2004 AMC 10B — Official Competition Problems (January 2004)
📅 2004 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
Each row of the Misty Moon Amphitheater has 33 seats. Rows 12 through 22 are reserved for a youth club. How many seats are reserved for this club? (A) \ 297 (B) \ 330 (C) \ 363 (D) \ 396 (E) \ 726
💡 解题思路
There are $22-12+1=11$ rows of $33$ seats, giving $11\times 33=\boxed{\mathrm{(C)}\ 363}$ seats.
2
第 2 题
数字运算
How many two-digit positive integers have at least one 7 as a digit? (A) \ 10 (B) \ 18 (C) \ 19 (D) \ 20 (E) \ 30
💡 解题思路
Ten numbers $(70,71,\dots,79)$ have $7$ as the tens digit. Nine numbers $(17,27,\dots,97)$ have it as the ones digit. Number $77$ is in both sets.
3
第 3 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? (A) \ 3 (B) \ 6 (C) \ 9 (D) \ 12 (E) \ 15
💡 解题思路
At the fourth practice she made $48/2=24$ throws, at the third one it was $24/2=12$ , then we get $12/2=6$ throws for the second practice, and finally $6/2=3\Rightarrow\boxed{\mathrm{(A)}\ 3}$ throws
4
第 4 题
综合
A standard six-sided die is rolled, and P is the product of the five numbers that are visible. What is the largest number that is certain to divide P ? (A) \ 6 (B) \ 12 (C) \ 24 (D) \ 144 (E) \ 720
💡 解题思路
The product of all six numbers is $6!=720$ . The products of numbers that can be visible are $720/1$ , $720/2$ , ..., $720/6$ . The answer to this problem is their greatest common divisor -- which is
5
第 5 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,
6
第 6 题
几何·面积
Which of the following numbers is a perfect square? (A) \ 98! · 99! (B) \ 98! · 100! (C) \ 99! · 100! (D) \ 99! · 101! (E) \ 100! · 101!
💡 解题思路
Using the fact that $n! = n\cdot (n-1)!$ , we can write:
7
第 7 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 $\boxed{\mathrm{(A)}\ 5}$ .
8
第 8 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
9
第 9 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 $S_{\square}=100$ .
10
第 10 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 $n=\boxed{\mathrm{(D)\ }10}$ .
11
第 11 题
概率
Two eight-sided dice each have faces numbered 1 through 8 . When the dice are rolled, each face has an equal probability of appearing on the top. What is the probability that the product of the two top numbers is greater than their sum? (A)\ \frac{1}{2} (B)\ \frac{47}{64} (C)\ \frac{3}{4} (D)\ \frac{55}{64} (E)\ \frac{7}{8}
💡 解题思路
We have $1\times n = n < 1 + n$ , hence if at least one of the numbers is $1$ , the sum is larger. There $15$ such possibilities.
12
第 12 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 $\pi(b^2-c^2)$ .
13
第 13 题
概率
In the United States, coins have the following thicknesses: penny, 1.55 mm; nickel, 1.95 mm; dime, 1.35 mm; quarter, 1.75 mm. If a stack of these coins is exactly 14 mm high, how many coins are in the stack? (A) \ 7 (B) \ 8 (C) \ 9 (D) \ 10 (E) \ 11
💡 解题思路
All numbers in this solution will be in hundredths of a millimeter.
14
第 14 题
分数与比例
A bag initially contains red marbles and blue marbles only, with more blue than red. Red marbles are added to the bag until only \frac{1}{3} of the marbles in the bag are blue. Then yellow marbles are added to the bag until only \frac{1}{5} of the marbles in the bag are blue. Finally, the number of blue marbles in the bag is doubled. What fraction of the marbles now in the bag are blue? (A) \ \frac{1}{5} (B) \ \frac{1}{4} (C) \ \frac{1}{3} (D) \ \frac{2}{5} (E) \ \frac{1}{2}
💡 解题思路
We can ignore most of the problem statement. The only important information is that immediately before the last step blue marbles formed $\frac{1}{5}$ of the marbles in the bag. This means that there
15
第 15 题
概率
Patty has 20 coins consisting of nickels and dimes. If her nickels were dimes and her dimes were nickels, she would have 70 cents more. How much are her coins worth?
💡 解题思路
She has $n$ nickels and $d=20-n$ dimes. Their total cost is $5n+10d=5n+10(20-n)=200-5n$ cents. If the dimes were nickels and vice versa, she would have $10n+5d=10n+5(20-n)=100+5n$ cents. This value sh
16
第 16 题
几何·面积
Three circles of radius 1 are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle? (A) \ \frac{2 + √(6)}{3} (B) \ 2 (C) \ \frac{2 + 3√(2)}{2} (D) \ \frac{3 + 2√(3)}{3} (E) \ \frac{3 + √(3)}{2}
💡 解题思路
The situation is shown in the picture below. The radius we seek is $SD = AD + AS$ . Clearly $AD=1$ . The point $S$ is the center of the equilateral triangle $ABC$ , thus $AS$ is $2/3$ of the altitude
17
第 17 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 $\overline{ba}=10b+a$ .
