2022 AMC amc12b 真题 答案 详解

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中英双语真题

2022 AMC 12B

1. Define x⊕y to be |x-y| for all real numbers x and y. What is the value of (10 ⊕ (20⊕3))−((10⊕2)⊕3)?

对于所有的实数x和y,x⊕y定义为|x-y|.问(10⊕(20⊕3))−((10⊕2)⊕3)的值是多少?

(A) -2 (B) -1 (C) 0 (D) 1 (E) 2

1. In rhombus ABCD, point P lies on segment AD so that BP ⊥ AD, AP = 3, and PD = 2. What is the area of ABCD? (Note: The figure is not drawn to scale.)

在菱形ABCD中，点P位于线段AD上使得BP⊥AD，AP=3，PD=2.问ABCD的面积是多少?(注：图形未按比例绘制。)

(A) 3√5 (B) 10 (C) 6√5 (D) 20 (E) 25

1. How many of the first ten numbers of the sequence 121, 11211, 1112111,... are prime numbers?

在数列121, 11211, 1112111,...的前十项中有多少个数是素数?

(A) 0 (B) 1 (C) 2 (D) 3 (E) 4

1. For how many values of the constant k will the polynomial x^2 + kx + 36 have two distinct integer roots?

使得多项式x^2+kx+36有两个不同的整数根的常数k的取值有多少种?

(A) 6 (B) 8 (C) 9 (D) 14 (E) 16

英文真题

2022 AMC 12B Problems Problem 1 Define $x \diamond y$ to be $\vert x - y \vert$ for all real numbers $x$ and $y$. What is the value of $(1 \diamond (2 \diamond 3)) - ((1 \diamond 2) \diamond 3)$? (A) $-2$ (B) $-1$ (C) $0$ (D) $1$ (E) $2$ Problem 2 In rhombus $ABCD$, point $P$ lies on segment $\overline{AD}$ so that $\overline{BP} \bot \overline{AD}, AP = 3,$ and $PD = 2.$ What is the area of $ABCD?$ (Note: The figure is not drawn to scale.) (A) $3\sqrt{5}$ (B) $10$ (C) $6\sqrt{5}$ (D) $20$ (E) $25$ Problem 3 How many of the first ten numbers of the sequence $121,11211,1112111,\ldots$ are prime numbers? (A) $0$ (B) $1$ (C) $2$ (D) $3$ (E) $4$ Problem 4 For how many values of the constant $k$ will the polynomial $x^2 + kx + 36$ have two distinct integer roots? (A) $6$ (B) $8$ (C) $9$ (D) $14$ (E) $16$ Problem 5 The point $(-1,-2)$ is rotated $270^\circ$ counterclockwise about the point $(3,1).$ What are the coordinates of its new position? (A) $(-3,-4)$ (B) $(0,5)$ (C) $(2,-1)$ (D) $(4,3)$ (E) $(6,-3)$ Problem 6 Consider the following $100$ sets of $10$ elements each: ${1,2,3,\ldots,10},$ ${11,12,13,\ldots,20},$ ${21,22,23,\ldots,30},$ $\vdots$ ${991,992,993,\ldots,1000}.$ How many of these sets contain exactly two multiples of $7?$ (A) $40$ (B) $42$ (C) $43$ (D) $49$ (E) $50$

真题文字详解(英文)

Problem 1 The following problem is from both the 2022 AMC 10B #1 and 2022 AMC 12B #1, so both problems redirect to this page. Define x ◊ y to be |x - y| for all real numbers x and y. What is the value of (1 ◊ (2 ◊ 3)) − ((1 ◊ 2) ◊ 3)? (A) -2 (B) -1 (C) 0 (D) 1 (E) 2 Solution 1 We have (1 ◊ (2 ◊ 3)) − ((1 ◊ 2) ◊ 3) = |1 − |2 − 3|| − ||1 − 2| − 3| = |1 − 1| − |1 − 3| = 0 − 2 = (A) -2. ~MRENTHUSIASM Solution 2 Observe that the ◊ function is simply the positive difference between two numbers. Thus, we evaluate: the difference between 2 and 3 is 1; the difference between 1 and 1 is 0; the difference between 1 and 2 is 1; the difference between 1 and 3 is 2; and finally, 0 − 2 = (A) -2. Problem 2 The following problem is from both the 2022 AMC 10B #2 and 2022 AMC 12B #2, so both problems redirect to this page. In rhombus ABCD, point P lies on segment AD such that BP ⊥ AD, AP = 3, and PD = 2. What is the area of ABCD? (Note: The figure is not drawn to scale.) (A) 3√5 (B) 10 (C) 6√5 (D) 20 (E) 25 Solution 1 AD = AP + PD = 3 + 2 = 5. ABCD is a rhombus, so AB = AD = 5. △APB is a 3-4-5 right triangle, hence BP = 4. The area of the rhombus is base times height: bh = (AD)(BP) = 5 · 4 = (D) 20. Figure redrawn to scale. Solution 2 (The Area Of A Triangle)

真题文字详解(中英双语)

Problem 1 The following problem is from both the 2022 AMC 10B #1 and 2022 AMC 12B #1, so both problems redirect to this page. 以下问题来自2022年AMC 10B第1题和2022年AMC 12B第1题，因此这两个问题都指向这一页。 Define x ◊ y to be |x - y| for all real numbers x and y. What is the value of 定义 x ◊ y 为|x - y|对于所有实数x和y，求其值。 (1 ◊ (2 ◊ 3)) − ((1 ◊ 2) ◊ 3)? (A) -2 (B) -1 (C) 0 (D) 1 (E) 2 Solution 1 We have (1 ◊ (2 ◊ 3)) − ((1 ◊ 2) ◊ 3) = |1 − |2 − 3|| − ||1 − 2| − 3| = |1 − 1| − |1 − 3| = 0 − 2 = (A) -2. ~MRENTHUSIASM 我们有 ~MRENTHUSIASM Solution 2 Observe that the ◊ function is simply the positive difference between two numbers. Thus, we evaluate: the difference between 2 and 3 is 1; the difference between 1 and 1 is 0; the difference between 1 and 2 is 1; the difference between 1 and 3 is 2; and finally, 0 − 2 = (A) -2. 观察到◊函数仅仅是两个数之间的正差。因此我们进行计算：2 和 3 之间的差是 1；1 和 1 之间的差是 0；1 和 2 之间的差是 1；1 和 3 之间的差是 2；最后是 0 - 2 = (A) -2. Problem 2 The following problem is from both the 2022 AMC 10B #2 and 2022 AMC 12B #2, so both problems redirect to this page. 以下问题来自2022年AMC 10B第2题和2022年AMC 12B第2题，因此这两个问题都指向这一页。 In rhombus ABCD, point P lies on segment AD so that BP ⊥ AD, AP = 3, and PD = 2. What is the area of ABCD? (Note: The figure is not drawn to scale.) 在菱形ABCD中，点P位于线段AD上，使得BP⊥AD，AP=3，PD=2。ABCD的面积是多少？（注意：图形没有按比例绘制。） (A) 3√5 (B) 10 (C) 6√5 (D) 20 (E) 25 Solution 1 AD = AP + PD = 3 + 2 = 5. ABCD is a rhombus, so AB = AD = 5. ABCD是一个菱形，所以AB = AD = 5. Figure redrawn to scale.