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Difference between revisions of "2023 AIME I Problems/Problem 15"

(Created page with "==Problem 15== Find the largest prime number <math>p<1000</math> for which there exists a complex number <math>z</math> satisfying * the real and imaginary parts of <math>z</m...")
 
Line 7: Line 7:
 
==Answer: 349==
 
==Answer: 349==
 
Suppose <math>z=a+bi</math>; notice that <math>\arg(z^{3})\approx 45^{\circ}</math>, so by De Moivre’s theorem <math>\arg(z)\approx 15^{\circ}</math> and <math>\tfrac{b}{a}\approx tan(15^{\circ})=2-\sqrt{3}</math>. Now just try pairs <math>(a, b)=(t, \left(2-\sqrt{3}\right)t)</math> going down from <math>t=\lfloor\sqrt{1000}\rfloor</math>, writing down the value of <math>a^{2}+b^{2}</math> on the right; and eventually we arrive at <math>(a, b)=(18, 5)</math> the first time <math>a^{2}+b^{2}</math> is prime. Therefore, <math>p=\boxed{349}</math>.
 
Suppose <math>z=a+bi</math>; notice that <math>\arg(z^{3})\approx 45^{\circ}</math>, so by De Moivre’s theorem <math>\arg(z)\approx 15^{\circ}</math> and <math>\tfrac{b}{a}\approx tan(15^{\circ})=2-\sqrt{3}</math>. Now just try pairs <math>(a, b)=(t, \left(2-\sqrt{3}\right)t)</math> going down from <math>t=\lfloor\sqrt{1000}\rfloor</math>, writing down the value of <math>a^{2}+b^{2}</math> on the right; and eventually we arrive at <math>(a, b)=(18, 5)</math> the first time <math>a^{2}+b^{2}</math> is prime. Therefore, <math>p=\boxed{349}</math>.
 +
 +
==Solution==
 +
 +
Denote <math>z = a + i b</math>. Thus, <math>a^2 + b^2 = p</math>.
 +
 +
Thus,
 +
\[
 +
z^3 = a \left( a^2 - 3 b^2 \right) + i b \left( - b^2 + 3 a^2 \right) .
 +
\]
 +
 +
Because <math>p</math>, <math>{\rm Re} \left( z^3 \right)</math>, <math>{\rm Im} \left( z^3 \right)</math> are three sides of a triangle, we have <math>{\rm Re} \left( z^3 \right) > 0</math> and <math>{\rm Im} \left( z^3 \right) > 0</math>.
 +
Thus,
 +
\begin{align*}
 +
a \left( a^2 - 3 b^2 \right) & > 0 , \hspace{1cm} (1) \\
 +
b \left( - b^2 + 3 a^2 \right) & > 0. \hspace{1cm} (2)
 +
\end{align*}
 +
 +
Because <math>p</math>, <math>{\rm Re} \left( z^3 \right)</math>, <math>{\rm Im} \left( z^3 \right)</math> are three sides of a triangle, we have the following triangle inequalities:
 +
\begin{align*}
 +
{\rm Re} \left( z^3 \right) + {\rm Im} \left( z^3 \right) & > p \hspace{1cm} (3) \\
 +
p + {\rm Re} \left( z^3 \right) & > {\rm Im} \left( z^3 \right) \hspace{1cm} (4) \\
 +
p + {\rm Im} \left( z^3 \right) & > {\rm Re} \left( z^3 \right) \hspace{1cm} (5)
 +
\end{align*}
 +
 +
We notice that <math>| z^3 | = p^{3/2}</math>, and <math>{\rm Re} \left( z^3 \right)</math>, <math>{\rm Im} \left( z^3 \right)</math>, and <math>| z^3 |</math> form a right triangle. Thus, <math>{\rm Re} z^3 + {\rm Im} z^3 > p^{3/2}</math>.
 +
Because <math>p > 1</math>, <math>p^{3/2} > p</math>.
 +
Therefore, (3) holds.
 +
 +
Conditions (4) and (5) can be written in the joint form as
 +
\[
 +
\left| {\rm Re} \left( z^3 \right) - {\rm Im} \left( z^3 \right) \right| < p . \hspace{1cm} (4)
 +
\]
 +
 +
We have
 +
\begin{align*}
 +
{\rm Re} \left( z^3 \right) - {\rm Im} \left( z^3 \right)
 +
& = \left( a^3 - 3 a b^2 \right) - \left( - b^3 + 3 a^2 b \right) \\
 +
& = \left( a + b \right) \left( a^2 - 4 ab + b^2 \right)
 +
\end{align*}
 +
and <math>p = a^2 + b^2</math>.
 +
 +
Thus, (5) can be written as
 +
\[
 +
\left| \left( a + b \right) \left( a^2 - 4 ab + b^2 \right)  \right|
 +
< a^2 + b^2 . \hspace{1cm} (6)
 +
\]
 +
 +
Therefore, we need to jointly solve (1), (2), (6).
