2002B 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
统计
The arithmetic mean of the nine numbers in the set \{9, 99, 999, 9999, \ldots, 999999999\} is a 9 -digit number M , all of whose digits are distinct. The number M doesn't contain the digit (A)\ 0 (B)\ 2 (C)\ 4 (D)\ 6 (E)\ 8
💡 解题思路
We wish to find $\frac{9+99+\cdots +999999999}{9}$ , or $\frac{9(1+11+111+\cdots +111111111)}{9}=123456789$ . This doesn't have the digit 0, so the answer is $\boxed{\mathrm{(A)}\ 0}$
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第 2 题
方程
What is the value of (3x - 2)(4x + 1) - (3x - 2)4x + 1 when x=4 ? (A)\ 0 (B)\ 1 (C)\ 10 (D)\ 11 (E)\ 12
💡 解题思路
By the distributive property,
3
第 3 题
数论
For how many positive integers n is n^2 - 3n + 2 a prime number? (A)\ none (B)\ one (C)\ two (D)\ more\ than\ two,\ but\ finitely\ many (E)\ infinitely\ many
💡 解题思路
Factoring, we get $n^2 - 3n + 2 = (n-2)(n-1)$ . Either $n-1$ or $n-2$ is odd, and the other is even. Their product must yield an even number. The only prime that is even is $2$ , which is when $n$ is
4
第 4 题
逻辑推理
Let n be a positive integer such that \frac 12 + \frac 13 + \frac 17 + \frac 1n is an integer. Which of the following statements is not true: (A)\ 2\ divides\ n (B)\ 3\ divides\ n (C)\ 6\ divides\ n (D)\ 7\ divides\ n (E)\ n > 84
Let v, w, x, y, and z be the degree measures of the five angles of a pentagon . Suppose that v < w < x < y < z and v, w, x, y, and z form an arithmetic sequence . Find the value of x . (A)\ 72 (B)\ 84 (C)\ 90 (D)\ 108 (E)\ 120
💡 解题思路
The sum of the degree measures of the angles of a pentagon (as a pentagon can be split into $5- 2 = 3$ triangles) is $3 \cdot 180 = 540^{\circ}$ . If we let $v = x - 2d, w = x - d, y = x + d, z = x+2d
6
第 6 题
方程
Suppose that a and b are nonzero real numbers, and that the equation x^2 + ax + b = 0 has solutions a and b . Then the pair (a,b) is (A)\ (-2,1) (B)\ (-1,2) (C)\ (1,-2) (D)\ (2,-1) (E)\ (4,4)
💡 解题思路
Since $(x-a)(x-b) = x^2 - (a+b)x + ab = x^2 + ax + b = 0$ , it follows by comparing coefficients that $-a - b = a$ and that $ab = b$ . Since $b$ is nonzero, $a = 1$ , and $-1 - b = 1 \Longrightarrow b
7
第 7 题
几何·面积
The product of three consecutive positive integers is 8 times their sum. What is the sum of their squares ? (A)\ 50 (B)\ 77 (C)\ 110 (D)\ 149 (E)\ 194
💡 解题思路
Let the three consecutive positive integers be $a-1$ , $a$ , and $a+1$ . Since the mean is $a$ , the sum of the integers is $3a$ . So $8$ times the sum is just $24a$ . With this, we now know that $a(a
8
第 8 题
行程问题
Suppose July of year N has five Mondays. Which of the following must occur five times in the August of year N ? (Note: Both months have 31 days.) \textrm{(A)}\ Monday \textrm{(B)}\ Tuesday \textrm{(C)}\ Wednesday \textrm{(D)}\ Thursday \textrm{(E)}\ Friday
💡 解题思路
If there are five Mondays, there are only three possibilities for their dates: $(1,8,15,22,29)$ , $(2,9,16,23,30)$ , and $(3,10,17,24,31)$ .
