2007 AMC 10B — Official Competition Problems (February 2007)
📅 2007 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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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
几何·面积
Isabella's house has 3 bedrooms. Each bedroom is 12 feet long, 10 feet wide, and 8 feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy 60 square feet in each bedroom. How many square feet of walls must be painted? (A)\ 678 (B)\ 768 (C)\ 786 (D)\ 867 (E)\ 876
💡 解题思路
There are four walls in each bedroom (she can't paint floors or ceilings). Therefore, we calculate the number of square feet of walls there is in one bedroom: \[2\cdot(12\cdot8+10\cdot8)-60=2\cdot176-
2
第 2 题
分数与比例
Define the operation \star by a \star b = (a+b)b. What is (3 \star 5) - (5 \star 3)?
A college student drove his compact car 120 miles home for the weekend and averaged 30 miles per gallon. On the return trip the student drove his parents' SUV and averaged only 20 miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip?
💡 解题思路
The trip was $240$ miles long and took $\dfrac{120}{30}+\dfrac{120}{20}=4+6=10$ gallons. Therefore, the average mileage was $\dfrac{240}{10}= \boxed{\textbf{(B) }24}$
4
第 4 题
几何·面积
The point O is the center of the circle circumscribed about \triangle ABC, with \angle BOC=120^\circ and \angle AOB=140^\circ, as shown. What is the degree measure of \angle ABC?
💡 解题思路
Because all the central angles of a circle add up to $360^\circ,$
5
第 5 题
逻辑推理
In a certain land, all Arogs are Brafs, all Crups are Brafs, all Dramps are Arogs, and all Crups are Dramps. Which of the following statements is implied by these facts?
💡 解题思路
It may be easier to visualize this by drawing some sort of diagram. From the first statement, you can draw an Arog circle inside of the Braf circle, since all Arogs are Brafs, but not all Brafs are Ar
6
第 6 题
方程
The 2007 AMC 10 will be scored by awarding 6 points for each correct response, 0 points for each incorrect response, and 1.5 points for each problem left unanswered. After looking over the 25 problems, Sarah has decided to attempt the first 22 and leave only the last 3 unanswered. How many of the first 22 problems must she solve correctly in order to score at least 100 points?
💡 解题思路
Sarah is leaving $3$ questions unanswered, guaranteeing her $3 \times 1.5 = 4.5$ points. She will either get $6$ points or $0$ points for the rest of the questions. Let $x$ be the number of questions
7
第 7 题
几何·角度
All sides of the convex pentagon ABCDE are of equal length, and \angle A= \angle B = 90^\circ. What is the degree measure of \angle E?
💡 解题思路
$AB = EC$ because they are opposite sides of a square. Also, $ED = DC = AB$ because all sides of the convex pentagon are of equal length. Since $ABCE$ is a square and $\triangle CED$ is an equilateral
8
第 8 题
统计
On the trip home from the meeting where this AMC10 was constructed, the Contest Chair noted that his airport parking receipt had digits of the form bbcac, where 0 \le a < b < c \le 9, and b was the average of a and c. How many different five-digit numbers satisfy all these properties?
💡 解题思路
Case $1$ : The numbers are separated by $1$ .
9
第 9 题
坐标几何
A cryptographic code is designed as follows. The first time a letter appears in a given message it is replaced by the letter that is 1 place to its right in the alphabet (asumming that the letter A is one place to the right of the letter Z ). The second time this same letter appears in the given message, it is replaced by the letter that is 1+2 places to the right, the third time it is replaced by the letter that is 1+2+3 places to the right, and so on. For example, with this code the word "banana" becomes "cbodqg". What letter will replace the last letter s in the message \["Lee's sis is a Mississippi miss, Chriss!"?\]
💡 解题思路
Since the letter that will replace the last $s$ does not depend on any letter except the other $s$ 's, you can disregard anything else. There are $12$ $s$ 's, so the last $s$ will be replaced by the l
10
第 10 题
几何·面积
Two points B and C are in a plane. Let S be the set of all points A in the plane for which \triangle ABC has area 1. Which of the following describes S?
💡 解题思路
Let $h$ be the length of the altitude of $\triangle ABC.$ Since segment $BC$ is the base of the triangle and cannot change, the area of the triangle is $\frac{1}{2}(BC)(h)=1$ and $h=\frac{2}{BC}.$ Thu
11
第 11 题
几何·面积
A circle passes through the three vertices of an isosceles triangle that has two sides of length 3 and a base of length 2 . What is the area of this circle?
💡 解题思路
Let $\triangle ABC$ have vertex $A$ and center $O$ , with foot of altitude from $A$ intersecting $BC$ at $D$ .
12
第 12 题
规律与数列
Tom's age is T years, which is also the sum of the ages of his three children. His age N years ago was twice the sum of their ages then. What is T/N ?
💡 解题思路
Tom's age $N$ years ago was $T-N$ . The sum of the ages of his three children $N$ years ago was $T-3N,$ since there are three children. If his age $N$ years ago was twice the sum of the children's age
13
第 13 题
几何·面积
Two circles of radius 2 are centered at (2,0) and at (0,2). What is the area of the intersection of the interiors of the two circles?
💡 解题思路
You can find the area of half the intersection by subtracting the isosceles triangle in the sector from the whole sector. This sector is one-fourth of the area of the circle with radius $2,$ and the i
14
第 14 题
应用题
Some boys and girls are having a car wash to raise money for a class trip to China. Initially 40\% of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then 30\% of the group are girls. How many girls were initially in the group?
