AMC10 2020 B

AMC10 2020 B · Q9

AMC10 2020 B · Q9. It mainly tests Quadratic equations, Manipulating equations.

How many ordered pairs of integers $(x, y)$ satisfy the equation $x^{2020} + y^2 = 2y$?

有多少个整数有序对$(x, y)$满足方程$x^{2020} + y^2 = 2y$？

(A) 1 1

(B) 2 2

(D) 4 4

(E) infinitely many 无穷多个

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): The given equation is equivalent to $x^{2020}+y^2-2y+1=1$, which is equivalent to $x^{2020}+(y-1)^2=1$. The only way to write 1 as the sum of the squares of two integers is as 0 + 1 or 1 + 0, so $x$ must be 0 or $\pm 1$. The complete set of solutions is $\{(0,0),(0,2),(1,1),(-1,1)\}$, which has 4 elements.

答案（D）：所给方程等价于 $x^{2020}+y^2-2y+1=1$，也等价于 $x^{2020}+(y-1)^2=1$。将 1 表示为两个整数平方和的唯一方式是 $0+1$ 或 $1+0$，因此 $x$ 必须为 0 或 $\pm 1$。解的全集为 $\{(0,0),(0,2),(1,1),(-1,1)\}$，共有 4 个元素。

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