text {(A)}\ \frac14 \text {(B)}\ \frac13 \text {(C)}\ \frac12 \text {(D)}\ \frac23 \text {(E)}\ \frac34
If \sum_{n = 0}^{\infty}{\cos^{2n}}\theta = 5 , what is the value of \cos{2\theta} ? \text {(A)} \frac15 \text {(B)} \frac25 \text {(C)} \frac {\sqrt5}{5} \text {(D)} \frac35 \text {(E)}\frac45
💡 解题思路
This is an infinite geometric series , which sums to $\frac{\cos^0 \theta}{1 - \cos^2 \theta} = 5 \Longrightarrow 1 = 5 - 5\cos^2 \theta \Longrightarrow \cos^2 \theta = \frac{4}{5}$ . Using the formul
22
第 22 题
行程问题
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 $2$ .
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第 23 题
规律与数列
A polynomial \[P(x) = c_{2004}x^{2004} + c_{2003}x^{2003} + ... + c_1x + c_0\] has real coefficients with c_{2004}\not = 0 and 2004 distinct complex zeroes z_k = a_k + b_ki , 1≤ k≤ 2004 with a_k and b_k real, a_1 = b_1 = 0 , and \[\sum_{k = 1}^{2004}{a_k} = \sum_{k = 1}^{2004}{b_k}.\] Which of the following quantities can be a non zero number? \text {(A)} c_0 \text {(B)} c_{2003} \text {(C)} b_2b_3...b_{2004} \text {(D)} \sum_{k = 1}^{2004}{a_k} \text {(E)}\sum_{k = 1}^{2004}{c_k}
💡 解题思路
We have to evaluate the answer choices and use process of elimination:
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第 24 题
几何·面积
A plane contains points A and B with AB = 1 . Let S be the union of all disks of radius 1 in the plane that cover \overline{AB} . What is the area of S ? \textbf {(A) } 2π + \sqrt3 \textbf {(B) } \frac {8π}{3} \textbf {(C) } 3π - \frac {\sqrt3}{2} \textbf {(D) } \frac {10π}{3} - \sqrt3 \textbf {(E) }4π - 2\sqrt3
💡 解题思路
As the red circles move about segment $AB$ , they cover the area we are looking for. On the left side, the circle must move around pivoted on $B$ . On the right side, the circle must move pivoted on $
25
第 25 题
整数运算
For each integer n≥ 4 , let a_n denote the base- n number 0.\overline{133}_n . The product a_4a_5·s a_{99} can be expressed as \frac {m}{n!} , where m and n are positive integers and n is as small as possible. What is m ? \text {(A)} 98 \text {(B)} 101 \text {(C)} 132 \text {(D)} 798 \text {(E)}962
💡 解题思路
This is an infinite geometric series with common ratio $\frac{1}{x^3}$ and initial term $x^{-1} + 3x^{-2} + 3x^{-3}$ , so $a_x = \left(\frac{1}{x} + \frac{3}{x^2} + \frac{3}{x^3}\right)\left(\frac{1}{