| 1. Find all primes which can be written both as a sum of two primes and as a difference of two primes. |
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| 2. P is a fixed point in the plane. A, B, C are points such that PA = 3, PB = 5, PC = 7 and the area ABC is as large as possible. Show that P must be the orthocenter of ABC. |
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| 3. Let N be the natural numbers and N' = N ∪ {0}. Find all functions f:N→N' such that f(xy) = f(x) + f(y), f(30) = 0 and f(x) = 0 for all x = 7 mod 10. |
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| 4. Two triangles have the same incircle. Show that if a circle passes through five of the six vertices of the two triangles, then it also passes through the sixth. | |
| 5. A figure on a computer screen shows n points on a sphere, no four coplanar. Some pairs of points are joined by segments. Each segment is colored red or blue. For each point there is a key that switches the colors of all segments with that point as endpoint. For every three points there is a sequence of key presses that makes the three segments between them red. Show that it is possible to make all the segments on the screen red. Find the smallest number of key presses that can turn all the segments red, starting from the worst case. |
To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
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(C) John Scholes
jscholes@kalva.demon.co.uk
26 August 2003
Last corrected/updated 26 Aug 03