2007A 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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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
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 brick has volume $8000 cm^3$ . The base of the aquarium has area $4000 cm^2$ . For every cm the water rises, the volume increases by $4000 cm^3$ ; therefore, when the volume increases by $8000 cm^
3
第 3 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 $n$ be the smaller term. Then $n+2=3n \Longrightarrow 2n = 2 \Longrightarrow n=1$
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第 4 题
统计
Kate rode her bicycle for 30 minutes at a speed of 16 mph, then walked for 90 minutes at a speed of 4 mph. What was her overall average speed in miles per hour? (A)\ 7 (B)\ 9 (C)\ 10 (D)\ 12 (E)\ 14
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 6 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 7 题
规律与数列
Let a, b, c, d , and e be five consecutive terms in an arithmetic sequence, and suppose that a+b+c+d+e=30 . Which of a, b, c, d, or e can be found? \textrm{(A)} \ a \textrm{(B)}\ b \textrm{(C)}\ c \textrm{(D)}\ d \textrm{(E)}\ e
💡 解题思路
Let $f$ be the common difference between the terms.
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第 8 题
几何·面积
A star- polygon is drawn on a clock face by drawing a chord from each number to the fifth number counted clockwise from that number. That is, chords are drawn from 12 to 5, from 5 to 10, from 10 to 3, and so on, ending back at 12. What is the degree measure of the angle at each vertex in the star polygon? (A)\ 20 (B)\ 24 (C)\ 30 (D)\ 36 (E)\ 60
💡 解题思路
We look at the angle between 12, 5, and 10. It subtends $\frac 16$ of the circle, or $60$ degrees (or you can see that the arc is $\frac 23$ of the right angle ). Thus, the angle at each vertex is an
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第 9 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 10 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 11 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 $S=111k=3 \cdot 37k$
12
第 12 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
13
第 13 题
坐标几何
A piece of cheese is located at (12,10) in a coordinate plane . A mouse is at (4,-2) and is running up the line y=-5x+18 . At the point (a,b) the mouse starts getting farther from the cheese rather than closer to it. What is a+b ? (A)\ 6 (B)\ 10 (C)\ 14 (D)\ 18 (E)\ 22
💡 解题思路
The point $(a,b)$ is the foot of the perpendicular from $(12,10)$ to the line $y=-5x+18$ . The perpendicular has slope $\frac{1}{5}$ , so its equation is $y=10+\frac{1}{5}(x-12)=\frac{1}{5}x+\frac{38}
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第 14 题
整数运算
Let a , b , c , d , and e be distinct integers such that (6-a)(6-b)(6-c)(6-d)(6-e)=45 What is a+b+c+d+e ? (A)\ 5 (B)\ 17 (C)\ 25 (D)\ 27 (E)\ 30
💡 解题思路
If $45$ is expressed as a product of five distinct integer factors, the absolute value of the product of any four is at least $|(-3)(-1)(1)(3)|=9$ , so no factor can have an absolute value greater tha
15
第 15 题
统计
The set \{3,6,9,10\} is augmented by a fifth element n , not equal to any of the other four. The median of the resulting set is equal to its mean . What is the sum of all possible values of n ? (A)\ 7 (B)\ 9 (C)\ 19 (D)\ 24 (E)\ 26
💡 解题思路
The median must either be $6, 9,$ or $n$ . Casework :
16
第 16 题
统计
How many three-digit numbers are composed of three distinct digits such that one digit is the average of the other two? (A)\ 96 (B)\ 104 (C)\ 112 (D)\ 120 (E)\ 256
💡 解题思路
We can find the number of increasing arithmetic sequences of length 3 possible from 0 to 9, and then find all the possible permutations of these sequences.
