2011A 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
应用题
A cell phone plan costs 20 dollars each month, plus 5 cents per text message sent, plus 10 cents for each minute used over 30 hours. In January Michelle sent 100 text messages and talked for 30.5 hours. How much did she have to pay?
💡 解题思路
The base price of Michelle's cell phone plan is $20$ dollars. If she sent $100$ text messages and it costs $5$ cents per text, then she must have spent $500$ cents for texting, or $5$ dollars. She tal
2
第 2 题
概率
There are 5 coins placed flat on a table according to the figure. What is the order of the coins from top to bottom? [图]
💡 解题思路
By careful inspection and common sense, the answer is $\textbf{(E)}$ .
3
第 3 题
工程问题
A small bottle of shampoo can hold 35 milliliters of shampoo, whereas a large bottle can hold 500 milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy?
💡 解题思路
To find how many small bottles we need, we can simply divide $500$ by $35$ . This simplifies to $\frac{100}{7}=14 \frac{2}{7}.$ Since the answer must be an integer greater than $14$ , we have to round
4
第 4 题
统计
At an elementary school, the students in third grade, fourth grade, and fifth grade run an average of 12 , 15 , and 10 minutes per day, respectively. There are twice as many third graders as fourth graders, and twice as many fourth graders as fifth graders. What is the average number of minutes run per day by these students?
💡 解题思路
Let us say that there are $f$ fifth graders. According to the given information, there must be $2f$ fourth graders and $4f$ third graders. The average time run by each student is equal to the total am
5
第 5 题
分数与比例
Last summer 30\% of the birds living on Town Lake were geese, 25\% were swans, 10\% were herons, and 35\% were ducks. What percent of the birds that were not swans were geese?
💡 解题思路
To simplify the problem, WLOG, let us say that there were a total of $100$ birds. The number of birds that are not swans is $75$ . The number of geese is $30$ . Therefore the percentage is just $\frac
6
第 6 题
综合
The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was 61 points. How many free throws did they make?
💡 解题思路
For the points made from two-point shots and from three-point shots to be equal, the numbers of made shots are in a $3:2$ ratio. Therefore, assume they made $3x$ and $2x$ two- and three- point shots,
7
第 7 题
统计
A majority of the 30 students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than 1 . The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was \17.71$ . What was the cost of a pencil in cents?
💡 解题思路
The total cost of the pencils can be found by $(\text{students}\cdot\text{pencils purchased by each}\cdot\text{price of each pencil})$ .
8
第 8 题
规律与数列
In the eight term sequence A , B , C , D , E , F , G , H , the value of C is 5 and the sum of any three consecutive terms is 30 . What is A+H ?
💡 解题思路
Let $A=x$ . Then from $A+B+C=30$ , we find that $B=25-x$ . From $B+C+D=30$ , we then get that $D=x$ . Continuing this pattern, we find $E=25-x$ , $F=5$ , $G=x$ , and finally $H=25-x$ . So $A+H=x+25-x=
9
第 9 题
综合
At a twins and triplets convention, there were 9 sets of twins and 6 sets of triplets, all from different families. Each twin shook hands with all the twins except his/her siblings and with half the triplets. Each triplet shook hands with all the triplets except his/her siblings and with half the twins. How many handshakes took place?
💡 解题思路
There are $18$ total twins and $18$ total triplets. Each of the twins shakes hands with the $16$ twins not in their family and $9$ of the triplets, a total of $25$ people. Each of the triplets shakes
10
第 10 题
几何·面积
A pair of standard 6 -sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?
💡 解题思路
For the circumference to be greater than the area, we must have $\pi d > \pi \left( \frac{d}{2} \right) ^2$ , or $d<4$ . Now since $d$ is determined by a sum of two dice, the only possibilities for $d
11
第 11 题
几何·面积
Circles A, B, and C each has radius 1 . Circles A and B share one point of tangency. Circle C has a point of tangency with the midpoint of \overline{AB}. What is the area inside circle C but outside circle A and circle B? [图]
A power boat and a raft both left dock A on a river and headed downstream. The raft drifted at the speed of the river current. The power boat maintained a constant speed with respect to the river. The power boat reached dock B downriver, then immediately turned and traveled back upriver. It eventually met the raft on the river 9 hours after leaving dock A. How many hours did it take the power boat to go from A to B ?
💡 解题思路
WLOG, let the speed of the river be 0. This is allowed because the problem never states that the speed of the current has to have a magnitude greater than 0. In this case, when the powerboat travels f
13
第 13 题
几何·面积
Triangle ABC has side-lengths AB = 12, BC = 24, and AC = 18. The line through the incenter of \triangle ABC parallel to \overline{BC} intersects \overline{AB} at M and \overline{AC} at N. What is the perimeter of \triangle AMN?
💡 解题思路
Let $O$ be the incenter of $\triangle{ABC}$ . Because $\overline{MO} \parallel \overline{BC}$ and $\overline{BO}$ is the angle bisector of $\angle{ABC}$ , we have
14
第 14 题
概率
Suppose a and b are single-digit positive integers chosen independently and at random. What is the probability that the point (a,b) lies above the parabola y=ax^2-bx ?
