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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.

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Ladder and Cube

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Rolling Along the Trail

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

The possible scores that can be obtained are: $1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,25,30,36$.

The third and fourth scores differ by $11$. The only pairs of numbers that do this are $(1,12)$, $(4,15)$, $(5,16)$, $(9,20)$ and $(25,36)$.

When these pairs are completed into sequences, they become:

Of these, only the fourth one uses only accessible numbers, so the sequence is $10,15,9,20,12$.

This problem is taken from the UKMT Mathematical Challenges.
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