18
第 18 题
几何·面积
In the right triangle \triangle ACE , we have AC=12 , CE=16 , and EA=20 . Points B , D , and F are located on AC , CE , and EA , respectively, so that AB=3 , CD=4 , and EF=5 . What is the ratio of the area of \triangle DBF to that of \triangle ACE ? [图] (A) \ \frac{1}{4} (B) \ \frac{9}{25} (C) \ \frac{3}{8} (D) \ \frac{11}{25} (E) \ \frac{7}{16}
💡 解题思路
Let $x = [DBF]$ . Because $\triangle ACE$ is divided into four triangles, $[ACE] = [BCD] + [ABF] + [DEF] + x$ .
19
第 19 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
20
第 20 题
几何·面积
In \triangle ABC points D and E lie on BC and AC , respectively. If AD and BE intersect at T so that \frac{AT}{DT}=3 and \frac{BT}{ET}=4 , what is \frac{CD}{BD} ? [图] (A) \ \frac{1}{8} (B) \ \frac{2}{9} (C) \ \frac{3}{10} (D) \ \frac{4}{11} (E) \ \frac{5}{12}
💡 解题思路
We use the square bracket notation $[\cdot]$ to denote area.
21
第 21 题
规律与数列
Let 1 , 4 , 7 , \ldots and 9 , 16 , 23 , \ldots be two arithmetic progressions. The set S is the union of the first 2004 terms of each sequence. How many distinct numbers are in S ? (A) \ 3722 (B) \ 3732 (C) \ 3914 (D) \ 3924 (E) \ 4007
💡 解题思路
The terms in the first sequence are defined by $n \equiv 1 \pmod3$ , while the terms in the second sequence are defined by $n \equiv 2 \pmod 7.$ We seek to find the solutions to this system of modular
22
第 22 题
几何·面积
A triangle with sides of 5, 12, and 13 has both an inscribed and a circumscribed circle. What is the distance between the centers of those circles? (A) \ \frac{3√(5)}{2} (B) \ \frac{7}{2} (C) \ √(15) (D) \ \frac{√(65)}{2} (E) \ \frac{9}{2}
💡 解题思路
[asy] import geometry; unitsize(0.6 cm); pair A, B, C, D, E, F, I, O; A = (5^2/13,5*12/13); B = (0,0); C = (13,0); I = incenter(A,B,C); D = (I + reflect(B,C)*(I))/2; E = (I + reflect(C,A)*(I))/2; F =
23
第 23 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? (A) \ \frac{1}{4} (B) \ \frac{5}{16} (C) \ \frac{3}{8} (D) \ \frac{7}{16} (E) \ \frac{1}{2}
💡 解题思路
Label the six sides of the cube by numbers $1$ to $6$ as on a classic dice. Then the "four vertical faces" can be: $\{1,2,5,6\}$ , $\{1,3,4,6\}$ , or $\{2,3,4,5\}$ .
24
第 24 题
几何·面积
In triangle ABC we have AB=7 , AC=8 , BC=9 . Point D is on the circumscribed circle of the triangle so that AD bisects angle BAC . What is the value of \frac{AD}{CD} ? (A) \dfrac{9}{8} (B) \dfrac{5}{3} (C) 2 (D) \dfrac{17}{7} (E) \dfrac{5}{2}
💡 解题思路
Set $\overline{BD}$ 's length as $x$ . $\overline{CD}$ 's length must also be $x$ since $\angle BAD$ and $\angle DAC$ intercept arcs of equal length (because $\angle BAD=\angle DAC$ ). Using Ptolemy's
25
第 25 题
几何·面积
A circle of radius 1 is internally tangent to two circles of radius 2 at points A and B , where AB is a diameter of the smaller circle. What is the area of the region, shaded in the picture, that is outside the smaller circle and inside each of the two larger circles? (A) \ \frac{5}{3} π - 3\sqrt 2 (B) \ \frac{5}{3} π - 2\sqrt 3 (C) \ \frac{8}{3} π - 3\sqrt 3 (D) \ \frac{8}{3} π - 3\sqrt 2 (E) \ \frac{8}{3} π - 2\sqrt 3 [图]
💡 解题思路
The area of the small circle is $\pi$ . We can add it to the shaded region, compute the area of the new region, and then subtract the area of the small circle from the result.