2004B 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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1
第 1 题
综合
At each basketball practice last week, Jenny made twice as many free throws as she made at the previous practice. At her fifth practice she made 48 free throws. How many free throws did she make at the first practice? (\mathrm {A}) 3 (\mathrm {B}) 6 (\mathrm {C}) 9 (\mathrm {D}) 12 (\mathrm {E}) 15
💡 解题思路
Each day Jenny makes half as many free throws as she does at the next practice. Hence on the fourth day she made $\frac{1}{2} \cdot 48 = 24$ free throws, on the third $12$ , on the second $6$ , and on
2
第 2 题
综合
In the expression c· a^b-d , the values of a , b , c , and d are 0 , 1 , 2 , and 3 , although not necessarily in that order. What is the maximum possible value of the result? (A)\ 5 (B)\ 6 (C)\ 8 (D)\ 9 (E)\ 10
💡 解题思路
If $a=0$ or $c=0$ , the expression evaluates to $-d<0$ . If $b=0$ , the expression evaluates to $c-d\leq 2$ . Case $d=0$ remains. In that case, we want to maximize $c\cdot a^b$ where $\{a,b,c\}=\{1,2,
3
第 3 题
整数运算
If x and y are positive integers for which 2^x3^y=1296 , what is the value of x+y ? (\mathrm {A})\ 8 (\mathrm {B})\ 9 (\mathrm {C})\ 10 (\mathrm {D})\ 11 (\mathrm {E})\ 12
💡 解题思路
$1296 = 2^4 3^4$ and $4+4=\boxed{8} \Longrightarrow \mathrm{(A)}$ .
4
第 4 题
概率
An integer x , with 10≤ x≤ 99 , is to be chosen. If all choices are equally likely, what is the probability that at least one digit of x is a 7? (\mathrm {A}) \dfrac{1}{9} (\mathrm {B}) \dfrac{1}{5} (\mathrm {C}) \dfrac{19}{90} (\mathrm {D}) \dfrac{2}{9} (\mathrm {E}) \dfrac{1}{3}
💡 解题思路
The digit $7$ can be either the tens digit ( $70, 71, \dots, 79$ : $10$ possibilities), or the ones digit ( $17, 27, \dots, 97$ : $9$ possibilities), but we counted the number $77$ twice. This means t
5
第 5 题
规律与数列
On a trip from the United States to Canada, Isabella took d U.S. dollars. At the border she exchanged them all, receiving 10 Canadian dollars for every 7 U.S. dollars. After spending 60 Canadian dollars, she had d Canadian dollars left. What is the sum of the digits of d ? (A)\ 5 (B)\ 6 (C)\ 7 (D)\ 8 (E)\ 9
💡 解题思路
Isabella had $60+d$ Canadian dollars. Setting up an equation we get $d=\frac{7}{10}\cdot(60+d)$ , which solves to $d=140$ , and the sum of digits of $d$ is $\boxed{\mathrm{(A)}\ 5}$ .
6
第 6 题
综合
Minneapolis-St. Paul International Airport is 8 miles southwest of downtown St. Paul and 10 miles southeast of downtown Minneapolis. Which of the following is closest to the number of miles between downtown St. Paul and downtown Minneapolis? (A)\ 13 (B)\ 14 (C)\ 15 (D)\ 16 (E)\ 17
💡 解题思路
The directions "southwest" and "southeast" are orthogonal. Thus the described situation is a right triangle with legs $8$ miles and $10$ miles long. The hypotenuse length is $\sqrt{8^2 + 10^2}\approx1
7
第 7 题
几何·面积
A square has sides of length 10 , and a circle centered at one of its vertices has radius 10 . What is the area of the union of the regions enclosed by the square and the circle? (A)\ 200+25π (B)\ 100+75π (C)\ 75+100π (D)\ 100+100π (E)\ 100+125π https://youtu.be/IGN4XxJIbE0 ~Education, the Study of Everything
💡 解题思路
The area of the circle is $S_{\bigcirc}=100\pi$ ; the area of the square is $S_{\square}=100$ .
8
第 8 题
综合
A grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains 100 cans, how many rows does it contain? (A)\ 5 (B)\ 8 (C)\ 9 (D)\ 10 (E)\ 11
💡 解题思路
The sum of the first $n$ odd numbers is $n^2$ . As in our case $n^2=100$ , we have $n=\boxed{\mathrm{(D)\ }10}$ .
