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Indonesia Regional MO 2006 Part B

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October 2, 2021
algebranumber theorycombinatoricsgeometryIndonesia Regional MO

Problem Statement

[url=https://artofproblemsolving.com/community/c6h2372254p19397375]p1. Suppose that triangle ABCABC is a right angled triangle at BB. The altitude from BB intersects the side ACAC at point DD. If the points EE and FF are the midpoints of BDBD and CDCD, respectively, prove that AE⊥BFAE \perp BF.
p2. Let mm be a natural number that satisfy 1003<m<20061003 < m < 2006. Given the set of natural numbers S={1,2,3,...,m}S = \{1, 2, 3,..., m\}, how many members of SS must be selected so that there is always at least one pair of selected elements has sum 20062006 ?
p3. Let d=gcd(7n+5,5n+4)d = gcd (7n + 5, 5n + 4), where n is a natural number. (a) Prove that for every natural number nn, d=1d = 1 or 33. (b) Prove that d=3d = 3 if and only if n=3k+1n = 3k + 1, for a natural number kk.
p4. Win has two coins. He will do the following procedure over and over again as long as he still has coins: toss all the coins he has at the same time; every coin that appears with a number side will be given to Albert. Determine the probability that Win will repeat this procedure more than three times.
p5. Let a,b,ca, b, c be natural numbers. If all the three roots of the equation x2−2ax+b=0x^2 - 2ax + b = 0 x2−2bx+c=0x^2 -2bx + c = 0 x2−2cx+a=0x^2 - 2cx + a = 0 are natural numbers, determine a,ba, b and cc.
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