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Difference between revisions of "2013 AMC 12A Problems/Problem 25"

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==Solution==
 
==Solution==
  
−
Suppose <math>f(z)=z^2+iz+1=c=a+bi</math>. We look for <math>z</math> with <math>\text{Im}(z)>0</math> such that <math>a,b</math> are integers where <math>|a|, |b|\leq 10</math>.
+
Suppose <math>f(z)=z^2+iz+1=c=a+bi</math>. We look for <math>z</math> with <math>\operatorname{Im}(z)>0</math> such that <math>a,b</math> are integers where <math>|a|, |b|\leq 10</math>.
  
 
First, use the quadratic formula:
 
First, use the quadratic formula:
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Generally, consider the imaginary part of a radical of a complex number: <math>\sqrt{u}</math>, where <math>u = v+wi = r e^{i\theta}</math>.
 
Generally, consider the imaginary part of a radical of a complex number: <math>\sqrt{u}</math>, where <math>u = v+wi = r e^{i\theta}</math>.
  
−
<math>Im (\sqrt{u}) = Im(\pm \sqrt{r} e^{i\theta/2}) = \pm \sqrt{r} \sin(i\theta/2) = \pm \sqrt{r}\sqrt{\frac{1-\cos\theta}{2}} = \pm \sqrt{\frac{r-v}{2}}</math>.
+
<math>\operatorname{Im}(\sqrt{u}) = \operatorname{Im}(\pm \sqrt{r} e^{i\theta/2}) = \pm \sqrt{r} \sin(\theta/2) = \pm \sqrt{r}\sqrt{\frac{1-\cos\theta}{2}} = \pm \sqrt{\frac{r-v}{2}}</math>.
  
 
Now let <math>u= -5/4 + c</math>, then <math>v = -5/4 + a</math>, <math>w=b</math>, <math>r=\sqrt{v^2 + w^2}</math>.
 
Now let <math>u= -5/4 + c</math>, then <math>v = -5/4 + a</math>, <math>w=b</math>, <math>r=\sqrt{v^2 + w^2}</math>.
  
−
Note that <math>Im(z)>0</math> if and only if <math>\pm \sqrt{\frac{r-v}{2}}>\frac{1}{2}</math>. The latter is true only when we take the positive sign, and that <math>r-v > 1/2</math>,
+
Note that <math>\operatorname{Im}(z)>0</math> if and only if <math>\pm \sqrt{\frac{r-v}{2}}>\frac{1}{2}</math>. The latter is true only when we take the positive sign, and that <math>r-v > 1/2</math>,
  
 
or <math>v^2 + w^2 > (1/2 + v)^2 = 1/4 + v + v^2</math>, <math>w^2 > 1/4 + v</math>, or <math>b^2 > a-1</math>.
 
or <math>v^2 + w^2 > (1/2 + v)^2 = 1/4 + v + v^2</math>, <math>w^2 > 1/4 + v</math>, or <math>b^2 > a-1</math>.
  
−
In other words, for all <math>z</math>, <math>f(z)=a+bi</math> satisfies <math>b^2 > a-1</math>, and there is one and only one <math>z</math> that makes it true. Therefore we are just going to count the number of ordered pairs <math>(a,b)</math> such that <math>a</math>, <math>b</math> are integers of magnitude no greater than <math>10</math>, and that <math>b^2 \geq a</math>.
+
In other words, when <math>b^2 > a-1</math>, the equation <math>f(z)=a+bi</math> has unique solution <math>z</math> in the region <math>\operatorname{Im}(z)>0</math>; and when <math>b^2 \leq a-1</math> there is no solution. Therefore the number of desired solution <math>z</math> is the same as the number of ordered pairs <math>(a,b)</math> such that integers <math>|a|, |b|\leq 10</math>, and that <math>b^2 \geq a</math>.
  
