3. Chuk and Gek need to divide a triangular cake between themselves. Gek sets the condition that he will cut his portion with a straight cut, and Chuk agrees to this on the condition that he can mark a point $P$ in advance, through which the cut must pass. The cake has the same thickness everywhere and is uniform in taste. What strategies should Chuk and Gek adopt to maximize their portions? How much extra will Gek get if both adopt optimal strategies?
Solution. Chuk's strategy is to choose point $P$ at the centroid of the triangle, while Gek's strategy is to cut parallel to one of the sides of the triangle. If both adhere to these strategies, the cake will be divided between Chuk and Gek in the ratio $4: 5$.