 +
From (1) and (2), we have either <math>a, b >0</math>, or <math>a, b < 0</math>.
 +
In (6), by symmetry, without loss of generality, we assume <math>a, b > 0</math>.
 +
 +
Thus, (1) and (2) are reduced to
 +
\[
 +
a > \sqrt{3} b . \hspace{1cm} (7)
 +
\]
 +
 +
Let <math>a = \lambda b</math>. Plugging this into (6), we get
 +
\begin{align*}
 +
\left| \left( \left( \lambda - 2 \right)^2 - 3 \right)  \right|
 +
< \frac{1}{b} \frac{\lambda^2 + 1}{\lambda + 1} .  \hspace{1cm} (8)
 +
\end{align*}
 +
 +
Because <math>p= a^2 + b^2</math> is a prime, <math>a</math> and <math>b</math> are relatively prime.
 +
 +
Therefore, we can use (7), (8), <math>a^2 + b^2 <1000</math>, and <math>a</math> and <math>b</math> are relatively prime to solve the problem.
 +
 +
To facilitate efficient search, we apply the following criteria:
 +
\begin{enumerate}
 +
\item To satisfy (7) and <math>a^2 + b^2 < 1000</math>, we have <math>1 \leq b \leq 15</math>.
 +
In the outer layer, we search for <math>b</math> in a decreasing order.
 +
In the inner layer, for each given <math>b</math>, we search for <math>a</math>.
 +
\item Given <math>b</math>, we search for <math>a</math> in the range <math>\sqrt{3} b < a < \sqrt{1000 - b^2}</math>.
 +
\item We can prove that for <math>b \geq 9</math>, there is no feasible <math>a</math>.
 +
The proof is as follows.
 +
 +
For <math>b \geq 9</math>, to satisfy <math>a^2 + b^2 < 1000</math>, we have <math>a \leq 30</math>.
 +
Thus, <math>\sqrt{3} < \lambda \leq \frac{30}{9}</math>.
 +
Thus, the R.H.S. of (8) has the following upper bound
 +
\begin{align*}
 +
\frac{1}{b} \frac{\lambda^2 + 1}{\lambda + 1}
 +
& < \frac{1}{b} \frac{\lambda^2 + \lambda}{\lambda + 1} \\
 +
& = \frac{\lambda}{b} \\
 +
& \leq \frac{\frac{30}{9}}{9} \\
 +
& < \frac{10}{27} .
 +
\end{align*}
 +
 +
Hence, to satisfy (8), a necessary condition is
 +
\begin{align*}
 +
\left| \left( \left( \lambda - 2 \right)^2 - 3 \right)  \right|
 +
< \frac{10}{27} .
 +
\end{align*}
 +
 +
However, this cannot be satisfied for <math>\sqrt{3} < \lambda \leq \frac{30}{9}</math>.
 +
Therefore, there is no feasible solution for <math>b \geq 9</math>.
 +
Therefore, we only need to consider <math>b \leq 8</math>.
 +
 +
\item We eliminate <math>a</math> that are not relatively prime to <math>b</math>.
 +
 +
\item We use the following criteria to quickly eliminate <math>a</math> that make <math>a^2 + b^2</math> a composite number.
 +
\begin{enumerate}
 +
\item For <math>b \equiv 1 \pmod{2}</math>, we eliminate <math>a</math> satisfying <math>a \equiv 1 \pmod{2}</math>.
 +
\item For <math>b \equiv \pm 1 \pmod{5}</math> (resp. <math>b \equiv \pm 2 \pmod{5}</math>), we eliminate <math>a</math> satisfying <math>a \equiv \pm 2 \pmod{5}</math> (resp. <math>a \equiv \pm 1 \pmod{5}</math>).
 +
\end{enumerate}
 +
 +
\item For the remaining <math>\left( b, a \right)</math>, check whether (8) and the condition that <math>a^2 + b^2</math> is prime are both satisfied.
 +
 +
The first feasible solution is <math>b = 5</math> and <math>a = 18</math>.
 +
Thus, <math>a^2 + b^2 = 349</math>.
 +
 +
\item For the remaining search, given <math>b</math>, we only search for <math>a \geq \sqrt{349 - b^2}</math>.
 +
\end{enumerate}
 +
 +
Following the above search criteria, we find the final answer as <math>b = 5</math> and <math>a = 18</math>.
 +
Thus, the largest prime <math>p</math> is <math>p = a^2 + b^2 = \boxed{\textbf{(349) }}</math>.
 +
 +
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Revision as of 12:28, 8 February 2023

