(x12 - x3x5)(x22 - x3x5) ≤ 0
(x22 - x4x1)(x32 - x4x1) ≤ 0
(x32 - x5x2)(x42 - x5x2) ≤ 0
(x42 - x1x3)(x52 - x1x3) ≤ 0
(x52 - x2x4)(x12 - x2x4) ≤ 0
| A1. Given any set of ten distinct numbers in the range 10, 11, ... , 99, prove that we can always find two disjoint subsets with the same sum. |
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| A2. Given n > 4, prove that every cyclic quadrilateral can be dissected into n cyclic quadrilaterals. |
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| A3. Prove that (2m)!(2n)! is a multiple of m!n!(m+n)! for any non-negative integers m and n. |
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B1. Find all positive real solutions to:
(x12 - x3x5)(x22 - x3x5) ≤ 0
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| B2. f and g are real-valued functions defined on the real line. For all x and y, f(x + y) + f(x - y) = 2f(x)g(y). f is not identically zero and |f(x)| ≤ 1 for all x. Prove that |g(x)| ≤ 1 for all x. |
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| B3. Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane. |
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