Putnam 2017, Problem B-1

Let \( L_1 \) and \( L_2 \) be distinct lines in the plane. Prove that \( L_1 \) and \( L_2 \) intersect if and only if, for every real number \( \lambda \neq 0 \) and every point \( P \) not on \( L_1 \) or \( L_2 \), there exist points \( A_1 \) on \( L_1 \) and \( A_2 \) on \( L_2 \) such that \( \overrightarrow{PA_2} = \lambda \overrightarrow{PA_1} \).

Solution:
Recall that \( L_1 \) and \( L_2 \) intersect if and only if they are not parallel. In one direction, suppose that \( L_1 \) and \( L_2 \) intersect. Then for any \( P \) and \( \lambda \), the dilation (homothety) of the plane by a factor of \( \lambda \) with center \( P \) carries \( L_1 \) to another line parallel to \( L_1 \) and hence not parallel to \( L_2 \). Let \( A_2 \) be the unique intersection of \( L_2 \) with the image of \( L_1 \), and let \( A_1 \) be the point on \( L_1 \) whose image under the dilation is \( A_2 \); then \( \overrightarrow{PA_2} = \lambda \overrightarrow{PA_1} \).  

In the other direction, suppose that \( L_1 \) and \( L_2 \) are parallel. Let \( P \) be any point in the region between \( L_1 \) and \( L_2 \) and take \( \lambda = 1 \). Then for any point \( A_1 \) on \( L_1 \) and any point \( A_2 \) on \( L_2 \), the vectors \( \overrightarrow{PA_1} \) and \( \overrightarrow{PA_2} \) have components perpendicular to \( L_1 \) pointing in opposite directions; in particular, the two vectors cannot be equal.  

### Reinterpretation
(by Karl Mahlburg) In terms of vectors, we may find vectors \( \vec{v}_1, \vec{v}_2 \) and scalars \( c_1, c_2 \) such that \( L_i = \{ \vec{x} \in \mathbb{R}^2 : \vec{v}_i \cdot \vec{x} = c_i \} \). The condition in the problem amounts to finding a vector \( \vec{w} \) and a scalar \( t \) such that \( P + \vec{w} \in L_1, P + \lambda \vec{w} \in L_2 \); this comes down to solving the linear system  
\[
\vec{v}_1 \cdot (P + \vec{w}) = c_1, \quad \vec{v}_2 \cdot (P + \lambda \vec{w}) = c_2
\]  
which is nondegenerate and solvable for all \( \lambda \) if and only if \( \vec{v}_1, \vec{v}_2 \) are linearly independent.