| 1. The convex polygon P0P1 ... Pn is divided into triangles by drawing non-intersecting diagonals. Show that the triangles can be labeled with the numbers 1, 2, ... , n-1 so that the triangle labeled i contains the vertex Pi (for each i). | |
| 2. For each prime dividing a positive integer n, take the largest power of the prime not exceeding n and form the sum of these prime powers. For example, if n = 100, the sum is 26 + 52 = 89. Show that there are infinitely many n for which the sum exceeds n. | |
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3. Vertex A of the triangle ABC is reflected in the opposite side to give A'. The points B' and C' are defined similarly. Show that the area of A'B'C' is less than 5 times the area of ABC.
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The original problems are in Hungarian. Many thanks to Péter Dombi for the translation.
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(C) John Scholes
jscholes@kalva.demon.co.uk
19 Apr 2003
Last corrected/updated 19 Apr 2003