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Golden Thoughts

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

From All Corners

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.

Star Gazing

Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.

One and Three

Age 14 to 16 Challenge Level:
How about using the fact that each boat makes a journey to the first meeting and the distance each travels is related to its speed? The boats continue without any change of speed, from the first to the next meeting and again those distances are related to the speeds, speeds which are still as they were before.