Difference between revisions of "2022 AIME I Problems/Problem 4"

Revision as of 22:12, 17 February 2022

Problem

Let $w = \dfrac{\sqrt{3} + i}{2}$ and $z = \dfrac{-1 + i\sqrt{3}}{2},$ where $i = \sqrt{-1}.$ Find the number of ordered pairs $(r,s)$ of positive integers not exceeding $100$ that satisfy the equation $i \cdot w^r = z^s.$

Solution

We rewrite $w$ and $z$ in polar form: $\begin{align*} w &= e^{i\cdot\frac{\pi}{6}}, \\ z &= e^{i\cdot\frac{2\pi}{3}}. \end{align*}$ The equation $i \cdot w^r = z^s$ becomes $\begin{align*} e^{i\cdot\frac{\pi}{2}} \cdot \left(e^{i\cdot\frac{\pi}{6}}\right)^r &= \left(e^{i\cdot\frac{2\pi}{3}}\right)^s \\ e^{i\left(\frac{\pi}{2}+\frac{\pi}{6}r\right)} &= e^{i\left(\frac{2\pi}{3}s\right)} \\ \frac{\pi}{2}+\frac{\pi}{6}r &= \frac{2\pi}{3}s+2\pi k \\ 3+r &= 4s+12k \\ 3+r &= 4(s+3k). \end{align*}$ for some integer $k.$

Since $4\leq 3+r\leq 103$ and $4\mid 3+r,$ we conclude that $\begin{align*} 3+r &\in \{4,8,12,\ldots,100\}, \\ s+3k &\in \{1,2,3,\ldots,25\}. \end{align*}$ Note that the values for $s+3k$ and the values for $r$ have one-to-one correspondence.

We apply casework to the values for $s+3k:$

$s+3k\equiv0\pmod{3}$

There are $8$ values for $s+3k,$ so there are $8$ values for $r.$ It follows that $s\equiv0\pmod{3},$ so there are $33$ values for $s.$

There are $8\cdot33=264$ ordered pairs $(r,s)$ in this case.

$s+3k\equiv1\pmod{3}$

There are $9$ values for $s+3k,$ so there are $9$ values for $r.$ It follows that $s\equiv1\pmod{3},$ so there are $34$ values for $s.$

There are $9\cdot34=306$ ordered pairs $(r,s)$ in this case.

$s+3k\equiv2\pmod{3}$

There are $8$ values for $s+3k,$ so there are $8$ values for $r.$ It follows that $s\equiv2\pmod{3},$ so there are $33$ values for $s.$

There are $8\cdot33=264$ ordered pairs $(r,s)$ in this case.

Together, the answer is $264+306+264=\boxed{834}.$

~MRENTHUSIASM

Solution 2

First we recognize that $w = cis(30^{\circ})$ and $z = cis(12^{\circ})$ because the cosine and sine sums of those angles give the values of $w$ and $z$ , respectively. By Demoivre's theorem, $cis(\theta)^n = cis(n\theta)$ . When you multiply by $i$ , we can think of that as rotating the complex number 90 degrees counterclockwise in the complex plane. Therefore, by the equation we know that $30r + 90$ and $120s$ land on the same angle.

This means that:

$30r + 90 \equiv 120s \pmod 360$

Which we can simplify to

$r+3 \equiv 4s \pmod 12$ .

$\begin{tabular}[t]{|l|ccccc|c|} \multicolumn{7}{c}{USAMTS Scores Round 1}\\\hline Name&\#1&\#2&\#3&\#4&\#5&Total\\\hline John Doe&5&5&3&2&1&16\\ Jane Doe&5&5&5&4&5&24\\ Richard Feynman&5&5&5&5&5&25\\\hline \end{tabular}$

Video Solution (Mathematical Dexterity)

https://www.youtube.com/watch?v=XiEaCq5jf5s

@@ Line 56: / Line 56: @@
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+\multicolumn{7}{c}{USAMTS Scores Round 1}\\\hline
-\begin{document}
+Name&\#1&\#2&\#3&\#4&\#5&Total\\\hline
-Table \ref{table:1} is an example of a referenced \LaTeX{} element.
+John Doe&5&5&3&2&1&16\\
+Jane Doe&5&5&5&4&5&24\\
-\begin{table}[h!]
+Richard Feynman&5&5&5&5&5&25\\\hline
-\centering
-\begin{tabular}{||c c c c||}
- \hline
- Col1 & Col2 & Col2 & Col3 \\ [0.5ex]
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-& 6 & 87837 & 787 \\
-& 7 & 78 & 5415 \\
-& 545 & 778 & 7507 \\
-& 545 & 18744 & 7560 \\
-& 88 & 788 & 6344 \\ [1ex]
- \hline
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-\caption{Table to test captions and labels.}
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