2012 AMC 10B — Official Competition Problems (February 2012)
📅 2012 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 third-grade classroom at Pearl Creek Elementary has 18 students and 2 pet rabbits. How many more students than rabbits are there in all 4 of the third-grade classrooms?
💡 解题思路
In each class, there are $18-2=16$ more students than rabbits. So for all classrooms, the difference between students and rabbits is $16 \times 4 = \boxed{\textbf{(C)}\ 64}$
2
第 2 题
几何·面积
A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle? [图]
💡 解题思路
Note that the diameter of the circle is equal to the shorter side of the rectangle. Since the radius is $5$ , the diameter is $2\cdot 5 = 10$ . Since the sides of the rectangle are in a $2:1$ ratio, t
3
第 3 题
坐标几何
The point in the xy -plane with coordinates (1000, 2012) is reflected across the line y=2000 . What are the coordinates of the reflected point?
💡 解题思路
The line $y = 2000$ is a horizontal line located $12$ units beneath the point $(1000, 2012)$ . When a point is reflected about a horizontal line, only the $y$ - coordinate will change. The $x$ - coord
4
第 4 题
综合
When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same with his marbles, he has 3 marbles left over. Ringo and Paul pool their marbles and place them into as many bags as possible, with 6 marbles per bag. How many marbles will be leftover?
💡 解题思路
In total, there were $3+4=7$ marbles left from both Ringo and Paul.We know that $7 \equiv 1 \pmod{6}$ . This means that there would be $1$ marble leftover, or $\boxed{A}$ .
5
第 5 题
应用题
Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is 10%. She leaves a 15% tip on the price of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of 27.50 dollars for dinner. What is the cost of her dinner without tax or tip in dollars?
💡 解题思路
Let $x$ be the cost of her dinner.
6
第 6 题
逻辑推理
In order to estimate the value of x-y where x and y are real numbers with x > y > 0 , Xiaoli rounded x up by a small amount, rounded y down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct?
💡 解题思路
Let's define $z$ as the amount rounded up by and down by.
7
第 7 题
综合
For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide?
💡 解题思路
Let $x$ be the number of acorns that both animals had.
8
第 8 题
规律与数列
What is the sum of all integer solutions to 1<(x-2)^2<25 ?
💡 解题思路
$(x-2)^2$ = perfect square.
9
第 9 题
规律与数列
Two integers have a sum of 26. When two more integers are added to the first two integers the sum is 41. Finally when two more integers are added to the sum of the previous four integers the sum is 57. What is the minimum number of odd integers among the 6 integers?
💡 解题思路
Out of the first two integers, it's possible for both to be even: for example, $10 + 16 = 26.$ But the next two integers, when added, increase the sum by $15,$ which is odd, so one of them must be odd
10
第 10 题
方程
How many ordered pairs of positive integers (M,N) satisfy the equation \frac{M}{6}=\frac{6}{N}?
💡 解题思路
Cross-multiplying gives $MN=36.$ We write $36$ as a product of two positive integers: \begin{align*} 36 &= 1\cdot36 \\ &= 2\cdot18 \\ &= 3\cdot12 \\ &= 4\cdot9 \\ &= 6\cdot6. \end{align*} The products
11
第 11 题
综合
A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible?
💡 解题思路
Desserts must be chosen for $7$ days: Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday.
12
第 12 题
行程问题
Point B is due east of point A . Point C is due north of point B . The distance between points A and C is 10\sqrt 2 , and \angle BAC = 45^\circ . Point D is 20 meters due north of point C . The distance AD is between which two integers?
It takes Clea 60 seconds to walk down an escalator when it is not operating, and only 24 seconds to walk down the escalator when it is operating. How many seconds does it take Clea to ride down the operating escalator when she just stands on it?
💡 解题思路
Let $s$ be the speed of the escalator and $c$ be the speed of Clea. Using $d = v t$ , the first statement can be translated to the equation $d = 60c$ . The second statement can be translated to $d = 2
14
第 14 题
几何·面积
Two equilateral triangles are contained in square whose side length is 2\sqrt 3 . The bases of these triangles are the opposite side of the square, and their intersection is a rhombus. What is the area of the rhombus? (A) \frac{3}{2} (B) \sqrt 3 (C) 2\sqrt 2 - 1 (D) 8\sqrt 3 - 12 (E) \frac{4\sqrt 3}{3}
💡 解题思路
Observe that the rhombus is made up of two congruent equilateral triangles with side length equal to GF. Since AE has length $\sqrt{3}$ and triangle AEF is a 30-60-90 triangle, it follows that EF has
15
第 15 题
综合
In a round-robin tournament with 6 teams, each team plays one game against each other team, and each game results in one team winning and one team losing. At the end of the tournament, the teams are ranked by the number of games won. What is the maximum number of teams that could be tied for the most wins at the end of the tournament?
