|ai - a1| x1 + |ai - a2| x2 + |ai - a3| x3 + |ai - a4| x4 = 1, i = 1, 2, 3, 4, where ai are distinct reals.
| A1. Problems A, B and C were posed in a mathematical contest. 25 competitors solved at least one of the three. Amongst those who did not solve A, twice as many solved B as C. The number solving only A was one more than the number solving A and at least one other. The number solving just A equalled the number solving just B plus the number solving just C. How many solved just B? |
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| A2. Prove that if BC + AC = tan C/2 (BC tan A + AC tan B), then the triangle ABC is isosceles. |
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| A3. Prove that a point in space has the smallest sum of the distances to the vertices of a regular tetrahedron iff it is the center of the tetrahedron. |
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| B1. Prove that 1/sin 2x + 1/sin 4x + ... + 1/sin 2nx = cot x - cot 2nx for any natural number n and any real x (with sin 2nx non-zero). |
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B2. Solve the equations:
|ai - a1| x1 + |ai - a2| x2 + |ai - a3| x3 + |ai - a4| x4 = 1, i = 1, 2, 3, 4, where ai are distinct reals. |
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| B3. Take any points K, L, M on the sides BC, CA, AB of the triangle ABC. Prove that at least one of the triangles AML, BKM, CLK has area ≤ 1/4 area ABC. |
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© John Scholes
jscholes@kalva.demon.co.uk
21 Sep 1998
Last corrected/updated 26 Sep 2003