| 1. A triangle has sides 6, 8, 10. Show that there is a unique line which bisects the area and the perimeter. |
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| 2. Is there an integer which is doubled by moving its first digit to the end? [For example, 241 does not work because 412 is not 2 x 241.] |
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| 3. A regular 1985-gon is inscribed in a circle (so each vertex lies on the circle). Another regular 1985-gon is circumcribed about the same circle (so that each side touches the circle). Show that the sum of the perimeters of the two polygons is at least twice the circumference of the circle. [Assume tan x >= x for 0 <= x < 90 deg.] |
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| 4. Show that n! is divisible by 2n-1 iff n is a power of 2. |
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| 5. Define the real sequence x1, x2, x3, ... by x1 = k, where 1 < k < 2, and xn+1 = xn - xn2/2 + 1. Show that |xn - √2| < 1/2n for n > 2. |
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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
19 June 2002