💡 解题思路
The total number of games (and wins) in the tournament is $\frac{6 \times 5}{2}= 15$ . A six-way tie is impossible as this would imply each team has 2.5 wins, so the maximum number of tied teams is fi
16
第 16 题
几何·面积
Three circles with radius 2 are mutually tangent. What is the total area of the circles and the region bounded by them, as shown in the figure? [图]
💡 解题思路
To determine the area of the figure, you can connect the centers of the circles to form an equilateral triangle with a side of length $4$ . We must find the area of this triangle to include the figure
17
第 17 题
分数与比例
Jesse cuts a circular paper disk of radius 12 along two radii to form two sectors, the smaller having a central angle of 120 degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller cone to that of the larger? (A) \frac{1}{8} (B) \frac{1}{4} (C) \frac{√(10)}{10} (D) \frac{√(5)}{6} (E) \frac{√(5)}{5}
💡 解题思路
Let's find the volume of the smaller cone first. We know that the circumference of the paper disk is $24\pi$ , so the circumference of the smaller cone would be $\dfrac{120}{360} \times 24\pi = 8\pi$
18
第 18 题
概率
Suppose that one of every 500 people in a certain population has a particular disease, which displays no symptoms. A blood test is available for screening for this disease. For a person who has this disease, the test always turns out positive. For a person who does not have the disease, however, there is a 2\% false positive rate--in other words, for such people, 98\% of the time the test will turn out negative, but 2\% of the time the test will turn out positive and will incorrectly indicate that the person has the disease. Let p be the probability that a person who is chosen at random from this population and gets a positive test result actually has the disease. Which of the following is closest to p ?
💡 解题思路
This question can be solved by considering all the possibilities:
19
第 19 题
几何·面积
In rectangle ABCD , AB=6 , AD=30 , and G is the midpoint of \overline{AD} . Segment AB is extended 2 units beyond B to point E , and F is the intersection of \overline{ED} and \overline{BC} . What is the area of quadrilateral BFDG ?
Bernardo and Silvia play the following game. An integer between 0 and 999 inclusive is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds 50 to it and passes the result to Bernardo. The winner is the last person who produces a number less than 1000 . Let N be the smallest initial number that results in a win for Bernardo. What is the sum of the digits of N ?
💡 解题思路
The last number that Bernardo says has to be between 950 and 999. Note that $1\rightarrow 2\rightarrow 52\rightarrow 104\rightarrow 154\rightarrow 308\rightarrow 358\rightarrow 716\rightarrow 766$ con
21
第 21 题
分数与比例
Four distinct points are arranged on a plane so that the segments connecting them have lengths a , a , a , a , 2a , and b . What is the ratio of b to a ?
💡 解题思路
When you see that there are lengths a and 2a, one could think of 30-60-90 triangles. Since all of the other's lengths are a, you could think that $b=\sqrt{3}a$ . Drawing the points out, it is possible
22
第 22 题
整数运算
Let ( a_1 , a_2 , ... a_{10} ) be a list of the first 10 positive integers such that for each 2\lei\le10 either a_i + 1 or a_i-1 or both appear somewhere before a_i in the list. How many such lists are there?
💡 解题思路
If we have 1 as the first number, then the only possible list is $(1,2,3,4,5,6,7,8,9,10)$ .
23
第 23 题
立体几何
A solid tetrahedron is sliced off a solid wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object?
💡 解题思路
This tetrahedron has the 4 vertices in these positions: on a corner (let's call this $A$ ) of the cube, and the other three corners ( $B$ , $C$ , and $D$ ) adjacent to this corner. We can find the hei
24
第 24 题
计数
Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those two girls but disliked by the third. In how many different ways is this possible?
💡 解题思路
Let the ordered triple $(a,b,c)$ denote that $a$ songs are liked by Amy and Beth, $b$ songs by Beth and Jo, and $c$ songs by Jo and Amy. The only possible triples are $(1,1,1), (2,1,1), (1,2,1)(1,1,2)
25
第 25 题
行程问题
A bug travels from A to B along the segments in the hexagonal lattice pictured below. The segments marked with an arrow can be traveled only in the direction of the arrow, and the bug never travels the same segment more than once. How many different paths are there? [图]