a ≤ cos A + √3 sin A.
(cm+1 - ck)(cm+2 - ck) ... (cm+n - ck)
is divisible by the product c1c2 ... cn.
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A1. The parallelogram ABCD has AB = a, AD = 1, angle BAD = A, and the triangle ABD has all angles acute. Prove that circles radius 1 and center A, B, C, D cover the parallelogram iff
a ≤ cos A + √3 sin A. |
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| A2. Prove that a tetrahedron with just one edge length greater than 1 has volume at most 1/8. |
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A3. Let k, m, n be natural numbers such that m + k + 1 is a prime greater than n + 1. Let cs = s(s+1). Prove that:
(cm+1 - ck)(cm+2 - ck) ... (cm+n - ck) is divisible by the product c1c2 ... cn. |
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| B1. A0B0C0 and A1B1C1 are acute-angled triangles. Construct the triangle ABC with the largest possible area which is circumscribed about A0B0C0 (BC contains A0, CA contains B0, and AB contains C0) and similar to A1B1C1. |
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| B2. a1, ... , a8 are reals, not all zero. Let cn = a1n + a2n + ... + a8n for n = 1, 2, 3, ... . Given that an infinite number of cn are zero, find all n for which cn is zero. |
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| B3. In a sports contest a total of m medals were awarded over n days. On the first day one medal and 1/7 of the remaining medals were awarded. On the second day two medals and 1/7 of the remaining medals were awarded, and so on. On the last day, the remaining n medals were awarded. How many medals were awarded, and over how many days? |
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