| 1. The positive integers are grouped as follows: 1, 2+3, 4+5+6, 7+8+9+10, ... Find the value of the nth sum. |
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| 2. Show that cos x2 + cos y2 - cos xy < 3 for reals x, y. |
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| 3. The equations 2x1 - x2 = 1, -x1 + 2x2 - x3 = 1, -x2 + 2x3 - x4 = 1, -x3 + 3x4 - x5 = 1, ... , -xn-2 + 2xn-1 - xn = 1, -xn-1 + 2xn = 1 have a solution in positive integers xi. Show that n must be even. |
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| 4. C, C' are concentric circles with radii R, R'. A rectangle has two adjacent vertices on C and the other two vertices on C'. Find its sides if its area is as large as possible. | |
| 5. Show that a unit square can be covered with three equal disks with radius < 1/√2. What is the smallest possible radius? | |
| 6. Show that the only real solution to x(x+y)2 = 9, x(y3 - x3) = 7 is x = 1, y = 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
8 Oct 2003
Last corrected/updated 8 Oct 03