| 1. A triangle has side lengths a, b, c. Prove that a(b - c)2 + b(c - a)2 + c(a - b)2 + 4abc > a3 + b3 + c3. | |
| 2. A class has n > 1 boys and n girls. For each arrangement X of the class in a line let f(X) be the number of ways of dividing the line into two non-empty segments, so that in each segment the number of boys and girls is equal. Let the number of arrangements with f(X) = 0 be A, and the number of arrangements with f(X) = 1 be B. Show that B = 2A. | |
| 3. ABCD is a square side 10. There are four points P1, P2, P3, P4 inside the square. Show that we can always construct line segments parallel to the sides of the square of total length 25 or less, so that each Pi is linked by the segments to both of the sides AB and CD. Show that for some points Pi it is not possible with a total length less than 25. |
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