Solution Consider any pentagon coloured red and blue in any way. Claim There is a line of symmetry from a vertex to the opposite side (with respect to vertex colour). WLOG let the number of red vertices be $\geq 3$. Case 1 (3 red). If all the vertices are adjacent, pick the middle. If there is a gap between the vertices (two together and one on its own), pick the non-together vertex. Case 2 (4 red). Pick the blue vertex. Case 3 (5 red). Pick any vertex. With the chosen vertex, the line of symmetry from there to the opposite side exists. Claim. The vertex of symmetry will always be blue after it has been recoloured. To be a line of symmetry, the colours on either side of the vertex must be the same, thus it is recoloured blue. Claim. A line of symmetry will always remain a line of symmetry across recolourings. Consider any vertex along with its corresponding vertex from the line of symmetry. As they are the same colour, and their pairs of corresponding neighbors have the same colour by symmetry, when they are recoloured, they are both changed to the same colour. Thus, the corresponding vertices will always be symmetry, and the line of symmetry remains across colourings. This finishes the problem, as the vertex of symmetry will be changed to blue, and will remain blue over recolourings.