(a) Prove that 0 ≤ bn < 2.
(b) Given c satisfying 0 ≤ c < 2, prove that we can find an so that bn > c for all sufficiently large n.
(AB + BC + CA)2 ≤ 6(AD2 + BD2 + CD2).
When do we have equality?
| A1. M is any point on the side AB of the triangle ABC. r, r1, r2 are the radii of the circles inscribed in ABC, AMC, BMC. q is the radius of the circle on the opposite side of AB to C, touching the three sides of AB and the extensions of CA and CB. Similarly, q1 and q2. Prove that r1r2q = rq1q2. |
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| A2. We have 0 ≤ xi < b for i = 0, 1, ... , n and xn > 0, xn-1 > 0. If a > b, and xnxn-1...x0 represents the number A base a and B base b, whilst xn-1xn-2...x0 represents the number A' base a and B' base b, prove that A'B < AB'. |
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A3. The real numbers a0, a1, a2, ... satisfy 1 = a0 ≤ a1 ≤ a2 ≤ ... . b1, b2, b3, ... are defined by bn = ∑1≤k≤n (1 - ak-1/ak)/√ak.
(a) Prove that 0 ≤ bn < 2. (b) Given c satisfying 0 ≤ c < 2, prove that we can find an so that bn > c for all sufficiently large n. |
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| B1. Find all positive integers n such that the set {n, n+1, n+2, n+3, n+4, n+5} can be partitioned into two subsets so that the product of the numbers in each subset is equal. |
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B2. In the tetrahedron ABCD, angle BDC = 90o and the foot of the perpendicular from D to ABC is the intersection of the altitudes of ABC. Prove that:
(AB + BC + CA)2 ≤ 6(AD2 + BD2 + CD2). When do we have equality? |
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| B3. Given 100 coplanar points, no 3 collinear, prove that at most 70% of the triangles formed by the points have all angles acute. |
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