9
第 9 题
规律与数列
If a,b,c,d are positive real numbers such that a,b,c,d form an increasing arithmetic sequence and a,b,d form a geometric sequence, then \frac ad is (A)\ \frac 1{12} (B)\ \frac 16 (C)\ \frac 14 (D)\ \frac 13 (E)\ \frac 12
💡 解题思路
We can let $a=1$ , $b=2$ , $c=3$ , and $d=4$ . $\frac{a}{d}=\boxed{\frac{1}{4}} \Longrightarrow \mathrm{(C)}$
10
第 10 题
规律与数列
How many different integers can be expressed as the sum of three distinct members of the set \{1,4,7,10,13,16,19\} ? (A)\ 13 (B)\ 16 (C)\ 24 (D)\ 30 (E)\ 35
💡 解题思路
Subtracting 10 from each number in the set, and dividing the results by 3, we obtain the set $\{-3, -2, -1, 0, 1, 2, 3\}$ . It is easy to see that we can get any integer between $-6$ and $6$ inclusive
11
第 11 题
数论
The positive integers A, B, A-B, and A+B are all prime numbers. The sum of these four primes is (A)\ even (B)\ divisible\ by\ 3 (C)\ divisible\ by\ 5 (D)\ divisible\ by\ 7 (E)\ prime
💡 解题思路
Since $A-B$ and $A+B$ must have the same parity , and since there is only one even prime number, it follows that $A-B$ and $A+B$ are both odd. Thus one of $A, B$ is odd and the other even. Since $A+B
12
第 12 题
几何·面积
For how many integers n is \dfrac n{20-n} the square of an integer? (A)\ 1 (B)\ 2 (C)\ 3 (D)\ 4 (E)\ 10
💡 解题思路
Let $x^2 = \frac{n}{20-n}$ , with $x \ge 0$ (note that the solutions $x < 0$ do not give any additional solutions for $n$ ). Then rewriting, $n = \frac{20x^2}{x^2 + 1}$ . Since $\text{gcd}(x^2, x^2 +
13
第 13 题
几何·面积
The sum of 18 consecutive positive integers is a perfect square . The smallest possible value of this sum is (A)\ 169 (B)\ 225 (C)\ 289 (D)\ 361 (E)\ 441
💡 解题思路
Let $a, a+1, \ldots, a + 17$ be the consecutive positive integers. Their sum, $18a + \frac{17(18)}{2} = 9(2a+17)$ , is a perfect square. Since $9$ is a perfect square, it follows that $2a + 17$ is a p
14
第 14 题
几何·面积
Four distinct circles are drawn in a plane . What is the maximum number of points where at least two of the circles intersect? (A)\ 8 (B)\ 9 (C)\ 10 (D)\ 12 (E)\ 16
💡 解题思路
For any given pair of circles, they can intersect at most $2$ times. Since there are ${4\choose 2} = 6$ pairs of circles, the maximum number of possible intersections is $6 \cdot 2 = 12$ . We can cons
15
第 15 题
数字运算
How many four-digit numbers N have the property that the three-digit number obtained by removing the leftmost digit is one ninth of N ? (A)\ 4 (B)\ 5 (C)\ 6 (D)\ 7 (E)\ 8
💡 解题思路
Let $N = \overline{abcd} = 1000a + \overline{bcd}$ , such that $\frac{N}{9} = \overline{bcd}$ . Then $1000a + \overline{bcd} = 9\overline{bcd} \Longrightarrow 125a = \overline{bcd}$ . Since $100 \le \
16
第 16 题
数论
Juan rolls a fair regular octahedral die marked with the numbers 1 through 8 . Then Amal rolls a fair six-sided die. What is the probability that the product of the two rolls is a multiple of 3? (A)\ \frac 1{12} (B)\ \frac 13 (C)\ \frac 12 (D)\ \frac 7{12} (E)\ \frac 23
💡 解题思路
On both dice, only the faces with the numbers $3,6$ are divisible by $3$ . Let $P(a) = \frac{2}{8} = \frac{1}{4}$ be the probability that Juan rolls a $3$ or a $6$ , and $P(b) = \frac{2}{6} = \frac 13
17
第 17 题
几何·面积
Andy's lawn has twice as much area as Beth's lawn and three times as much area as Carlos' lawn. Carlos' lawn mower cuts half as fast as Beth's mower and one third as fast as Andy's mower. If they all start to mow their lawns at the same time, who will finish first? (A)\ Andy (B)\ Beth (C)\ Carlos (D)\ Andy\ and \ Carlos\ tie\ for\ first. (E)\ All\ three\ tie.
💡 解题思路
We say Andy's lawn has an area of $x$ . Beth's lawn thus has an area of $\frac{x}{2}$ , and Carlos's lawn has an area of $\frac{x}{3}$ .
