Hi. My name is Murray and I am a grade 10 math student. The question is very difficult and goes as follows:

Inside a 1 m by 1 m square box in the xy-plane, there are finitely many line segments, whose lengths sum to exactly 10 m. Show that there exists a straight line in the plane which crosses at least six of these line segments. (Hint: first, show that there exists a straight line in the plane which crosses at least five of these line segments.)

This question is the last question on last year's mathcamp entrance quiz and I would appreciate any help you could give (my math teachers don't even know where to start)

Thanks alot

For the line L(i), let x(i) denote its length along the x-axis, and y(i) its length along the y-axis. Then the length of L(i) is the square root of x(i)² + y(i)², but most importantly we have length(L(i)) <= x(i) + y(i).

The sum of all lengths of lines is 10, which means that the sum of all x(i)s is at least 5, or the sum of all y(i)s is at least 5.

In the first case, you will be able to find a vertical straight line that crosses at least 5 lines. Do you see why? And do you know how to improve it to 6 lines?