Try this beautiful problem from the Pre-RMO II, 2019, Question 26, based on Distance travelled.

## Distance travelled – Problem 26

A friction-less board has the shape of an equilateral triangle of side length 1 meter with bouncing walls along the sides. A tiny super bouncy ball is fired from vertex A towards the side BC. The ball bounces off the walls of the board nine times before it hits a vertex for the first time. The bounces are such that the angle of incidence equals the angle of reflection. The distance travelled by the ball in meters is of the form \(\sqrt{N}\), where N is an integer

- is 107
- is 31
- is 840
- cannot be determined from the given information

**Key Concepts**

Equation

Algebra

Integers

## Check the Answer

But try the problem first…

Answer: is 31.

PRMO II, 2019, Question 26

Higher Algebra by Hall and Knight

## Try with Hints

First hint

x= length of line segment

and by cosine law on triangle of side x, 1, 5 and 120 (in degrees) as one angle gives

\(x^2=5^2+1^2-2 \times 5 \times 1 cos 120^\circ\)

\(=25+1+5\)

Second Hint

or, x=\(\sqrt{31}\)

or, N=31

Final Step

Folding the triangle continuously each time of reflection creates the above diagram. 9 points of reflection can be seen in the diagram. Thus root (N) is the length of line which is root (31). Thus N=31 is the answer.

## Other useful links

- https://www.cheenta.com/rational-number-and-integer-prmo-2019-question-9/
- https://www.youtube.com/watch?v=lBPFR9xequA

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