9
第 9 题
坐标几何
The point (-3,2) is rotated 90^\circ clockwise around the origin to point B . Point B is then reflected over the line x=y to point C . What are the coordinates of C ? (A)\ (-3,-2) (B)\ (-2,-3) (C)\ (2,-3) (D)\ (2,3) (E)\ (3,2)
💡 解题思路
The entire situation is in the picture below. The correct answer is $\boxed{\mathrm{(E)}\ (3,2)}$ .
10
第 10 题
几何·面积
An annulus is the region between two concentric circles. The concentric circles in the figure have radii b and c , with b>c . Let OX be a radius of the larger circle, let XZ be tangent to the smaller circle at Z , and let OY be the radius of the larger circle that contains Z . Let a=XZ , d=YZ , and e=XY . What is the area of the annulus? [图] (A) \ π a^2 (B) \ π b^2 (C) \ π c^2 (D) \ π d^2 (E) \ π e^2
💡 解题思路
The area of the large circle is $\pi b^2$ , the area of the small one is $\pi c^2$ , hence the shaded area is $\pi(b^2-c^2)$ .
11
第 11 题
统计
All the students in an algebra class took a 100 -point test. Five students scored 100 , each student scored at least 60 , and the mean score was 76 . What is the smallest possible number of students in the class? (A)\ 10 (B)\ 11 (C)\ 12 (D)\ 13 (E)\ 14
💡 解题思路
Let the number of students be $n\geq 5$ . Then the sum of their scores is at least $5\cdot 100 + (n-5)\cdot 60$ . At the same time, we need to achieve the mean $76$ , which is equivalent to achieving
12
第 12 题
规律与数列
In the sequence 2001 , 2002 , 2003 , \ldots , each term after the third is found by subtracting the previous term from the sum of the two terms that precede that term. For example, the fourth term is 2001 + 2002 - 2003 = 2000 . What is the 2004^\textrm{th} term in this sequence? (A) \ -2004 (B) \ -2 (C) \ 0 (D) \ 4003 (E) \ 6007
💡 解题思路
We already know that $a_1=2001$ , $a_2=2002$ , $a_3=2003$ , and $a_4=2000$ . Let's compute the next few terms to get the idea how the sequence behaves. We get $a_5 = 2002+2003-2000 = 2005$ , $a_6=2003
13
第 13 题
函数
If f(x) = ax+b and f^{-1}(x) = bx+a with a and b real, what is the value of a+b ? (A)\ -2 (B)\ -1 (C)\ 0 (D)\ 1 (E)\ 2
💡 解题思路
Since $f(f^{-1}(x))=x$ , it follows that $a(bx+a)+b=x$ , which implies $abx + a^2 +b = x$ . This equation holds for all values of $x$ only if $ab=1$ and $a^2+b=0$ .
14
第 14 题
几何·面积
In \triangle ABC , AB=13 , AC=5 , and BC=12 . Points M and N lie on AC and BC , respectively, with CM=CN=4 . Points J and K are on AB so that MJ and NK are perpendicular to AB . What is the area of pentagon CMJKN ? [图] (A)\ 15 (B)\ \frac{81}{5} (C)\ \frac{205}{12} (D)\ \frac{240}{13} (E)\ 20
💡 解题思路
The triangle $ABC$ is clearly a right triangle, its area is $\frac{5\cdot 12}2 = 30$ . If we knew the areas of triangles $AMJ$ and $BNK$ , we could subtract them to get the area of the pentagon.
15
第 15 题
数字运算
The two digits in Jack's age are the same as the digits in Bill's age, but in reverse order. In five years Jack will be twice as old as Bill will be then. What is the difference in their current ages? (A) \ 9 (B) \ 18 (C) \ 27 (D) \ 36 (E) \ 45
💡 解题思路
If Jack's current age is $\overline{ab}=10a+b$ , then Bill's current age is $\overline{ba}=10b+a$ .
