AMC 8 2022 Problem 20
AMC8 2022 年第 20 题
方程与不等式★★★☆☆
The grid below is to be filled with integers in such a way that the sum of the numbers in each row and the sum of the numbers in each column are the same. Four numbers are missing. The number x in the lower left corner is larger than the other three missing numbers. What is the smallest possible value of x ?
在下面的方格表中要填入整数，使得每行的各数之和与每列的各数之和都相同。现在有四个数缺失。左下角的数 比其他三个缺失的数都大。问 的最小可能值是多少？
- A.{-}1
- B.5
- C.6
- D.8
- E.9
正确答案 · Correct Answer
D
解析 · Solution
Solution 1
Let 𝑦, 𝑧 y,z, and 𝑤 w be the other missing numbers, as shown in the figure below. The sum of the numbers in each row and column is 12 12. It follows that 𝑥 + 𝑦 = 14 x+y=14, 𝑦 + 𝑧 = 13 y+z=13, 𝑧 + 𝑤 = 3 z+w=3, and 𝑥 + 𝑤 = 4 x+w=4. It is given that 𝑥 x is larger then the other missing numbers. The sum 𝑥 + 𝑦 x+y equals 14 14, so 𝑥 ≥ 8 x≥8. If 𝑥 𝑒 𝑞 𝑢 𝑎 𝑙 𝑠 8 x equals 8, then the values 𝑦 = 6 y=6, 𝑧 = 7 z=7, and 𝑤 = − 4 w=−4 satisfy the requirements of the problem. Therefore 𝑥 = 8 x=8 is the smallest integer that can be placed in the lower left corner. 𝑂 𝑅 OR Let 𝑥 x be the number in the lower left corner. The sum of the numbers in each row and column is 12 12, so the remaining entries can be expressed in terms of 𝑥 x, as shown below. Note that it is always the case thta 𝑥 > 𝑥 − 1 𝑎 𝑛 𝑑 14 − 𝑥 > 4 − 𝑥 x>x−1 and 14−x>4−x. This means that the only condition that must be satisfied is 𝑥 > 14 − 𝑥 x>14−x, which implies that 2 𝑥 > 14 2x>14, and hence 𝑥 > 7 x>7. Therefore 𝑥 = 8 x=8 is the smallest integer that can be placed in the lower left corner. Answer: 𝐷 D .
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