2009B 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
Each morning of her five-day workweek, Jane bought either a 50 -cent muffin or a 75 -cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy?
💡 解题思路
If Jane bought one more bagel but one fewer muffin, then her total cost for the week would increase by $25$ cents.
2
第 2 题
工程问题
Paula the painter had just enough paint for 30 identically sized rooms. Unfortunately, on the way to work, three cans of paint fell off her truck, so she had only enough paint for 25 rooms. How many cans of paint did she use for the 25 rooms? (A)\ 10 (B)\ 12 (C)\ 15 (D)\ 18 (E)\ 25
💡 解题思路
Losing three cans of paint corresponds to being able to paint five fewer rooms. So $\frac 35 \cdot 25 = \boxed{15}$ . The answer is $\mathrm{(C)}$ .
3
第 3 题
分数与比例
Twenty percent less than 60 is one-third more than what number? (A)\ 16 (B)\ 30 (C)\ 32 (D)\ 36 (E)\ 48
💡 解题思路
Twenty percent less than 60 is $\frac 45 \cdot 60 = 48$ . One-third more than a number n is $\frac 43n$ . Therefore $\frac 43n = 48$ and the number is $\boxed {36}$ . The answer is $\mathrm{(D)}$ .
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第 4 题
几何·面积
A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths 15 and 25 meters. What fraction of the yard is occupied by the flower beds? (A)\frac {1}{8} (B)\frac {1}{6} (C)\frac {1}{5} (D)\frac {1}{4} (E)\frac {1}{3}
💡 解题思路
Each triangle has leg length $\frac 12 \cdot (25 - 15) = 5$ meters and area $\frac 12 \cdot 5^2 = \frac {25}{2}$ square meters. Thus the flower beds have a total area of $25$ square meters. The entire
5
第 5 题
规律与数列
Kiana has two older twin brothers. The product of their three ages is 128. What is the sum of their three ages? (A)\ 10 (B)\ 12 (C)\ 16 (D)\ 18 (E)\ 24
💡 解题思路
The age of each person is a factor of $128 = 2^7$ . So the twins could be $2^0 = 1, 2^1 = 2, 2^2 = 4, 2^3 = 8$ years of age and, consequently Kiana could be $128$ , $32$ , $8$ , or $2$ years old, resp
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第 6 题
行程问题
By inserting parentheses, it is possible to give the expression \[2×3 + 4×5\] several values. How many different values can be obtained? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6
💡 解题思路
The three operations can be performed on any of $3! = 6$ orders. However, if the addition is performed either first or last, then multiplying in either order produces the same result. So at most four
7
第 7 题
应用题
In a certain year the price of gasoline rose by 20\% during January, fell by 20\% during February, rose by 25\% during March, and fell by x\% during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the nearest integer, what is x(A)\ 12 (B)\ 17 (C)\ 20 (D)\ 25 (E)\ 35
💡 解题思路
Let $p$ be the price at the beginning of January. The price at the end of March was $(1.2)(0.8)(1.25)p = 1.2p.$ Because the price at the end of April was $p$ , the price decreased by $0.2p$ during Apr
8
第 8 题
规律与数列
When a bucket is two-thirds full of water, the bucket and water weigh a kilograms. When the bucket is one-half full of water the total weight is b kilograms. In terms of a and b , what is the total weight in kilograms when the bucket is full of water? (A)\ \frac23a + \frac13b (B)\ \frac32a - \frac12b (C)\ \frac32a + b (D)\ \frac32a + 2b (E)\ 3a - 2b
💡 解题思路
Let $x$ be the weight of the bucket and let $y$ be the weight of the water in a full bucket. Then we are given that $x + \frac 23y = a$ and $x + \frac 12y = b$ . Hence $\frac 16y = a-b$ , so $y = 6a-6
9
第 9 题
几何·面积
Triangle ABC has vertices A = (3,0) , B = (0,3) , and C , where C is on the line x + y = 7 . What is the area of \triangle ABC ? (A)\ 6 (B)\ 8 (C)\ 10 (D)\ 12 (E)\ 14
💡 解题思路
Because the line $x + y = 7$ is parallel to $\overline {AB}$ , the area of $\triangle ABC$ is independent of the location of $C$ on that line. Therefore it may be assumed that $C$ is $(7,0)$ . In that
10
第 10 题
分数与比例
A particular 12 -hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a 1 , it mistakenly displays a 9 . For example, when it is 1:16 PM the clock incorrectly shows 9:96 PM. What fraction of the day will the clock show the correct time? (A)\ \frac 12 (B)\ \frac 58 (C)\ \frac 34 (D)\ \frac 56 (E)\ \frac {9}{10}
💡 解题思路
The clock will display the incorrect time for the entire hours of $1, 10, 11$ and $12$ . So the correct hour is displayed $\frac 23$ of the time. The minutes will not display correctly whenever either
11
第 11 题
综合
On Monday, Millie puts a quart of seeds, 25\% of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only 25\% of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet?
