Question from PRMO 2019.

Question

An ant leaves the anthill for its morning exercise. It walks 4 feet east and then makes a 160° turn to the right and walks 4 more feet. It then makes another 160° turn to the right and walks 4 more feet. If the ant continues this pattern until it reaches the anthill again, what is the distance in feet it would have travelled?

Well I have the solution but I am not able to understand it. Let $A_0 (0,0)$,$A_1 (4\cos0^\circ, 4\sin0^\circ)$, $A_2 (4\cos0^\circ + 4\cos160^\circ, 4\sin 0^\circ + 4\sin160^\circ)$,......, $A_n = (0,0)$ which gives $$4(\cos 0^\circ + \cos160^\circ +......+\cos(160(n-1))^\circ) = 0$$ and $$4(\sin 0^\circ +\sin 160^\circ +..... +\sin(160(n-1))^\circ) = 0$$ which in turn, gives $$\sin((80n)^\circ) = 0$$
$$n = 9$$ Then distance covered =$4×9$ =$36\space \text{feet}$.

Please explain how do we get that $\cos 0$ and $\cos 60$ etc. in it. This question has been asked in PRMO 2019 in India

Answer 1

For instance, consider the variation in the $x$-coordinate of the position of the ant. It starts by walking $4$ units in a direction making an angle $0^\circ$ with the horizontal, so $\Delta x_1=4\cos0^\circ$, then walks $4$ units in a direction making an angle $160^\circ$ with the horizontal, and $\Delta x_2 =4\cos 160^\circ$. For the third turn, the direction would be along ($0^\circ +160^\circ+160^\circ$), and $\Delta x_3 = 4\cos(160\cdot 2)$. In general, you can deduce that the angle along which the ant walks on the $n^{th}$ turn would be the sum of the angles turned before, plus $160^\circ$. This gives $\Delta x_n = 4\cos\left(160(n-1)\right)$. We need $$\sum_{i=1}^n \Delta x_i =0 \implies \sum_{i=1}^n \cos\left(160(i-1)\right)=0 \implies n=9$$ Similarly for the $\Delta y_i$.

Answer 2

I found an interesting method to solve this problem, which is very similar to the way we solved this problem from IOQM 2020-21: Credits: Allen Career Institute

We can use a similar analogy in this question by assuming two concentric circles, to one of which the paths followed by the ant are tangent and equal chords of other.

Here AB is 4 and angle formed is (180-160)=20

Then using similar calculations and method we get Total Number of chords = 360/gcd(160, 360)

*Just one problem: I am unable to see a way to prove that all these would be tangent to the inner circle. I first saw the answer and then thought about this method. It fortunately works, but I cannot understand how. I am open to comments and remarks.

Answer 3

You need not go into applying the coordinate system and trigonometry here.

Much of PRMO is based on deduction and observations. By observation, you can see that the ant will form a star en route its path to the starting point with each internal angle of the star being $20^{\circ}$. Hence you can calculate the number of sides in the star to be $\frac{180^{\circ}}{20^\circ}=9$,

$\because\space$ The sum of all the internal acute angles of a regular star $=180^{\circ}$, which can be quite easily proven.

Hope this methodology clears your doubt!