2022 AMC amc10b 真题 答案 详解

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中英双语真题

2022 AMC 10B

1. 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)?

对于所有的实数x和y,x⊕y定义为|x-y|.问(1⊕(2⊕3))−((1⊕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 three-digit positive integers have an odd number of even digits?

包含奇数个偶数数码的三位正整数有多少个？

(A) 150 (B) 250 (C) 350 (D) 450 (E) 550

1. A donkey suffers an attack of hiccups and the first hiccup happens at 4:00 one afternoon. Suppose that the donkey hiccups regularly every 5 seconds. At what time does the donkey's 700th hiccup occur?

一头驴因病连续打嗝，第一次打嗝发生在下午4:00.假设驴每隔5秒打嗝一次.那么驴第700次打嗝发生的时间是什么时候？

英文真题

2022 AMC 10B Problems Problem 1 Define $x \diamond y$ to be $|x-y|$ 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 $\overline{BP} \perp \overline{AD}$, $AP = 3$, 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 three-digit positive integers have an odd number of even digits?
(A) $150$ (B) $250$ (C) $350$ (D) $450$ (E) $550$

Problem 4 A donkey suffers an attack of hiccups and the first hiccup happens at $4:00$ one afternoon. Suppose that the donkey hiccups regularly every $5$ seconds. At what time does the donkey's $700^{\text{th}}$ hiccup occur?
(A) $15$ seconds after $4:58$
(B) $20$ seconds after $4:58$
(C) $25$ seconds after $4:58$
(D) $30$ seconds after $4:58$
(E) $35$ seconds after $4:58$

Problem 5 What is the value of
$\frac{(1+\frac13)(1+\frac15)(1+\frac17)}{\sqrt{(1-\frac12)(1-\frac14)(1-\frac16)}}?$
(A) $\sqrt{3}$ (B) $2$ (C) $\sqrt{15}$ (D) $4$ (E) $\sqrt{105}$

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. 以下问题既来自2022年AMC 10B第1题，也来自2022年AMC 12B第1题，因此这两个问题都指向这一页。 Define ( x \diamond y ) to be ( |x - y| ) for all real numbers ( x ) and ( y ). What is the value of 定义 ( x \diamond y ) 为 ( |x - y| )，对于所有实数 ( x ) 和 ( y )，求其值。 ( (1 \diamond (2 \diamond 3)) - ((1 \diamond 2) \diamond 3)? )
(A) -2 (B) -1 (C) 0 (D) 1 (E) 2 Solution 1 We have ( (1 \diamond (2 \diamond 3)) - ((1 \diamond 2) \diamond 3) = |1 - |2 - 3|| - ||1 - 2| - 3| )
( = |1 - 1| - |1 - 3| )
( = 0 - 2 )
( = \textbf{(A)} -2 ). ~MRENTHUSIASM 我们有 ~MRENTHUSIASM Solution 2 Observe that the ( \diamond ) 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 = \textbf{(A)} -2. 观察到 ( \diamond ) 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= \textbf{(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. 以下问题既来自2022年AMC 10B第2题，也来自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：图形未按比例绘制。） (A) ( 3\sqrt{5} ) (B) 10 (C) ( 6\sqrt{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.