8/((x1 + x2)(y1 + y2) - (z1 + z2)2) ≤ 1/(x1y1 - z12) + 1/(x2y2 - z22).
Give necessary and sufficient conditions for equality.
| A1. Prove that there are infinitely many positive integers m, such that n4 + m is not prime for any positive integer n. |
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| A2. Let f(x) = cos(a1 + x) + 1/2 cos(a2 + x) + 1/4 cos(a3 + x) + ... + 1/2n-1 cos(an + x), where ai are real constants and x is a real variable. If f(x1) = f(x2) = 0, prove that x1 - x2 is a multiple of π. |
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| A3. For each of k = 1, 2, 3, 4, 5 find necessary and sufficient conditions on a > 0 such that there exists a tetrahedron with k edges length a and the remainder length 1. |
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| B1. C is a point on the semicircle diameter AB, between A and B. D is the foot of the perpendicular from C to AB. The circle K1 is the in-circle of ABC, the circle K2 touches CD, DA and the semicircle, the circle K3 touches CD, DB and the semicircle. Prove that K1, K2 and K3 have another common tangent apart from AB. |
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| B2. Given n > 4 points in the plane, no three collinear. Prove that there are at least (n-3)(n-4)/2 convex quadrilaterals with vertices amongst the n points. |
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B3. Given real numbers x1, x2, y1, y2, z1, z2, satisfying x1 > 0, x2 > 0, x1y1 > z12, and x2y2 > z22, prove that:
8/((x1 + x2)(y1 + y2) - (z1 + z2)2) ≤ 1/(x1y1 - z12) + 1/(x2y2 - z22). Give necessary and sufficient conditions for equality. |
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