Problem 15

Find the largest prime number $p<1000$ for which there exists a complex number $z$ satisfying

  • the real and imaginary parts of $z$ are integers;
  • $|z|=\sqrt{p}$, and
  • there exists a triangle with side lengths $p$, the real part of $z^{3}$, and the imaginary part of $z^{3}$.

Answer: 349

Suppose $z=a+bi$; notice that $\arg(z^{3})\approx 45^{\circ}$, so by De Moivre’s theorem $\arg(z)\approx 15^{\circ}$ and $\tfrac{b}{a}\approx tan(15^{\circ})=2-\sqrt{3}$. Now just try pairs $(a, b)=(t, \left(2-\sqrt{3}\right)t)$ going down from $t=\lfloor\sqrt{1000}\rfloor$, writing down the value of $a^{2}+b^{2}$ on the right; and eventually we arrive at $(a, b)=(18, 5)$ the first time $a^{2}+b^{2}$ is prime. Therefore, $p=\boxed{349}$.

Solution

Denote $z = a + i b$. Thus, $a^2 + b^2 = p$.

Thus, \[ z^3 = a \left( a^2 - 3 b^2 \right) + i b \left( - b^2 + 3 a^2 \right) . \]

Because $p$, ${\rm Re} \left( z^3 \right)$, ${\rm Im} \left( z^3 \right)$ are three sides of a triangle, we have ${\rm Re} \left( z^3 \right) > 0$ and ${\rm Im} \left( z^3 \right) > 0$. Thus, \begin{align*} a \left( a^2 - 3 b^2 \right) & > 0 , \hspace{1cm} (1) \\ b \left( - b^2 + 3 a^2 \right) & > 0. \hspace{1cm} (2) \end{align*}

Because $p$, ${\rm Re} \left( z^3 \right)$, ${\rm Im} \left( z^3 \right)$ are three sides of a triangle, we have the following triangle inequalities: \begin{align*} {\rm Re} \left( z^3 \right) + {\rm Im} \left( z^3 \right) & > p \hspace{1cm} (3) \\ p + {\rm Re} \left( z^3 \right) & > {\rm Im} \left( z^3 \right) \hspace{1cm} (4) \\ p + {\rm Im} \left( z^3 \right) & > {\rm Re} \left( z^3 \right) \hspace{1cm} (5) \end{align*}

We notice that $| z^3 | = p^{3/2}$, and ${\rm Re} \left( z^3 \right)$, ${\rm Im} \left( z^3 \right)$, and $| z^3 |$ form a right triangle. Thus, ${\rm Re} z^3 + {\rm Im} z^3 > p^{3/2}$. Because $p > 1$, $p^{3/2} > p$. Therefore, (3) holds.

