2008B 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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题目涉及图形的部分，原题以文字描述代替（图形题建议配合原版试卷使用）
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?
💡 解题思路
If the basketball player makes $x$ three-point shots and $5-x$ two-point shots, he scores $3x+2(5-x)=10+x$ points. Clearly every value of $x$ yields a different number of total points. Since he can ma
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
应用题
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 $700,000$
4
第 4 题
几何·面积
On circle O , points C and D are on the same side of diameter \overline{AB} , \angle AOC = 30^\circ , and \angle DOB = 45^\circ . What is the ratio of the area of the smaller sector COD to the area of the circle? [图]
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?
💡 解题思路
The class could send $25$ carnations and no roses, $22$ carnations and $2$ roses, $19$ carnations and $4$ roses, and so on, down to $1$ carnation and $16$ roses. There are 9 total possibilities (from
6
第 6 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?
💡 解题思路
Every time the pedometer flips, Pete has walked $100,000$ steps. Therefore, Pete has walked a total of $100,000 \cdot 44 + 50,000 = 4,450,000$ steps, which is $4,450,000/1,800 = 2472.2$ miles, which i
7
第 7 题
应用题
For real numbers a and b , define a\textdollar b = (a - b)^2 . What is (x - y)^2\textdollar(y - x)^2 ?
💡 解题思路
$\left[ (x-y)^2 - (y-x)^2 \right]^2$
8
第 8 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
Since $\overline{AB}=4\overline{BD}$ and $\overline{AB}+\overline{BD}=\overline{AD}$ , $\overline{AB}=\frac{4}{5}\overline{AD}$ .
9
第 9 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 ?
💡 解题思路
Let $\alpha$ be the angle that subtends the arc $AB$ . By the law of cosines, $6^2=5^2+5^2-2\cdot 5\cdot 5\cos(\alpha)$ implies $\cos(\alpha) = 7/25$ .
10
第 10 题
时间问题
Bricklayer Brenda would take 9 hours to build a chimney alone, and bricklayer Brandon would take 10 hours to build it alone. When they work together they talk a lot, and their combined output is decreased by 10 bricks per hour. Working together, they build the chimney in 5 hours. How many bricks are in the chimney?
💡 解题思路
Let $h$ be the number of bricks in the chimney.
11
第 11 题
立体几何
A cone-shaped mountain has its base on the ocean floor and has a height of 8000 feet. The top \frac{1}{8} of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain in feet?
💡 解题思路
In a cone, radius and height each vary inversely with increasing height (i.e. the radius of the cone formed by cutting off the mountain at $4,000$ feet is half that of the original mountain). Therefor
12
第 12 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?
💡 解题思路
Letting $S_n$ be the nth partial sum of the sequence:
13
第 13 题
几何·面积
Vertex E of equilateral \triangle{ABE} is in the interior of unit square ABCD . Let R be the region consisting of all points inside ABCD and outside \triangle{ABE} whose distance from AD is between \frac{1}{3} and \frac{2}{3} . What is the area of R ?
💡 解题思路
The region is the shaded area:
14
第 14 题
几何·面积
A circle has a radius of \log_{10}{(a^2)} and a circumference of \log_{10}{(b^4)} . What is \log_{a}{b} ?
💡 解题思路
Let $C$ be the circumference of the circle, and let $r$ be the radius of the circle.
15
第 15 题
几何·面积
On each side of a unit square, an equilateral triangle of side length 1 is constructed. On each new side of each equilateral triangle, another equilateral triangle of side length 1 is constructed. The interiors of the square and the 12 triangles have no points in common. Let R be the region formed by the union of the square and all the triangles, and S be the smallest convex polygon that contains R . What is the area of the region that is inside S but outside R ?
💡 解题思路
[asy] real a = 1/2, b = sqrt(3)/2; draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((0,0)--(a,-b)--(1,0)--(1+b,a)--(1,1)--(a,1+b)--(0,1)--(-b,a)--(0,0)); draw((0,0)--(-1+a,-b)--(1+a,-b)--(1,0)--(1+b,-1+a
16
第 16 题
几何·面积
A rectangular floor measures a by b feet, where a and b are positive integers with b > a . An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width 1 foot around the painted rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair (a,b) ?
