Difference between revisions of "2022 AIME I Problems/Problem 4"
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e^{i\left(\frac{\pi}{2}+\frac{\pi}{6}r\right)} &= e^{i\left(\frac{2\pi}{3}s\right)}. | e^{i\left(\frac{\pi}{2}+\frac{\pi}{6}r\right)} &= e^{i\left(\frac{2\pi}{3}s\right)}. | ||
\end{align*}</cmath> | \end{align*}</cmath> | ||
− | Note that < | + | Note that |
+ | <cmath>\begin{align*} | ||
+ | \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*}</cmath> | ||
+ | for some integer <math>k.</math> | ||
+ | |||
+ | Since <math>4\leq 3+r\leq 103</math> and <math>4\mid 3+r,</math> we conclude that | ||
~MRENTHUSIASM ~bluesoul | ~MRENTHUSIASM ~bluesoul |
Revision as of 16:46, 17 February 2022
Problem
Let and where Find the number of ordered pairs of positive integers not exceeding that satisfy the equation
Solution
We rewrite and in polar form: The equation becomes Note that for some integer
Since and we conclude that
~MRENTHUSIASM ~bluesoul
See Also
2022 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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