(a) ai < bi for i = 1, 2, ... , n,
(b) q < bi+1/bi < 1/q for i = 1, 2, ... , n-1,
(c) b1 + b2 + ... + bn < (a1 + a2 + ... + an)(1 + q)/(1 - q).
| A1. OP1, OP2, ... , OP2n+1 are unit vectors in a plane. P1, P2, ... , P2n+1 all lie on the same side of a line through O. Prove that |OP1 + ... + OP2n+1| ≥ 1. |
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| A2. Can we find a finite set of non-coplanar points, such that given any two points, A and B, there are two others, C and D, with the lines AB and CD parallel and distinct? |
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| A3. a and b are real numbers for which the equation x4 + ax3 + bx2 + ax + 1 = 0 has at least one real solution. Find the least possible value of a2 + b2. |
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| B1. A soldier needs to sweep a region with the shape of an equilateral triangle for mines. The detector has an effective radius equal to half the altitude of the triangle. He starts at a vertex of the triangle. What path should he follow in order to travel the least distance and still sweep the whole region? |
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| B2. G is a set of non-constant functions f. Each f is defined on the real line and has the form f(x) = ax + b for some real a, b. If f and g are in G, then so is fg, where fg is defined by fg(x) = f(g(x)). If f is in G, then so is the inverse f-1. If f(x) = ax + b, then f-1(x) = x/a - b/a. Every f in G has a fixed point (in other words we can find xf such that f(xf) = xf. Prove that all the functions in G have a common fixed point. |
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B3. a1, a2, ... , an are positive reals, and q satisfies 0 < q < 1. Find b1, b2, ... , bn such that:
(a) ai < bi for i = 1, 2, ... , n, (b) q < bi+1/bi < 1/q for i = 1, 2, ... , n-1, (c) b1 + b2 + ... + bn < (a1 + a2 + ... + an)(1 + q)/(1 - q). |
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