1. At time t = 0, a lion L is standing at point O and a horse H is at point A running with speed v perpendicular to OA. The speed and direction of the horse does not change. The lion's strategy is to run with constant speed u at an angle 0 < φ < π/2 to the line LH. What is the condition on u and v for this strategy to result in the lion catching the horse? If the lion does not catch the horse, how close does he get? What is the choice of φ required to minimise this distance? | |
2. AB and CD are two fixed parallel chords of the circle S. M is a variable point on the circle. Q is the intersection of the lines MD and AB. X is the circumcenter of the triangle MCQ. Find the locus of X. What happens to X as M tends to (1) D, (2) C? Find a point E outside the plane of S such that the circumcenter of the tetrahedron MCQE has the same locus as X. | |
3. m an n are fixed positive integers and k is a fixed positive real. Show that the minimum value of x_{1}^{m} + x_{2}^{m} + x_{3}^{m} + ... + x_{n}^{m} for real x_{i} satisfying x_{1} + x_{2} + ... + x_{n} = k occurs at x_{1} = x_{2} = ... = x_{n}. |
To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
Vietnam home
© John Scholes
jscholes@kalva.demon.co.uk
16 July 2002