 
When <math>a\leq 0</math>, there is no restriction on <math>b</math> so there are <math>11\cdot 21 = 231</math> pairs;
 
When <math>a\leq 0</math>, there is no restriction on <math>b</math> so there are <math>11\cdot 21 = 231</math> pairs;
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when <math>a > 0</math>, there are <math>2(1+4+9+10+10+10+10+10+10+10)=2(84)=168</math> pairs.
 
when <math>a > 0</math>, there are <math>2(1+4+9+10+10+10+10+10+10+10)=2(84)=168</math> pairs.
  
−
So there are <math>231+168=399</math> in total.
+
So there are <math>231+168=\boxed{399}</math> in total.
 +
 
 +
==Solution 2 (motivated by coordinate geometry)==
 +
 
 +
We consider the function <math>f(z)</math> as a mapping from the 2-D complex plane onto itself. We complete the square of <math>f(z)=z^2+iz+1=(z+\frac{i}{2})^2+\frac{5}{4}</math>.
 +
 
 +
Now, we must decide the range of <math>f(z)</math> based on the domain of <math>z</math>, <math>\operatorname{Im}(z)>0</math>. To do this, we are interested in mapping the boundary line <math>\operatorname{Im}(z)=0</math>. To make the mapping simpler, let <math>f(z)=g(z)+\frac{5}{4}</math>, or <math>g(z)=(z+\frac{i}{2})^2</math>.
 +
 
 +
We intend to map of the line <math>\operatorname{Im}(z)=0</math> using the function <math>g(z)</math>. This transformation is equivalent to the polar equation <math>r=(\frac{1}{2}\csc(\frac{\theta}{2}))^2</math>. Using polar and trig identities, we can restate this equation as the rectangular form of a parabola,
 +
 
 +
<math>x=y^2-\frac{1}{4}</math>,
 +
 
 +
where <math>x=\operatorname{Re}(z)</math> and <math>y=\operatorname{Im}(z)</math>. So, we conclude that <math>f(z)</math> maps the line <math>\operatorname{Im}(z)=0</math> to the parabola
 +
 
 +
<math>x=y^2-\frac{1}{4}+\frac{5}{4}=y^2+1</math>.
 +
 
 +
A quick check reveals that the range of <math>f(z)</math> is to the left of the parabola, meaning that any point on or to the right of parabola cannot be reached.
 +
 
 +
Since the problem requires <math>|\operatorname{Re}(z)|</math> and <math>|\operatorname{Im}(z)|</math> to both be integers and at most 10, all that remains is counting all points with integer coordinates in the range of <math>f(z), \operatorname{Im}(z)>0</math>. To do this, we employ complementary counting.
 +
 
 +
The points of interest are <math>|\operatorname{Re}(z)|\leq 10</math> and <math>|\operatorname{Im}(z)|\leq 10</math>, resulting in a total of <math>441</math> points. For lattice points on or to the right of the parabola, there are <math>10</math> points for <math>x=0</math>, <math>9</math> points for <math>x=\pm 1</math>, <math>6</math> points for <math>x=\pm 2</math>, and <math>1</math> point for <math>x=\pm 3</math>. Summing it all together, our answer is <math>441-(10+2*9+2*6+2*1)=\boxed{399}</math>.
 +
 
 +
==Video Solution by Richard Rusczyk==
 +
https://artofproblemsolving.com/videos/amc/2013amc12a/365
  
 
== See also ==
 
== See also ==
 
{{AMC12 box|year=2013|ab=A|num-b=24|after=Last Question}}
 
{{AMC12 box|year=2013|ab=A|num-b=24|after=Last Question}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 14:22, 8 October 2023

Problem

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 $\text{Im}(z) > 0$ and both the real and the imaginary parts of $f(z)$ are integers with absolute value at most $10$?

$\textbf{(A)} \ 399 \qquad \textbf{(B)} \ 401 \qquad \textbf{(C)} \ 413 \qquad \textbf{(D)} \ 431 \qquad \textbf{(E)} \ 441$

Solution

Suppose $f(z)=z^2+iz+1=c=a+bi$. We look for $z$ with $\operatorname{Im}(z)>0$ such that $a,b$ are integers where $|a|, |b|\leq 10$.

