| 1. The crossed ladder problem. In an alley a ladder length 3 is propped up against one wall with its foot at the bottom of the other wall. It crosses another ladder length 2 leaning the other way a distance 1 from the ground. How wide is the alley? [In other words we are given a quadrilateral ABCD. Angles ADC and DCB are right angles. AC = 2, BD = 3 and AC, BD intersect at X a distance 1 from CD. Find CD.] |
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| 2. Find all solutions to 2a + 3b = 5c for positive integers a, b, c. | |
| 3. By considering ∫ xn/(1-x8)dx or otherwise, prove that π = ∑i=0∞ 1/16i (4/(8i+1) - 2/(8i+4) - 1/(8i+5) - 1/(8i+6) ). | |
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4. Ten hexagonal pieces are shown. How can they be placed on a hexagonal board to give the largest possible number of monochrome loops? [The pieces may not be turned over.]
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John Scholes
jscholes@kalva.demon.co.uk
1 Mar 2003
Last corrected/updated 1 Mar 2003