💡 解题思路
If $(a,b)$ lies above the parabola, then $b$ must be greater than $y(a)$ . We thus get the inequality $b>a^3-ba$ . Solving this for $b$ gives us $b>\frac{a^3}{a+1}$ . Now note that $\frac{a^3}{a+1}$ c
15
第 15 题
几何·面积
The circular base of a hemisphere of radius 2 rests on the base of a square pyramid of height 6 . The hemisphere is tangent to the other four faces of the pyramid. What is the edge-length of the base of the pyramid?
💡 解题思路
Let $ABCDE$ be the pyramid with $ABCD$ as the square base. Let $O$ and $M$ be the center of square $ABCD$ and the midpoint of side $AB$ respectively. Lastly, let the hemisphere be tangent to the trian
16
第 16 题
计数
Each vertex of convex pentagon ABCDE is to be assigned a color. There are 6 colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible?
💡 解题思路
We can do some casework when working our way around the pentagon from $A$ to $E$ . At each stage, there will be a makeshift diagram.
17
第 17 题
几何·面积
Circles with radii 1 , 2 , and 3 are mutually externally tangent. What is the area of the triangle determined by the points of tangency?
Suppose that |x+y|+|x-y|=2 . What is the maximum possible value of x^2-6x+y^2 ?
💡 解题思路
Plugging in some values, we see that the graph of the equation $|x+y|+|x-y| = 2$ is a square bounded by $x= \pm 1$ and $y = \pm 1$ .
19
第 19 题
规律与数列
At a competition with N players, the number of players given elite status is equal to 2^{1+\lfloor \log_{2} (N-1) \rfloor}-N . Suppose that 19 players are given elite status. What is the sum of the two smallest possible values of N ?
💡 解题思路
We start with $2^{1+\lfloor\log_{2}(N-1)\rfloor}-N = 19$ . After rearranging, we get $\lfloor\log_{2}(N-1)\rfloor = \log_{2} \left(\frac{N+19}{2}\right)$ .
20
第 20 题
函数
Let f(x)=ax^2+bx+c , where a , b , and c are integers. Suppose that f(1)=0 , 50<f(7)<60 , 70<f(8)<80 , 5000k<f(100)<5000(k+1) for some integer k . What is k ?
💡 解题思路
From $f(1) = 0$ , we know that $a+b+c = 0$ .
21
第 21 题
函数
Let f_{1}(x)=√(1-x) , and for integers n ≥ 2 , let f_{n}(x)=f_{n-1}(√(n^2 - x)) . If N is the largest value of n for which the domain of f_{n} is nonempty, the domain of f_{N} is \{c\} . What is N+c ?
💡 解题思路
The domain of $f_{1}(x)=\sqrt{1-x}$ is defined when $x\leq1$ . \[f_{2}(x)=f_{1}\left(\sqrt{4-x}\right)=\sqrt{1-\sqrt{4-x}}\]
22
第 22 题
几何·面积
Let R be a unit square region and n ≥ 4 an integer. A point X in the interior of R is called n-ray partitional if there are n rays emanating from X that divide R into n triangles of equal area. How many points are 100 -ray partitional but not 60 -ray partitional?
💡 解题思路
There must be four rays emanating from $X$ that intersect the four corners of the square region. Depending on the location of $X$ , the number of rays distributed among these four triangular sectors w
23
第 23 题
综合
Let f(z)= \frac{z+a}{z+b} and g(z)=f(f(z)) , where a and b are complex numbers. Suppose that | a | = 1 and g(g(z))=z for all z for which g(g(z)) is defined. What is the difference between the largest and smallest possible values of | b | ?
💡 解题思路
By algebraic manipulations, we obtain \[h(z)=g(g(z))=f(f(f(f(z))))=\frac{Pz+Q}{Rz+S}\] where \[P=(a+1)^2+a(b+1)^2\] \[Q=a(b+1)(b^2+2a+1)\] \[R=(b+1)(b^2+2a+1)\] \[S=a(b+1)^2+(a+b^2)^2\] In order for $
24
第 24 题
几何·面积
Consider all quadrilaterals ABCD such that AB=14 , BC=9 , CD=7 , and DA=12 . What is the radius of the largest possible circle that fits inside or on the boundary of such a quadrilateral?
💡 解题思路
Note as above that ABCD must be tangential to obtain the circle with maximal radius. Let $E$ , $F$ , $G$ , and $H$ be the points on $AB$ , $BC$ , $CD$ , and $DA$ respectively where the circle is tange
25
第 25 题
几何·面积
Triangle ABC has \angle BAC = 60^{\circ} , \angle CBA ≤ 90^{\circ} , BC=1 , and AC ≥ AB . Let H , I , and O be the orthocenter, incenter, and circumcenter of \triangle ABC , respectively. Assume that the area of pentagon BCOIH is the maximum possible. What is \angle CBA ?
💡 解题思路
By the Inscribed Angle Theorem, \[\angle BOC = 2\angle BAC = 120^\circ .\] Let $D$ and $E$ be the feet of the altitudes of $\triangle ABC$ from $B$ and $C$ , respectively. In $\triangle ACE$ we get $\