First, use the quadratic formula:

$z = \frac{1}{2} (-i \pm \sqrt{-1-4(1-c)}) = -\frac{i}{2} \pm \sqrt{ -\frac{5}{4} + c }$

Generally, consider the imaginary part of a radical of a complex number: $\sqrt{u}$, where $u = v+wi = r e^{i\theta}$.

$\operatorname{Im}(\sqrt{u}) = \operatorname{Im}(\pm \sqrt{r} e^{i\theta/2}) = \pm \sqrt{r} \sin(\theta/2) = \pm \sqrt{r}\sqrt{\frac{1-\cos\theta}{2}} = \pm \sqrt{\frac{r-v}{2}}$.

Now let $u= -5/4 + c$, then $v = -5/4 + a$, $w=b$, $r=\sqrt{v^2 + w^2}$.

Note that $\operatorname{Im}(z)>0$ if and only if $\pm \sqrt{\frac{r-v}{2}}>\frac{1}{2}$. The latter is true only when we take the positive sign, and that $r-v > 1/2$,

or $v^2 + w^2 > (1/2 + v)^2 = 1/4 + v + v^2$, $w^2 > 1/4 + v$, or $b^2 > a-1$.

In other words, when $b^2 > a-1$, the equation $f(z)=a+bi$ has unique solution $z$ in the region $\operatorname{Im}(z)>0$; and when $b^2 \leq a-1$ there is no solution. Therefore the number of desired solution $z$ is the same as the number of ordered pairs $(a,b)$ such that integers $|a|, |b|\leq 10$, and that $b^2 \geq a$.

When $a\leq 0$, there is no restriction on $b$ so there are $11\cdot 21 = 231$ pairs;

when $a > 0$, there are $2(1+4+9+10+10+10+10+10+10+10)=2(84)=168$ pairs.

So there are $231+168=\boxed{399}$ in total.

Solution 2 (motivated by coordinate geometry)

We consider the function $f(z)$ as a mapping from the 2-D complex plane onto itself. We complete the square of $f(z)=z^2+iz+1=(z+\frac{i}{2})^2+\frac{5}{4}$.

Now, we must decide the range of $f(z)$ based on the domain of $z$, $\operatorname{Im}(z)>0$. To do this, we are interested in mapping the boundary line $\operatorname{Im}(z)=0$. To make the mapping simpler, let $f(z)=g(z)+\frac{5}{4}$, or $g(z)=(z+\frac{i}{2})^2$.

We intend to map of the line $\operatorname{Im}(z)=0$ using the function $g(z)$. This transformation is equivalent to the polar equation $r=(\frac{1}{2}\csc(\frac{\theta}{2}))^2$. Using polar and trig identities, we can restate this equation as the rectangular form of a parabola,

$x=y^2-\frac{1}{4}$,

where $x=\operatorname{Re}(z)$ and $y=\operatorname{Im}(z)$. So, we conclude that $f(z)$ maps the line $\operatorname{Im}(z)=0$ to the parabola

$x=y^2-\frac{1}{4}+\frac{5}{4}=y^2+1$.

A quick check reveals that the range of $f(z)$ is to the left of the parabola, meaning that any point on or to the right of parabola cannot be reached.

Since the problem requires $|\operatorname{Re}(z)|$ and $|\operatorname{Im}(z)|$ to both be integers and at most 10, all that remains is counting all points with integer coordinates in the range of $f(z), \operatorname{Im}(z)>0$. To do this, we employ complementary counting.

The points of interest are $|\operatorname{Re}(z)|\leq 10$ and $|\operatorname{Im}(z)|\leq 10$, resulting in a total of $441$ points. For lattice points on or to the right of the parabola, there are $10$ points for $x=0$, $9$ points for $x=\pm 1$, $6$ points for $x=\pm 2$, and $1$ point for $x=\pm 3$. Summing it all together, our answer is $441-(10+2*9+2*6+2*1)=\boxed{399}$.

Video Solution by Richard Rusczyk

https://artofproblemsolving.com/videos/amc/2013amc12a/365

See also

2013 AMC 12A (Problems • Answer Key • Resources)
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