💡 解题思路
On Monday, day 1, the birds find $\frac 14$ quart of millet in the feeder. On Tuesday they find \[\frac 14 + \frac 34 \cdot \frac 14\] quarts of millet. On Wednesday, day 3, they find \[\frac 14 + \fr
12
第 12 题
规律与数列
The fifth and eighth terms of a geometric sequence of real numbers are 7! and 8! respectively. What is the first term? (A)\ 60 (B)\ 75 (C)\ 120 (D)\ 225 (E)\ 315
💡 解题思路
Let the $n$ th term of the series be $ar^{n-1}$ . Because \[\frac {8!}{7!} = \frac {ar^7}{ar^4} = r^3 = 8,\] it follows that $r = 2$ and the first term is $a = \frac {7!}{r^4} = \frac {7!}{16} = \boxe
13
第 13 题
几何·面积
Triangle ABC has AB = 13 and AC = 15 , and the altitude to \overline{BC} has length 12 . What is the sum of the two possible values of BC ? (A)\ 15 (B)\ 16 (C)\ 17 (D)\ 18 (E)\ 19
💡 解题思路
Let $D$ be the foot of the altitude to $\overline{BC}$ . Then $BD = \sqrt {13^2 - 12^2} = 5$ and $DC = \sqrt {15^2 - 12^2} = 9$ . Thus $BC = BD + BC = 5 + 9 = 14$ . Otherwise, assume that the triangle
14
第 14 题
几何·面积
Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from (c,0) to (3,3) , divides the entire region into two regions of equal area. What is c ? [图]
💡 解题思路
For $c\geq 1.5$ the shaded area is at most $1.5$ , which is too little. Hence $c<1.5$ , and therefore the point $(2,1)$ is indeed inside the shaded part, as shown in the picture.
15
第 15 题
方程
Assume 0 < r < 3 . Below are five equations for x . Which equation has the largest solution x ?
💡 解题思路
(B) Intuitively, $x$ will be largest for that option for which the value in the parentheses is smallest.
16
第 16 题
分数与比例
Trapezoid ABCD has AD||BC , BD = 1 , \angle DBA = 23^{\circ} , and \angle BDC = 46^{\circ} . The ratio BC: AD is 9: 5 . What is CD ? (A)\ \frac 79 (B)\ \frac 45 (C)\ \frac {13}{15} (D)\ \frac 89 (E)\ \frac {14}{15}
💡 解题思路
Extend $\overline {AB}$ and $\overline {DC}$ to meet at $E$ . Then
17
第 17 题
概率
Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of the opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube? (A)\frac 18 (B)\frac {3}{16} (C)\frac 14 (D)\frac 38 (E)\frac 12
💡 解题思路
There are two possible stripe orientations for each of the six faces of the cube, so there are $2^6 = 64$ possible stripe combinations. There are three pairs of parallel faces so, if there is an encir
18
第 18 题
坐标几何
Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every 90 seconds, and Robert runs clockwise and completes a lap every 80 seconds. Both start from the same line at the same time. At some random time between 10 minutes and 11 minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. What is the probability that both Rachel and Robert are in the picture? (A)\frac {1}{16} (B)\frac 18 (C)\frac {3}{16} (D)\frac 14 (E)\frac {5}{16}
💡 解题思路
After $10$ minutes $(600$ seconds $),$ Rachel will have completed $6$ laps and be $30$ seconds from completing her seventh lap. Because Rachel runs one-fourth of a lap in $22.5$ seconds, she will be i
19
第 19 题
数论
For each positive integer n , let f(n) = n^4 - 360n^2 + 400 . What is the sum of all values of f(n) that are prime numbers?
