BC/PD + CA/PE + AB/PF.
F(n,r) = (n+1)/(r+1).
(b) For which n > 2 is there exactly one set having this property?
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A1. P is a point inside the triangle ABC. D, E, F are the feet of the perpendiculars from P to the lines BC, CA, AB respectively. Find all P which minimise:
BC/PD + CA/PE + AB/PF. |
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A2. Take r such that 1 ≤ r ≤ n, and consider all subsets of r elements of the set {1, 2, ... , n}. Each subset has a smallest element. Let F(n,r) be the arithmetic mean of these smallest elements. Prove that:
F(n,r) = (n+1)/(r+1). |
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| A3. Determine the maximum value of m2 + n2, where m and n are integers in the range 1, 2, ... , 1981 satisfying (n2 - mn - m2)2 = 1. |
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B1. (a) For which n > 2 is there a set of n consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining n - 1 numbers?
(b) For which n > 2 is there exactly one set having this property? |
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| B2. Three circles of equal radius have a common point O and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incenter and the circumcenter of the triangle are collinear with the point O. |
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| B3. The function f(x,y) satisfies: f(0,y) = y + 1, f(x+1,0) = f(x,1), f(x+1,y+1) = f(x,f(x+1,y)) for all non-negative integers x, y. Find f(4, 1981). |
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