16
第 16 题
函数
A function f is defined by f(z) = i\overline{z} , where i=√(-1) and \overline{z} is the complex conjugate of z . How many values of z satisfy both |z| = 5 and f(z) = z ? (A)\ 0 (B)\ 1 (C)\ 2 (D)\ 4 (E)\ 8
💡 解题思路
Let $z = a+bi$ , so $\overline{z} = a-bi$ . By definition, $z = a+bi = f(z) = i(a-bi) = b+ai$ , which implies that all solutions to $f(z) = z$ lie on the line $y=x$ on the complex plane. The graph of
17
第 17 题
方程
For some real numbers a and b , the equation \[8x^3 + 4ax^2 + 2bx + a = 0\] has three distinct positive roots. If the sum of the base- 2 logarithms of the roots is 5 , what is the value of a ? (A)\ -256 (B)\ -64 (C)\ -8 (D)\ 64 (E)\ 256
💡 解题思路
Let the three roots be $x_1,x_2,x_3$ . \[\log_2 x_1 + \log_2 x_2 + \log_2 x_3 = \log_2 x_1x_2x_3= 5 \Longrightarrow x_1x_2x_3 = 32\] By Vieta’s formulas , \[8(x-x_1)(x-x_2)(x-x_3) = 8x^3 + 4ax^2 + 2bx
18
第 18 题
函数
Points A and B are on the parabola y=4x^2+7x-1 , and the origin is the midpoint of AB . What is the length of AB ? (A)\ 2\sqrt5 (B)\ 5+\frac{\sqrt2}{2} (C)\ 5+\sqrt2 (D)\ 7 (E)\ 5\sqrt2
💡 解题思路
Let the coordinates of $A$ be $(x_A,y_A)$ . As $A$ lies on the parabola, we have $y_A=4x_A^2+7x_A-1$ . As the origin is the midpoint of $AB$ , the coordinates of $B$ are $(-x_A,-y_A)$ . We need to cho
19
第 19 题
立体几何
A truncated cone has horizontal bases with radii 18 and 2 . A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere? (A)\ 6 (B)\ 4√(5) (C)\ 9 (D)\ 10 (E)\ 6√(3)
💡 解题思路
Consider a trapezoid (label it $ABCD$ as follows) cross-section of the truncate cone along a diameter of the bases:
20
第 20 题
概率
Each face of a cube is painted either red or blue, each with probability 1/2 . The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color?
💡 解题思路
There are $2^6$ possible colorings of the cube. Consider the color that appears with greater frequency. The property obviously holds true if $5$ or $6$ of the faces are colored the same, which for eac
21
第 21 题
坐标几何
The graph of 2x^2 + xy + 3y^2 - 11x - 20y + 40 = 0 is an ellipse in the first quadrant of the xy -plane. Let a and b be the maximum and minimum values of \frac yx over all points (x,y) on the ellipse. What is the value of a+b ? (A)\ 3 (B)\ √(10) (C)\ \frac 72 (D)\ \frac 92 (E)\ 2√(14)
💡 解题思路
$\frac yx$ represents the slope of a line passing through the origin. It follows that since a line $y = mx$ intersects the ellipse at either $0, 1,$ or $2$ points, the minimum and maximum are given wh
22
第 22 题
几何·面积
The square is a multiplicative magic square. That is, the product of the numbers in each row, column, and diagonal is the same. If all the entries are positive integers, what is the sum of the possible values of g ?
💡 解题思路
All the unknown entries can be expressed in terms of $b$ . Since $100e = beh = ceg = def$ , it follows that $h = \frac{100}{b}, g = \frac{100}{c}$ , and $f = \frac{100}{d}$ . Comparing rows $1$ and $3
23
第 23 题
规律与数列
The polynomial x^3 - 2004 x^2 + mx + n has integer coefficients and three distinct positive zeros. Exactly one of these is an integer, and it is the sum of the other two. How many values of n are possible? (A)\ 250,\!000 (B)\ 250,\!250 (C)\ 250,\!500 (D)\ 250,\!750 (E)\ 251,\!000
💡 解题思路
Let the roots be $r,s,r + s$ , and let $t = rs$ . Then
24
第 24 题
几何·面积
In \triangle ABC , AB = BC , and \overline{BD} is an altitude . Point E is on the extension of \overline{AC} such that BE = 10 . The values of \tan \angle CBE , \tan \angle DBE , and \tan \angle ABE form a geometric progression , and the values of \cot \angle DBE,\cot \angle CBE,\cot \angle DBC form an arithmetic progression . What is the area of \triangle ABC ? (A)\ 16 (B)\ \frac {50}3 (C)\ 10√(3) (D)\ 8√(5) (E)\ 18
💡 解题思路
Let $\alpha = DBC$ . Then the first condition tells us that \[\tan^2 DBE = \tan(DBE - \alpha)\tan(DBE + \alpha) = \frac {\tan^2 DBE - \tan^2 \alpha}{1 - \tan ^2 DBE \tan^2 \alpha},\] and multiplying o
25
第 25 题
数字运算
Given that 2^{2004} is a 604 - digit number whose first digit is 1 , how many elements of the set S = \{2^0,2^1,2^2,\ldots ,2^{2003}\} have a first digit of 4 ? (A)\ 194 (B)\ 195 (C)\ 196 (D)\ 197 (E)\ 198
💡 解题思路
Given $n$ digits, there must be exactly one power of $2$ with $n$ digits such that the first digit is $1$ . Thus $S$ contains $603$ elements with a first digit of $1$ . For each number in the form of