17
第 17 题
综合
Suppose that \sin a + \sin b = √(\frac{5){3}} and \cos a + \cos b = 1 . What is \cos (a - b) ? (A)\ √(\frac{5){3}} - 1 (B)\ \frac 13 (C)\ \frac 12 (D)\ \frac 23 (E)\ 1
💡 解题思路
We can make use the of the trigonometric Pythagorean identities : square both equations and add them up:
18
第 18 题
函数
The polynomial f(x) = x^{4} + ax^{3} + bx^{2} + cx + d has real coefficients, and f(2i) = f(2 + i) = 0. What is a + b + c + d?(A)\ 0 (B)\ 1 (C)\ 4 (D)\ 9 (E)\ 16
💡 解题思路
A fourth degree polynomial has four roots . Since the coefficients are real(meaning that complex roots come in conjugate pairs), the remaining two roots must be the complex conjugates of the two given
19
第 19 题
几何·面积
Triangles ABC and ADE have areas 2007 and 7002, respectively, with B = (0,0),C = (223,0),D = (680,380), and E = (689,389). What is the sum of all possible x coordinates of A ? (A)\ 282 (B)\ 300 (C)\ 600 (D)\ 900 (E)\ 1200
💡 解题思路
From $k = [ABC] = \frac 12bh$ , we have that the height of $\triangle ABC$ is $h = \frac{2k}{b} = \frac{2007 \cdot 2}{223} = 18$ . Thus $A$ lies on the lines $y = \pm 18 \quad \mathrm{(1)}$ .
20
第 20 题
立体几何
Corners are sliced off a unit cube so that the six faces each become regular octagons. What is the total volume of the removed tetrahedra? (A)\ \frac{5√(2)-7}{3} (B)\ \frac{10-7√(2)}{3} (C)\ \frac{3-2√(2)}{3} (D)\ \frac{8√(2)-11}{3} (E)\ \frac{6-4√(2)}{3}
💡 解题思路
Since the sides of a regular polygon are equal in length, we can call each side $x$ . Examine one edge of the unit cube: each contains two slanted diagonal edges of an octagon and one straight edge. T
21
第 21 题
坐标几何
The sum of the zeros , the product of the zeros, and the sum of the coefficients of the function f(x)=ax^{2}+bx+c are equal. Their common value must also be which of the following? \textrm{(A)}\ \textrm{the\ coefficient\ of\ }x^{2}~~~ \textrm{(B)}\ \textrm{the\ coefficient\ of\ }x\textrm{(C)}\ \textrm{the\ y-intercept\ of\ the\ graph\ of\ }y=f(x)\textrm{(D)}\ \textrm{one\ of\ the\ x-intercepts\ of\ the\ graph\ of\ }y=f(x)\textrm{(E)}\ \textrm{the\ mean\ of\ the\ x-intercepts\ of\ the\ graph\ of\ }y=f(x)
💡 解题思路
By Vieta's Formulas , the sum of the roots of a quadratic equation is $\frac {-b}a$ , the product of the zeros is $\frac ca$ , and the sum of the coefficients is $a + b + c$ . Setting equal the first
22
第 22 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
23
第 23 题
几何·面积
Square ABCD has area 36, and \overline{AB} is parallel to the x-axis . Vertices A,B , and C are on the graphs of y = \log_{a}x,y = 2\log_{a}x, and y = 3\log_{a}x, respectively. What is a?(A)\ \sqrt [6]{3} (B)\ \sqrt {3} (C)\ \sqrt [3]{6} (D)\ \sqrt {6} (E)\ 6
💡 解题思路
Let $x$ be the x-coordinate of $B$ and $C$ , and $x_2$ be the x-coordinate of $A$ and $y$ be the y-coordinate of $A$ and $B$ . Then $2\log_ax= y \Longrightarrow a^{y/2} = x$ and $\log_ax_2 = y \Longri
24
第 24 题
方程
For each integer n>1 , let F(n) be the number of solutions to the equation \sin{x}=\sin{(nx)} on the interval [0,π] . What is \sum_{n=2}^{2007} F(n) ? (A)\ 2014524 (B)\ 2015028 (C)\ 2015033 (D)\ 2016532 (E)\ 2017033
💡 解题思路
By looking at various graphs, we obtain that, for most of the graphs
25
第 25 题
整数运算
Call a set of integers spacy if it contains no more than one out of any three consecutive integers. How many subsets of \{1,2,3,\ldots,12\}, including the empty set , are spacy? (A)\ 121 (B)\ 123 (C)\ 125 (D)\ 127 (E)\ 129
💡 解题思路
Let $S_{n}$ denote the number of spacy subsets of $\{ 1, 2, ... n \}$ . We have $S_{0} = 1, S_{1} = 2, S_{2} = 3$ .