AMC12 2013 A

AMC12 2013 A · Q25

AMC12 2013 A · Q25. It mainly tests Complex numbers (rare), Algebra misc.

Let $f:\mathbb{C}\to\mathbb{C}$ be defined by $f(z)=z^2+iz+1$. How many complex numbers $z$ are there such that $\operatorname{Im}(z)>0$ and both the real and the imaginary parts of $f(z)$ are integers with absolute value at most $10$?

设 $f:\mathbb{C}\to\mathbb{C}$ 定义为 $f(z)=z^2+iz+1$。满足 $\operatorname{Im}(z)>0$ 且 $f(z)$ 的实部与虚部均为整数、并且它们的绝对值都不超过 $10$ 的复数 $z$ 有多少个？

(A) 399 399

(B) 401 401

(D) 431 431

(E) 441 441

Answer

Correct choice: (A)

正确答案：（A）

Solution

Answer (A): Let $H=\{z\in\mathbb{C}:\operatorname{Im}(z)>0\}$. If $z_1,z_2\in H$ and $f(z_1)=f(z_2)$, then $$ z_1^2-z_2^2+i(z_1-z_2)=(z_1-z_2)(z_1+z_2+i)=0. $$ Because $\operatorname{Im}(z_1)>0$ and $\operatorname{Im}(z_2)>0$, it follows that $z_1+z_2+i\ne0$. Thus $z_1=z_2$; that is, the function $f$ is one-to-one on $H$. Let $r$ be a positive real number. Note that $f(r)=r^2+1+ir$ describes the top part of the parabola $x=y^2+1$. Similarly, $f(-r)=r^2+1-ir$ describes the bottom part of the parabola $x=y^2+1$. Because $f(i)=-1$, it follows that the image set $f(H)$ equals $\{w\in\mathbb{C}:\operatorname{Re}(w)<(\operatorname{Im}(w))^2+1\}$. Thus the set of complex numbers $w\in f(H)$ with integer real and imaginary parts of absolute value at most $10$ is equal to $$ S=\{w=a+ib\in\mathbb{C}:a,b\in\mathbb{Z},\ |a|\le10,\ |b|\le10,\ \text{and }a<b^2+1\}. $$ Because $f$ is one-to-one, the required answer is $|f^{-1}(S)|=|S|$ and $$ |S|=21^2-\sum_{b=-3}^{3}\sum_{a=b^2+1}^{10}1 =441-\sum_{b=-3}^{3}(10-b^2) =441-(1+6+9+10+9+6+1)=399. $$

答案 (A)：设 $H=\{z\in\mathbb{C}:\operatorname{Im}(z)>0\}$。若 $z_1,z_2\in H$ 且 $f(z_1)=f(z_2)$，则 $$ z_1^2-z_2^2+i(z_1-z_2)=(z_1-z_2)(z_1+z_2+i)=0. $$ 由于 $\operatorname{Im}(z_1)>0$ 且 $\operatorname{Im}(z_2)>0$，可得 $z_1+z_2+i\ne0$。因此 $z_1=z_2$；即函数 $f$ 在 $H$ 上是一一对应的。令 $r$ 为正实数。注意 $f(r)=r^2+1+ir$ 描述抛物线 $x=y^2+1$ 的上半部分。同理，$f(-r)=r^2+1-ir$ 描述抛物线 $x=y^2+1$ 的下半部分。由于 $f(i)=-1$，可知像集 $f(H)$ 等于 $\{w\in\mathbb{C}:\operatorname{Re}(w)<(\operatorname{Im}(w))^2+1\}$。因此，满足实部与虚部为整数且绝对值至多为 $10$ 的 $w\in f(H)$ 构成的集合为 $$ S=\{w=a+ib\in\mathbb{C}:a,b\in\mathbb{Z},\ |a|\le10,\ |b|\le10,\ \text{且 }a<b^2+1\}. $$ 因为 $f$ 是一一对应的，所求为 $|f^{-1}(S)|=|S|$，并且 $$ |S|=21^2-\sum_{b=-3}^{3}\sum_{a=b^2+1}^{10}1 =441-\sum_{b=-3}^{3}(10-b^2) =441-(1+6+9+10+9+6+1)=399. $$

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