💡 解题思路
To find the answer it was enough to play around with $f$ . One can easily find that $f(1)=41$ is a prime, then $f$ becomes negative for $n$ between $2$ and $18$ , and then $f(19)=761$ is again a prime
20
第 20 题
综合
A convex polyhedron Q has vertices V_1,V_2,\ldots,V_n , and 100 edges. The polyhedron is cut by planes P_1,P_2,\ldots,P_n in such a way that plane P_k cuts only those edges that meet at vertex V_k . In addition, no two planes intersect inside or on Q . The cuts produce n pyramids and a new polyhedron R . How many edges does R have? (A)\ 200 (B)\ 2n (C)\ 300 (D)\ 400 (E)\ 4n The process described in this problem exists in practice and is known as a truncation .
💡 解题思路
Each edge of $Q$ is cut by two planes, so $R$ has $200$ vertices. Three edges of $R$ meet at each vertex, so $R$ has a total of $\frac 12 \cdot 3 \cdot 200 = \boxed {300}$ edges.
21
第 21 题
计数
Ten women sit in 10 seats in a line. All of the 10 get up and then reseat themselves using all 10 seats, each sitting in the seat she was in before or a seat next to the one she occupied before. In how many ways can the women be reseated?
💡 解题思路
Notice that either a woman stays in her own seat after the rearrangement, or two adjacent women swap places. Thus, our answer is counting the number of ways to arrange 1x1 and 2x1 blocks to form a 1x1
22
第 22 题
几何·面积
Parallelogram ABCD has area 1,\!000,\!000 . Vertex A is at (0,0) and all other vertices are in the first quadrant. Vertices B and D are lattice points on the lines y = x and y = kx for some integer k > 1 , respectively. How many such parallelograms are there? (A lattice point is any point whose coordinates are both integers.)
💡 解题思路
The area of any parallelogram $ABCD$ can be computed as the size of the vector product of $\overrightarrow{AB}$ and $\overrightarrow{AD}$ .
23
第 23 题
概率
A region S in the complex plane is defined by \[S = \{x + iy: - 1\le x\le1, - 1\le y\le1\}.\] A complex number z = x + iy is chosen uniformly at random from S . What is the probability that (\frac34 + \frac34i)z is also in S ?
💡 解题思路
First, turn $\frac34 + \frac34i$ into polar form as $\frac{3\sqrt{2}}{4}e^{\frac{\pi}{4}i}$ . Restated using geometric probabilities, we are trying to find the portion of a square enlarged by a factor
24
第 24 题
函数
For how many values of x in [0,π] is \sin^{ - 1}(\sin 6x) = \cos^{ - 1}(\cos x) ? Note: The functions \sin^{ - 1} = \arcsin and \cos^{ - 1} = \arccos denote inverse trigonometric functions.
💡 解题思路
First of all, we have to agree on the range of $\sin^{-1}$ and $\cos^{-1}$ . This should have been a part of the problem statement -- but as it is missing, we will assume the most common definition: $
25
第 25 题
几何·面积
The set G is defined by the points (x,y) with integer coordinates, 3\le|x|\le7 , 3\le|y|\le7 . How many squares of side at least 6 have their four vertices in G ? [图]
💡 解题思路
We need to find a reasonably easy way to count the squares.