2007 AMC 10A — Official Competition Problems (February 2007)
📅 2007 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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第 1 题
计数
One ticket to a show costs \20 at full price. Susan buys 4 tickets using a coupon that gives her a 25% discount. Pam buys 5 tickets using a coupon that gives her a 30% discount. How many more dollars does Pam pay than Susan? (A)\ 2 (B)\ 5 (C)\ 10 (D)\ 15 (E)\ 20Answer: (C) Susan pays (4)(0.75)(20) = 60 dollars. Pam pays (5)(0.70)(20) = 70 dollars, so she pays 70-60=10 more dollars than Susan.
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第 2 题
综合
Define a@b = ab - b^{2} and a\#b = a + b - ab^{2} . What is \frac {6@2}{6\#2} ? (A)\ - \frac {1}{2} (B)\ - \frac {1}{4} (C)\ \frac {1}{8} (D)\ \frac {1}{4} (E)\ \frac {1}{2}
💡 解题思路
$6@2$ must be equal to $6*2-2^2$ which is 8. $6\# 2$ is equal to $6+2-6*2^2$ which is $8-24 = -16$ . Therefore $\frac{6@2}{6\# 2}$ must be equal to $\frac{8}{-16} = -\frac{1}{2}$ . Therefore the solut
3
第 3 题
行程问题
An aquarium has a rectangular base that measures 100 cm by 40 cm and has a height of 50 cm. It is filled with water to a height of 40 cm. A brick with a rectangular base that measures 40 cm by 20 cm and a height of 10 cm is placed in the aquarium. By how many centimeters does the water rise? (A)\ 0.5 (B)\ 1 (C)\ 1.5 (D)\ 2 (E)\ 2.5
💡 解题思路
The volume of the brick is $40 \times 20 \times 10 = 8000$ . Thus the water volume rose $8000 = 100 \times 40 \times h \Longrightarrow h = 2\ \mathrm{(D)}$ .
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第 4 题
规律与数列
The larger of two consecutive odd integers is three times the smaller. What is their sum? (A)\ 4 (B)\ 8 (C)\ 12 (D)\ 16 (E)\ 20
💡 解题思路
Let the two consecutive odd integers be $a$ , $a+2$ . Then $a+2 = 3a$ , so $a = 1, a + 2 = 3$ and their sum is $4\ \mathrm{(A)}$ .
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第 5 题
应用题
The school store sells 7 pencils and 8 notebooks for \mathdollar 4.15 . It also sells 5 pencils and 3 notebooks for \mathdollar 1.77 . How much do 16 pencils and 10 notebooks cost? (A)\mathdollar 1.76 (B)\mathdollar 5.84 (C)\mathdollar 6.00 (D)\mathdollar 6.16 (E)\mathdollar 6.32
💡 解题思路
Let $p$ be cost of one pencil in dollars and $n$ the cost of one notebook in dollars. Then
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第 6 题
分数与比例
At Euclid High School, the number of students taking the AMC 10 was 60 in 2002, 66 in 2003, 70 in 2004, 76 in 2005, 78 and 2006, and is 85 in 2007. Between what two consecutive years was there the largest percentage increase? (A)\ 2002\ and\ 2003 (B)\ 2003\ and\ 2004 (C)\ 2004\ and\ 2005 (D)\ 2005\ and\ 2006 (E)\ 2006\ and\ 2007
💡 解题思路
We compute the percentage increases:
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第 7 题
应用题
Last year Mr. Jon Q. Public received an inheritance. He paid 20\% in federal taxes on the inheritance, and paid 10\% of what he had left in state taxes. He paid a total of \textdollar10500 for both taxes. How many dollars was his inheritance? (\mathrm {A})\ 30000 (\mathrm {B})\ 32500 (\mathrm {C})\ 35000 (\mathrm {D})\ 37500 (\mathrm {E})\ 40000
💡 解题思路
After paying his taxes, he has $0.8*0.9=0.72$ of his earnings left. Since $10500$ is $0.28$ of his income, he got a total of $\frac{10500}{0.28}=37500\ \mathrm{(D)}$ .