💡 解题思路
$A_{outer}=ab$
17
第 17 题
几何·面积
Let A , B and C be three distinct points on the graph of y=x^2 such that line AB is parallel to the x -axis and \triangle ABC is a right triangle with area 2008 . What is the sum of the digits of the y -coordinate of C ?
💡 解题思路
Supposing $\angle A=90^\circ$ , $AC$ is perpendicular to $AB$ and, it follows, to the $x$ -axis, making $AC$ a segment of the line $x=m$ . But that would mean that the coordinates of $C$ are $(m, m^2)
18
第 18 题
几何·面积
A pyramid has a square base ABCD and vertex E . The area of square ABCD is 196 , and the areas of \triangle ABE and \triangle CDE are 105 and 91 , respectively. What is the volume of the pyramid?
💡 解题思路
Let $h$ be the height of the pyramid and $a$ be the distance from $h$ to $CD$ . The side length of the base is $14$ . The heights of $\triangle ABE$ and $\triangle CDE$ are $2\cdot105\div14=15$ and $2
19
第 19 题
函数
A function f is defined by f(z) = (4 + i) z^2 + \alpha z + \gamma for all complex numbers z , where \alpha and \gamma are complex numbers and i^2 = - 1 . Suppose that f(1) and f(i) are both real. What is the smallest possible value of | \alpha | + |\gamma | ?
💡 解题思路
We need only concern ourselves with the imaginary portions of $f(1)$ and $f(i)$ (both of which must be 0). These are:
20
第 20 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(
21
第 21 题
几何·面积
Two circles of radius 1 are to be constructed as follows. The center of circle A is chosen uniformly and at random from the line segment joining (0,0) and (2,0) . The center of circle B is chosen uniformly and at random, and independently of the first choice, from the line segment joining (0,1) to (2,1) . What is the probability that circles A and B intersect?
💡 解题思路
Circles centered at $A$ and $B$ will overlap if $A$ and $B$ are closer to each other than if the circles were tangent. The circles are tangent when the distance between their centers is equal to the s
22
第 22 题
概率
A parking lot has 16 spaces in a row. Twelve cars arrive, each of which requires one parking space, and their drivers chose spaces at random from among the available spaces. Auntie Em then arrives in her SUV, which requires 2 adjacent spaces. What is the probability that she is able to park?
💡 解题思路
Auntie Em won't be able to park only when none of the four available spots touch. We can form a bijection between all such cases and the number of ways to pick four spots out of 13: since none of the
23
第 23 题
规律与数列
The sum of the base- 10 logarithms of the divisors of 10^n is 792 . What is n ? (A)\ 11 (B)\ 12 (C)\ 13 (D)\ 14 (E)\ 15
💡 解题思路
Every factor of $10^n$ will be of the form $2^a \times 5^b , a\leq n , b\leq n$ . Not all of these base ten logarithms will be rational, but we can add them together in a certain way to make it ration
24
第 24 题
几何·面积
Let A_0=(0,0) . Distinct points A_1,A_2,\dots lie on the x -axis, and distinct points B_1,B_2,\dots lie on the graph of y=√(x) . For every positive integer n,\ A_{n-1}B_nA_n is an equilateral triangle. What is the least n for which the length A_0A_n≥100 ?
💡 解题思路
Let $a_n=|A_{n-1}A_n|$ . We need to rewrite the recursion into something manageable. The two strange conditions, $B$ 's lie on the graph of $y=\sqrt{x}$ and $A_{n-1}B_nA_n$ is an equilateral triangle,
25
第 25 题
几何·面积
Let ABCD be a trapezoid with AB||CD, AB=11, BC=5, CD=19, and DA=7 . Bisectors of \angle A and \angle D meet at P , and bisectors of \angle B and \angle C meet at Q . What is the area of hexagon ABQCDP ?
💡 解题思路
Note: In the image AB and CD have been swapped from their given lengths in the problem. However, this doesn't affect any of the solving.