Difference between revisions of "2020 AIME I Problems/Problem 8"

(Solution 1 (Coordinates))
(Solution 2 (Complex))
Line 26: Line 26:
 
We put the ant in the complex plane, with its first move going in the positive real direction.
 
We put the ant in the complex plane, with its first move going in the positive real direction.
 
Take
 
Take
<cmath>|\sum_{k=0}^{\infty} (5\frac{e^{k\pi i / 3}}{2^i})|^2</cmath>
+
<cmath>|\sum_{k=0}^{\infty} (5\frac{e^{k\pi i / 3}}{2^k})|^2</cmath>
 
and this is an infinite geometric series. Summing using <math>\frac{a}{1-r}</math> gives <math>\boxed{103}.</math> ~awang11
 
and this is an infinite geometric series. Summing using <math>\frac{a}{1-r}</math> gives <math>\boxed{103}.</math> ~awang11
  

Revision as of 16:23, 12 March 2020

Note: Please do not post problems here until after the AIME.

Problem

Solution 1 (Coordinates)

We plot this on the coordinate grid with point $O$ as the origin. We will keep a tally of the x-coordinate and y-coordinate separately.

First move: The ant moves right $5$. Second move: We use properties of a $30-60-90$ triangle to get $\frac{5}{4}$ right, $\frac{5\sqrt{3}}{4}$ up. Third move: $\frac{5}{8}$ left, $\frac{5\sqrt{3}}{8}$ up. Fourth move: $\frac{5}{8}$ left. Fifth move: $\frac{5}{32}$ left, $\frac{5\sqrt{3}}{32}$ down. Sixth move: $\frac{5}{64}$ right, $\frac{5\sqrt{3}}{64}$ down.

Total of x-coordinate: $5 + \frac{5}{4} - \frac{5}{8} - \frac{5}{8}  - \frac{5}{32} + \frac{5}{64} = \frac{315}{64}$. Total of y-coordinate: $0 + \frac{5\sqrt{3}}{4} + \frac{5\sqrt{3}}{8} + 0 - \frac{5\sqrt{3}}{32} - \frac{5\sqrt{3}}{64} = \frac{105\sqrt{3}}{64}$.

After this cycle of six moves, all moves repeat with a factor of $(\frac{1}{2})^6 = \frac{1}{64}$. Using the formula for a geometric series, multiplying each sequence by $\frac{1}{1-\frac{1}{64}} = \frac{64}{63}$ will give us the point $P$.

$\frac{315}{64} \cdot \frac{64}{63} = 5$, $\frac{105\sqrt{3}}{64} \cdot \frac{64}{63} = \frac{5\sqrt{3}}{3}$. Therefore, the coordinates of point $P$ are $(5,\frac{5\sqrt{3}}{3})$, so using the Pythagorean Theorem, $OP^2 = \frac{100}{3}$, for an answer of $\boxed{103}$.

-molocyxu

Solution 2 (Complex)

We put the ant in the complex plane, with its first move going in the positive real direction. Take \[|\sum_{k=0}^{\infty} (5\frac{e^{k\pi i / 3}}{2^k})|^2\] and this is an infinite geometric series. Summing using $\frac{a}{1-r}$ gives $\boxed{103}.$ ~awang11

See Also

2020 AIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 7
Followed by
Problem 9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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