A1. Let x_{1}, x_{2}, ... , x_{n} be real numbers in the interval [0, π] such that (1 + cos x_{1}) + (1 + cos x_{2}) + ... + (1 + cos x_{n}) is an odd integer. Show that sin x_{1} + sin x_{2} + ... + sin x_{n} ≥ 1. | |
A2. Let x_{1}, x_{2}, ... , x_{n} be positive reals with sum s. Show that (x_{1} + 1/x_{1})^{2} + (x_{2} + 1/x_{2})^{2} + ... + (x_{n} + 1/x_{n})^{2} ≥ n(n/s + s/n)^{2}. | |
A3. P is a point inside the triangle A_{1}A_{2}A_{3}. The ray A_{i}P meets the opposite side at B_{i}. C_{i} is the midpoint of A_{i}B_{i} and D_{i} is the midpoint of PB_{i}. Show that area C_{1}C_{2}C_{3} = area D_{1}D_{2}D_{3}. | |
B1. Show that for any tetrahedron it is possible to find two perpendicular planes such that if the projection of the tetrahedron onto the two planes has areas A and A', then A'/A > √2. | |
B2. Does there exist real m such that the equation x^{3} - 2x^{2} - 2x + m has three different rational roots? | |
B3. Given n > 1 and real s > 0, find the maximum of x_{1}x_{2} + x_{2}x_{3} + x_{3}x_{4} + ... + x_{n-1}x_{n} for non-negative reals x_{i} such that x_{1} + x_{2} + ... + x_{n} = s. |
To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
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© John Scholes
jscholes@kalva.demon.co.uk
23 July 2002