Problem
ALG-B1-M06-P021 Intersections of lines
Nine functions \(f_i(t)=u_i+v_i t\) are given, where \(u_1
Look at the lowest graph at \(t=0\) and the highest graph at \(t=0\).
Answer: no.
Since the graphs are lines and the coefficients \(v_i\) are distinct, any two graphs intersect at most once. Consider the order of the graphs by height for a very large \(t\), after all positive intersections.
The graph \(f_1\) was the lowest at \(t=0\). Each of its meetings moves it one position upward in the current order. If it has exactly four meetings, then finally it is fifth from the bottom.
Similarly, \(f_9\) was the highest at \(t=0\). Each of its meetings moves it one position downward. If it has exactly four meetings, then finally it is also fifth from the bottom. Two different graphs cannot occupy the same position. Contradiction.
Source analysis: the main objects are ordered moving objects; the obvious approach is to try to construct an example; the hidden observation is that an extreme object with four position changes must end in the middle; the key ideas are one-time intersection and extreme positions. The context is changed to an algebraic model of lines.