| 1. Each of the numbers 1, 2, ... , 10 is colored red or blue. 5 is red and at least one number is blue. If m, n are different colors and m+n ≤ 10, then m+n is blue. If m, n are different colors and mn ≤ 10, then mn is red. Find all the colors. |
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| 2. p(x) is a polynomial such that p(y2+1) = 6y4 - y2 + 5. Find p(y2-1). |
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| 3. Are there any integral solutions to n2 + (n+1)2 + (n+2)2 = m2? |
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| 4. The vertices of a triangle are three-dimensional lattice points. Show that its area is at least ½. |
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| 5. Let f(n) be defined on the positive integers and satisfy: f(prime) = 1; f(ab) = a f(b) + f(a) b. Show that f is unique and find all n such that n = f(n). |
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| 6. Solve y(x+y)2 = 9, y(x3-y3) = 7. |
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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