18
第 18 题
概率
A point P is randomly selected from the rectangular region with vertices (0,0),(2,0),(2,1),(0,1) . What is the probability that P is closer to the origin than it is to the point (3,1) ? (A)\ \frac 12 (B)\ \frac 23 (C)\ \frac 34 (D)\ \frac 45 (E)\ 1
💡 解题思路
Assume that the point $P$ is randomly chosen within the rectangle with vertices $(0,0)$ , $(3,0)$ , $(3,1)$ , $(0,1)$ . In this case, the region for $P$ to be closer to the origin than to point $(3,1)
19
第 19 题
逻辑推理
If a,b, and c are positive real numbers such that a(b+c) = 152, b(c+a) = 162, and c(a+b) = 170 , then abc is (A)\ 672 (B)\ 688 (C)\ 704 (D)\ 720 (E)\ 750
💡 解题思路
Adding up the three equations gives $2(ab + bc + ca) = 152 + 162 + 170 = 484 \Longrightarrow ab + bc + ca = 242$ . Subtracting each of the above equations from this yields, respectively, $bc = 90, ca
20
第 20 题
几何·面积
Let \triangle XOY be a right-angled triangle with m\angle XOY = 90^{\circ} . Let M and N be the midpoints of legs OX and OY , respectively. Given that XN = 19 and YM = 22 , find XY . (A)\ 24 (B)\ 26 (C)\ 28 (D)\ 30 (E)\ 32
💡 解题思路
Let $OM = x$ , $ON = y$ . By the Pythagorean Theorem on $\triangle XON, MOY$ respectively, \begin{align*} (2x)^2 + y^2 &= 19^2\\ x^2 + (2y)^2 &= 22^2\end{align*}
Since $2002 = 11 \cdot 13 \cdot 14$ , it follows that \begin{eqnarray*} a_n =\left\{ \begin{array}{lr} 11, & \text{if\ }n=13 \cdot 14 \cdot k, \quad k = 1,2,\cdots 10;\\ 13, & \text{if\ }n=14 \cdot 11
22
第 22 题
整数运算
For all integers n greater than 1 , define a_n = \frac{1}{\log_n 2002} . Let b = a_2 + a_3 + a_4 + a_5 and c = a_{10} + a_{11} + a_{12} + a_{13} + a_{14} . Then b- c equals (A)\ -2 (B)\ -1 (C)\ \frac{1}{2002} (D)\ \frac{1}{1001} (E)\ \frac 12
💡 解题思路
By the change of base formula, $a_n = \frac{1}{\frac{\log 2002}{\log n}} = \left(\frac{1}{\log 2002}\right) \log n$ . Thus \begin{align*}b- c &= \left(\frac{1}{\log 2002}\right)(\log 2 + \log 3 + \log
23
第 23 题
几何·面积
In \triangle ABC , we have AB = 1 and AC = 2 . Side \overline{BC} and the median from A to \overline{BC} have the same length. What is BC ? (A)\ \frac{1+√(2)}{2} (B)\ \frac{1+√(3)}2 (C)\ √(2) (D)\ \frac 32 (E)\ √(3)
💡 解题思路
[asy] unitsize(4cm); pair A, B, C, D, M; A = (1.768,0.935); B = (1.414,0); C = (0,0); D = (1.768,0); M = (0.707,0); draw(A--B--C--cycle); draw(A--D); draw(D--B); draw(A--M); label("$A$",A,N); label("$
24
第 24 题
几何·面积
A convex quadrilateral ABCD with area 2002 contains a point P in its interior such that PA = 24, PB = 32, PC = 28, PD = 45 . Find the perimeter of ABCD . (A)\ 4√(2002) (B)\ 2√(8465) (C)\ 2(48+√(2002)) (D)\ 2√(8633) (E)\ 4(36 + √(113))
💡 解题思路
We have \[[ABCD] = 2002 \le \frac 12 (AC \cdot BD)\] (This is true for any convex quadrilateral: split the quadrilateral along $AC$ and then using the triangle area formula to evaluate $[ACB]$ and $[A
25
第 25 题
几何·面积
Let f(x) = x^2 + 6x + 1 , and let R denote the set of points (x,y) in the coordinate plane such that \[f(x) + f(y) \le 0 and f(x)-f(y) \le 0\] The area of R is closest to
💡 解题思路
The first condition gives us that \[x^2 + 6x + 1 + y^2 + 6y + 1 \le 0 \Longrightarrow (x+3)^2 + (y+3)^2 \le 16\]