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The 12th Vietnam Mathematical Olympiad
1974年第十二届越南数学奥林匹克
A1. Find all positive integers n and b with 0 < b < 10 such that if an is the positive integer with n digits, all of them 1, then a2n - b an is a square.
A2. (1) How many positive integers n are such that n is divisible by 8 and n+1 is divisible by 25 ?
(2) How many positive integers n are such that n is divisible by 21 and n+1 is divisible by 165 ?
(3) Find all integers n such that n is divisible by 9, n+1 is divisible by 25 and n+2 is divisible by 4.
B1. ABC is a triangle. AH is the altitude. P, Q are the feet of the perpendiculars from P to AB, AC respectively. M is a variable point on PQ. The line through M perpendicular to MH meets the lines AB, AC at R, S respectively. Show that ARHS is cyclic. If M' is another position of M with corresponding points R', S', show that the ratio RR'/SS' is constant. Find the conditions on ABC such that if M moves at constant speed along PQ, then the speeds of R along AB and S along AC are the same. The point K on the line HM is on the other side of M to H and satisfies KM = HM. The line through K perpendicular to PQ meets the line RS at D. Show that if ∠A = 90o, then ∠BHR = ∠DHR.
B2. C is a cube side 1. The 12 lines containing the sides of the cube meet at plane p in 12 points. What can you say about the 12 points?
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