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第 8 题
几何·面积
Triangles ABC and ADC are isosceles with AB=BC and AD=DC . Point D is inside triangle ABC , angle ABC measures 40 degrees, and angle ADC measures 140 degrees. What is the degree measure of angle BAD ? (A)\ 20 (B)\ 30 (C)\ 40 (D)\ 50 (E)\ 60
💡 解题思路
We angle chase and find out that:
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第 9 题
方程
Real numbers a and b satisfy the equations 3^{a} = 81^{b + 2} and 125^{b} = 5^{a - 3} . What is ab ? (A)\ -60 (B)\ -17 (C)\ 9 (D)\ 12 (E)\ 60
💡 解题思路
\[81^{b+2} = 3^{4(b+2)} = 3^a \Longrightarrow a = 4b+8\]
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第 10 题
统计
The Dunbar family consists of a mother, a father, and some children. The average age of the members of the family is 20 , the father is 48 years old, and the average age of the mother and children is 16 . How many children are in the family? (A)\ 2 (B)\ 3 (C)\ 4 (D)\ 5 (E)\ 6
💡 解题思路
Let $n$ be the number of children. Then the total ages of the family is $48 + 16(n+1)$ , and the total number of people in the family is $n+2$ . So
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第 11 题
规律与数列
The numbers from 1 to 8 are placed at the vertices of a cube in such a manner that the sum of the four numbers on each face is the same. What is this common sum? (A)\ 14 (B)\ 16 (C)\ 18 (D)\ 20 (E)\ 24
💡 解题思路
The sum of the numbers on one face of the cube is equal to the sum of the numbers on the opposite face of the cube; these $8$ numbers represent all of the vertices of the cube. Thus the answer is $\fr
12
第 12 题
计数
Two tour guides are leading six tourists. The guides decide to split up. Each tourist must choose one of the guides, but with the stipulation that each guide must take at least one tourist. How many different groupings of guides and tourists are possible? (A)\ 56 (B)\ 58 (C)\ 60 (D)\ 62 (E)\ 64 https://youtu.be/0W3VmFp55cM?t=3352 ~ pi_is_3.14
💡 解题思路
Each tourist has to pick in between the $2$ guides, so for $6$ tourists there are $2^6$ possible groupings. However, since each guide must take at least one tourist, we subtract the $2$ cases where a
13
第 13 题
分数与比例
The trip from Carville to Nikpath requires 4\frac 12 hours when traveling at an average speed of 70 miles per hour. How many hours does the trip require when traveling at an average speed of 60 miles per hour? Express your answer as a decimal to the nearest hundredth. Let x represent the distance from home to the stadium, and let r represent the distance from Yan to home. Our goal is to find \frac{r}{x-r} . If Yan walks directly to the stadium, then assuming he walks at a rate of 1 , it will take him x-r units of time. Similarly, if he walks back home it will take him r + \frac{x}{7} units of time. Because the two times are equal, we can create the following equation: x-r = r + \frac{x}{7} . We get x-2r=\frac{x}{7} , so \frac{6}{7}x = 2r , and \frac{x}{r} = \frac{7}{3} . This minus one is the reciprocal of what we want to find: \frac{7}{3}-1 = \frac{4}{3} , so the answer is [(B)\ \frac{3]{4}}
💡 解题思路
Let the distance from Yan's initial position to the stadium be $a$ and the distance from Yan's initial position to home be $b$ . We are trying to find $b/a$ , and we have the following identity given
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第 14 题
几何·面积
A triangle with side lengths in the ratio 3 : 4 : 5 is inscribed in a circle with radius 3. What is the area of the triangle? (A)\ 8.64 (B)\ 12 (C)\ 5π (D)\ 17.28 (E)\ 18
💡 解题思路
Since 3-4-5 is a Pythagorean triple , the triangle is a right triangle . Since the hypotenuse is a diameter of the circumcircle , the hypotenuse is $2r = 6$ . Then the other legs are $\frac{24}5=4.8$
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第 15 题
几何·面积
Four circles of radius 1 are each tangent to two sides of a square and externally tangent to a circle of radius 2 , as shown. What is the area of the square? (A)\ 32 (B)\ 22 + 12\sqrt {2} (C)\ 16 + 16\sqrt {3} (D)\ 48 (E)\ 36 + 16\sqrt {2}
💡 解题思路
Draw a square connecting the centers of the four small circles of radius $1$ . This square has a diagonal of length $6$ , as it includes the diameter of the big circle of radius $2$ and two radii of t
16
第 16 题
概率
Integers a, b, c, and d , not necessarily distinct, are chosen independently and at random from 0 to 2007, inclusive. What is the probability that ad-bc is even ? (A)\ \frac 38 (B)\ \frac 7{16} (C)\ \frac 12 (D)\ \frac 9{16} (E)\ \frac 58
💡 解题思路
The only time when $ad-bc$ is even is when $ad$ and $bc$ are of the same parity . The chance of $ad$ being odd is $\frac 12 \cdot \frac 12 = \frac 14$ , since the only way to have $ad$ be odd is to ha
17
第 17 题
整数运算
Suppose that m and n are positive integers such that 75m = n^{3} . What is the minimum possible value of m + n ? (A)\ 15 (B)\ 30 (C)\ 50 (D)\ 60 (E)\ 5700
💡 解题思路
$3 \cdot 5^2m$ must be a perfect cube, so each power of a prime in the factorization for $3 \cdot 5^2m$ must be divisible by $3$ . Thus the minimum value of $m$ is $3^2 \cdot 5 = 45$ , which makes $n
18
第 18 题
几何·面积
Consider the 12 -sided polygon ABCDEFGHIJKL , as shown. Each of its sides has length 4 , and each two consecutive sides form a right angle. Suppose that \overline{AG} and \overline{CH} meet at M . What is the area of quadrilateral ABCM ? [图] (A)\ \frac {44}{3} (B)\ 16 (C)\ \frac {88}{5} (D)\ 20 (E)\ \frac {62}{3}
💡 解题思路
We can obtain the solution by calculating the area of rectangle $ABGH$ minus the combined area of triangles $\triangle AHG$ and $\triangle CGM$ .