Conditions (4) and (5) can be written in the joint form as \[ \left| {\rm Re} \left( z^3 \right) - {\rm Im} \left( z^3 \right) \right| < p . \hspace{1cm} (4) \]

We have \begin{align*} {\rm Re} \left( z^3 \right) - {\rm Im} \left( z^3 \right) & = \left( a^3 - 3 a b^2 \right) - \left( - b^3 + 3 a^2 b \right) \\ & = \left( a + b \right) \left( a^2 - 4 ab + b^2 \right) \end{align*} and $p = a^2 + b^2$.

Thus, (5) can be written as \[ \left| \left( a + b \right) \left( a^2 - 4 ab + b^2 \right) \right| < a^2 + b^2 . \hspace{1cm} (6) \]

Therefore, we need to jointly solve (1), (2), (6). From (1) and (2), we have either $a, b >0$, or $a, b < 0$. In (6), by symmetry, without loss of generality, we assume $a, b > 0$.

Thus, (1) and (2) are reduced to \[ a > \sqrt{3} b . \hspace{1cm} (7) \]

Let $a = \lambda b$. Plugging this into (6), we get \begin{align*} \left| \left( \left( \lambda - 2 \right)^2 - 3 \right) \right| < \frac{1}{b} \frac{\lambda^2 + 1}{\lambda + 1} . \hspace{1cm} (8) \end{align*}

Because $p= a^2 + b^2$ is a prime, $a$ and $b$ are relatively prime.

Therefore, we can use (7), (8), $a^2 + b^2 <1000$, and $a$ and $b$ are relatively prime to solve the problem.

To facilitate efficient search, we apply the following criteria: \begin{enumerate} \item To satisfy (7) and $a^2 + b^2 < 1000$, we have $1 \leq b \leq 15$. In the outer layer, we search for $b$ in a decreasing order. In the inner layer, for each given $b$, we search for $a$. \item Given $b$, we search for $a$ in the range $\sqrt{3} b < a < \sqrt{1000 - b^2}$. \item We can prove that for $b \geq 9$, there is no feasible $a$. The proof is as follows.

For $b \geq 9$, to satisfy $a^2 + b^2 < 1000$, we have $a \leq 30$. Thus, $\sqrt{3} < \lambda \leq \frac{30}{9}$. Thus, the R.H.S. of (8) has the following upper bound \begin{align*} \frac{1}{b} \frac{\lambda^2 + 1}{\lambda + 1} & < \frac{1}{b} \frac{\lambda^2 + \lambda}{\lambda + 1} \\ & = \frac{\lambda}{b} \\ & \leq \frac{\frac{30}{9}}{9} \\ & < \frac{10}{27} . \end{align*}

Hence, to satisfy (8), a necessary condition is \begin{align*} \left| \left( \left( \lambda - 2 \right)^2 - 3 \right) \right| < \frac{10}{27} . \end{align*}

However, this cannot be satisfied for $\sqrt{3} < \lambda \leq \frac{30}{9}$. Therefore, there is no feasible solution for $b \geq 9$. Therefore, we only need to consider $b \leq 8$.

\item We eliminate $a$ that are not relatively prime to $b$.

\item We use the following criteria to quickly eliminate $a$ that make $a^2 + b^2$ a composite number. \begin{enumerate} \item For $b \equiv 1 \pmod{2}$, we eliminate $a$ satisfying $a \equiv 1 \pmod{2}$. \item For $b \equiv \pm 1 \pmod{5}$ (resp. $b \equiv \pm 2 \pmod{5}$), we eliminate $a$ satisfying $a \equiv \pm 2 \pmod{5}$ (resp. $a \equiv \pm 1 \pmod{5}$). \end{enumerate}

\item For the remaining $\left( b, a \right)$, check whether (8) and the condition that $a^2 + b^2$ is prime are both satisfied.

The first feasible solution is $b = 5$ and $a = 18$. Thus, $a^2 + b^2 = 349$.

\item For the remaining search, given $b$, we only search for $a \geq \sqrt{349 - b^2}$. \end{enumerate}

Following the above search criteria, we find the final answer as $b = 5$ and $a = 18$. Thus, the largest prime $p$ is $p = a^2 + b^2 = \boxed{\textbf{(349) }}$.

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

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