| A1. Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another. |
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| A2. Find all natural numbers n the product of whose decimal digits is n2 - 10n - 22. |
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A3. a, b, c are real with a non-zero. x1, x2, ... , xn satisfy the n equations:
axi2 + bxi + c = xi+1, for 1 ≤ i < n axn2 + bxn + c = x1 Prove that the system has zero, 1 or >1 real solutions according as (b - 1)2 - 4ac is <0, =0 or >0. |
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| B1. Prove that every tetrahedron has a vertex whose three edges have the right lengths to form a triangle. |
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B2. Let f be a real-valued function defined for all real numbers, such that for some a > 0 we have
f(x + a) = 1/2 + √(f(x) - f(x)2) for all x. Prove that f is periodic, and give an example of such a non-constant f for a = 1. |
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B3. For every natural number n evaluate the sum
[(n+1)/2] + [(n+2)/4] + [(n+4)/8] + ... + [(n+2k)/2k+1] + ... , where [x] denotes the greatest integer ≤ x. |
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