19
第 19 题
几何·面积
A paint brush is swept along both diagonals of a square to produce the symmetric painted area, as shown. Half the area of the square is painted. What is the ratio of the side length of the square to the brush width? [图] (A)\ 2\sqrt {2} + 1 (B)\ 3\sqrt {2} (C)\ 2\sqrt {2} + 2 (D)\ 3\sqrt {2} + 1 (E)\ 3\sqrt {2} + 2
💡 解题思路
Without loss of generality , let the side length of the square be $1$ unit. The area of the painted area is $\frac{1}2$ of the area of the larger square, so the total unpainted area is also $\frac{1}{
20
第 20 题
方程
Suppose that the number a satisfies the equation 4 = a + a^{ - 1} . What is the value of a^{4} + a^{ - 4} ?
💡 解题思路
Note that for all real numbers $k,$ we have $a^{2k} + a^{-2k} + 2 = (a^{k} + a^{-k})^2,$ from which \[a^{2k} + a^{-2k} = (a^{k} + a^{-k})^2-2.\] We apply this result twice to get the answer: \begin{al
21
第 21 题
几何·面积
A sphere is inscribed in a cube that has a surface area of 24 square meters. A second cube is then inscribed within the sphere. What is the surface area in square meters of the inner cube? (A)\ 3 (B)\ 6 (C)\ 8 (D)\ 9 (E)\ 12
💡 解题思路
We rotate the smaller cube around the sphere such that two opposite vertices of the cube are on opposite faces of the larger cube. Thus the main diagonal of the smaller cube is the side length of the
22
第 22 题
数论
A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with the terms 247, 475, and 756 and end with the term 824. Let S be the sum of all the terms in the sequence. What is the largest prime factor that always divides S ? (A)\ 3 (B)\ 7 (C)\ 13 (D)\ 37 (E)\ 43
💡 解题思路
A given digit appears as the hundreds digit, the tens digit, and the units digit of a term the same number of times. Let $k$ be the sum of the units digits in all the terms. Then $S=111k=3 \cdot 37k$
23
第 23 题
几何·面积
How many ordered pairs (m,n) of positive integers , with m \ge n , have the property that their squares differ by 96 ? (A)\ 3 (B)\ 4 (C)\ 6 (D)\ 9 (E)\ 12
💡 解题思路
\[m^2 - n^2 = (m+n)(m-n) = 96 = 2^{5} \cdot 3\]
24
第 24 题
几何·面积
Circles centered at A and B each have radius 2 , as shown. Point O is the midpoint of \overline{AB} , and OA = 2\sqrt {2} . Segments OC and OD are tangent to the circles centered at A and B , respectively, and EF is a common tangent . What is the area of the shaded region ECODF ? [图] (A)\ \frac {8\sqrt {2}}{3} (B)\ 8\sqrt {2} - 4 - π (C)\ 4\sqrt {2} (D)\ 4\sqrt {2} + \frac {π}{8} (E)\ 8\sqrt {2} - 2 - \frac {π}{2}
💡 解题思路
The area we are trying to find is simply $ABFE-(\overarc{AEC}+\triangle{ACO}+\triangle{BDO}+\overarc{BFD}).\overline{EF}\parallel\overline{AB}$ . Thus, $ABFE$ is a rectangle , and so its area is $b\ti
25
第 25 题
规律与数列
For each positive integer n , let S(n) denote the sum of the digits of n. For how many values of n is n + S(n) + S(S(n)) = 2007?(A)\ 1 (B)\ 2 (C)\ 3 (D)\ 4 (E)\ 5
💡 解题思路
For the sake of notation, let $T(n) = n + S(n) + S(S(n))$ . Obviously $n<2007$ . Then the maximum value of $S(n) + S(S(n))$ is when $n = 1999$ , and the sum becomes $28 + 10 = 38$ . So the minimum bou