| 1. A triangle area T is divided into six regions by lines drawn through a point inside the triangle parallel to the sides. The three triangular regions have areas T1, T2, T3. Show that √T =√T1 + √T2 + √T3. |
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| 2. Find n > 1 so that with stamp denominations n and n+2 it is possible to obtain any value ≥ 2n+2. |
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| 3. For x ≥ 1, define pn(x) = ½(x + √(x2-1) )n + ½(x - √(x2-1) )n. Show that pn(x) ≥ 1 and pmn(x) = pm(pn(x)). |
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| 4. The pentagon ABCDE is incribed in a circle. ∠A ≤ ∠B ≤ ∠C ≤ ∠D ≤ ∠E. Show that ∠C > π/2 and that this is the best possible lower bound. |
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| 5. Show that we can divide {1, 2, 3, ... , 2n} into two disjoint parts S, T such that ∑k∈S km = ∑k∈T km for m = 0, 1, 2, ... , n-1. |
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| 6. A rectangle is constructed from 1 x 6 rectangles. Show that one of its sides is a multiple of 6. |
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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
9 Oct 2003
Last corrected/updated 9 Oct 03