💡 解题思路
If we let $p$ be the number of people initially in the group, then $0.4p$ is the number of girls. If two girls leave and two boys arrive, the number of people in the group is still $p$ , but the numbe
15
第 15 题
几何·角度
The angles of quadrilateral ABCD satisfy \angle A=2 \angle B=3 \angle C=4 \angle D. What is the degree measure of \angle A, rounded to the nearest whole number?
💡 解题思路
The sum of the interior angles of any quadrilateral is $360^\circ.$ \begin{align*} 360 &= \angle A + \angle B + \angle C + \angle D\\ &= \angle A + \frac{1}{2}A + \frac{1}{3}A + \frac{1}{4}A\\ &= \fra
16
第 16 题
统计
A teacher gave a test to a class in which 10\% of the students are juniors and 90\% are seniors. The average score on the test was 84. The juniors all received the same score, and the average score of the seniors was 83. What score did each of the juniors receive on the test?
💡 解题思路
We can assume there are $10$ people in the class. Then there will be $1$ junior and $9$ seniors. The sum of everyone's scores is $10 \cdot 84 = 840$ . Since the average score of the seniors was $83$ ,
17
第 17 题
几何·面积
Point P is inside equilateral \triangle ABC . Points Q , R , and S are the feet of the perpendiculars from P to \overline{AB} , \overline{BC} , and \overline{CA} , respectively. Given that PQ=1 , PR=2 , and PS=3 , what is AB ? (A)\ 4 (B)\ 3√(3) (C)\ 6 (D)\ 4√(3) (E)\ 9
💡 解题思路
Drawing $\overline{PA}$ , $\overline{PB}$ , and $\overline{PC}$ , $\triangle ABC$ is split into three smaller triangles. The altitudes of these triangles are given in the problem as $PQ$ , $PR$ , and
18
第 18 题
几何·面积
A circle of radius 1 is surrounded by 4 circles of radius r as shown. What is r ?
💡 解题思路
You can express the line connecting the centers of an outer circle and the inner circle in two different ways. You can add the radius of both circles to get $r+1.$ You can also add the radius of two o
19
第 19 题
几何·面积
The wheel shown is spun twice, and the randomly determined numbers opposite the pointer are recorded. The first number is divided by 4, and the second number is divided by 5. The first remainder designates a column, and the second remainder designates a row on the checkerboard shown. What is the probability that the pair of numbers designates a shaded square?
💡 解题思路
When dividing each number on the wheel by $4,$ the remainders are $1, 1, 2, 2, 3,$ and $3.$ Each column on the checkerboard is equally likely to be chosen.
20
第 20 题
几何·面积
A set of 25 square blocks is arranged into a 5 × 5 square. How many different combinations of 3 blocks can be selected from that set so that no two are in the same row or column?
💡 解题思路
There are $25$ ways to choose the first square. The four remaining squares in its row and column and the square you chose exclude nine squares from being chosen next time.
21
第 21 题
几何·面积
Right \triangle ABC has AB=3, BC=4, and AC=5. Square XYZW is inscribed in \triangle ABC with X and Y on \overline{AC}, W on \overline{AB}, and Z on \overline{BC}. What is the side length of the square?
💡 解题思路
There are lots of similar triangles in the diagram, but we will only use $\triangle WBZ \sim \triangle ABC.$ If $h$ is the altitude from $B$ to $AC$ and $s$ is the sidelength of the square, then $h-s$
22
第 22 题
概率
A player chooses one of the numbers 1 through 4 . After the choice has been made, two regular four-sided (tetrahedral) dice are rolled, with the sides of the dice numbered 1 through 4. If the number chosen appears on the bottom of exactly one die after it has been rolled, then the player wins 1 dollar. If the number chosen appears on the bottom of both of the dice, then the player wins 2 dollars. If the number chosen does not appear on the bottom of either of the dice, the player loses 1 dollar. What is the expected return to the player, in dollars, for one roll of the dice?
💡 解题思路
There are $2 \cdot 3 \cdot 1 = 6$ ways for your number to show up once, $1 \cdot 1 = 1$ way for your number to show up twice, and $3 \cdot 3 = 9$ ways for your number to not show up at all. Think of t
23
第 23 题
几何·面积
A pyramid with a square base is cut by a plane that is parallel to its base and 2 units from the base. The surface area of the smaller pyramid that is cut from the top is half the surface area of the original pyramid. What is the altitude of the original pyramid?
💡 解题思路
Since the two pyramids are similar, the ratio of the altitudes is the square root of the ratio of the surface areas.
24
第 24 题
数论
Let n denote the smallest positive integer that is divisible by both 4 and 9, and whose base- 10 representation consists of only 4 's and 9 's, with at least one of each. What are the last four digits of n?
💡 解题思路
For a number to be divisible by $4,$ the last two digits have to be divisible by $4.$ That means the last two digits of this integer must be $4.$
25
第 25 题
数论
How many pairs of positive integers (a,b) are there such that a and b have no common factors greater than 1 and: \[\frac{a}{b} + \frac{14b}{9a}\] is an integer?
💡 解题思路
Combining the fraction, $\frac{9a^2 + 14b^2}{9ab}$ must be an integer.