2022 AMC amc8 真题 答案 详解

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中英双语真题

2022 AMC8
Problem 1
The Math Team designed a logo shaped like a multiplication symbol, shown below on a grid of 1-inch squares. What is the area of the logo in square inches?
在由边长为 1 英寸的正方形组成的方格纸上，数学小组设计了一个类似乘法符号的标志，如下图所示。问该标志的面积是多少平方英寸？

(A) 10 (B) 12 (C) 13 (D) 14 (E) 15

Problem 2
Consider these two operations: (a \spadesuit b = a^2 - b^2)
(a \star b = (a-b)^2)
What is the value of ((5 \spadesuit 3) \star 6)?
考虑这样两种运算：
(a \spadesuit b = a^2 - b^2)
(a \star b = (a-b)^2)
问 ((5 \spadesuit 3) \star 6) 的值是多少？

(A) -20 (B) 4 (C) 16 (D) 100 (E) 220

英文真题

2 MAA AMC 8 1. The Math Team designed a logo shaped like a multiplication symbol, shown below on a grid of 1-inch squares. What is the area of the logo in square inches? (A) 10 (B) 12 (C) 13 (D) 14 (E) 15 2. Consider these two operations: $$a \spadesuit b = a^2 - b^2$$ $$a \star b = (a-b)^2$$ What is the value of $(5 \spadesuit 3) \star 6$ ? (A) -20 (B) 4 (C) 16 (D) 100 (E) 220 3. When three positive integers $a$, $b$, and $c$ are multiplied together, their product is 100. Suppose $a < b < c$. In how many ways can the numbers be chosen? (A) 0 (B) 1 (C) 2 (D) 3 (E) 4

真题文字详解(英文)

Problem 1 The Math Team designed a logo shaped like a multiplication symbol, shown below on a grid of 1-inch squares. What is the area of the logo in square inches?
(A) 10 (B) 12 (C) 13 (D) 14 (E) 15

Solution 1 (A quick solution.) We can see that there are 4 whole squares, since the area of each square will be 1, 4 * 1 = 4. Next, there are 12 half squares, and 2 half squares are 1 whole square, so 12/2 = 6 whole squares. The area of this will be 6 * 1 = 6. Finally, we can add the 2 numbers. 4 + 6 = \boxed{10}.

Solution 2 Draw the following four lines as shown:
We see these lines split the figure into five squares with side length √2. Thus, the area is 5 · (√2)^2 = 5 · 2 = \boxed{10}.

Solution 3 There are 5 lattice points in the interior of the logo and 12 lattice points on the boundary of the logo. Because of Pick's Theorem, the area of the logo is 5 + 12/2 - 1 = \boxed{10}.

真题文字详解(中英双语)

Problem 1 The Math Team designed a logo shaped like a multiplication symbol, shown below on a grid of 1-inch squares. What is the area of the logo in square inches? 数学队设计了一个形状像乘法符号的标志，如下图所示在一个由1英寸正方形组成的网格上。这个标志的面积是多少平方英寸？
(A) 10 (B) 12 (C) 13 (D) 14 (E) 15

Solution 1 (A quick solution.) 解法1（快速解法。） We can see that there are 4 whole squares, since the area of each square will be 1, 4 * 1 = 4. Next, there are 12 half squares, and 2 half squares are 1 whole square, so 12/2 = 6 whole squares. The area of this will be 6 * 1 = 6. Finally, we can add the two numbers. 4 + 6 = (A) 10. 我们可以看到有4个完整正方形，因为每个正方形的面积是1，41=4。接下来有12个半块正方形，2个半块正方形等于1个整块正方形，所以12/2=6个整块正方形。这个部分的面积是61=6。最后，我们可以将这两个数相加。4+6=(A) 10。

Solution 2 Draw the following four lines as shown: 按照如下所示画出以下四条线：