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Indonesia Regional MO 2007 Part B

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October 2, 2021
algebranumber theorycombinatoricsgeometryIndonesia Regional MO

Problem Statement

[url=https://artofproblemsolving.com/community/c4h2372294p19397629]p1. Let ABCDABCD be a quadrilateral with AB=BC=CD=DAAB = BC = CD = DA. (a) Prove that point AA must be outside the triangle BCDBCD. (b) Prove that every pair of opposite sides of ABCDABCD is always parallel.
p2. Suppose aa and b b are two natural numbers, one of which is not a multiple of the other. Let's also say that lcm(a,b)lcm(a, b) is a 22-digit number, while gcd(a,b)gcd(a, b) can be obtained by reversing the order of numbers in lcm(a,b)lcm(a, b). Find the largest possible b b.
p3. Find all real numbers xx that satisfy x4−4x3+5x2−4x+1=0x^4 -4x^3 + 5x^2 - 4x + 1 = 0
[url=https://artofproblemsolving.com/community/c6h2372255p19397376]p4. In acute triangle ABCABC, ADAD, BEBE and CFCF are the altitudes, with D, E, F on sides BCBC, CACA, and ABAB, respectively. Prove that DE+DF≤BCDE + DF \le BC
p5. The numbers 1,2,3,...,15,161, 2, 3,..., 15, 16 are arranged on a 4×44 \times 4. For i=1,2,3,4i = 1, 2, 3, 4, let bib_i be the sum of the numbers in the ii-th row and kik_i is the sum of the numbers in column ii. Also, let d1d_1 and d2d_2 be the sum of the numbers on the two diagonals. The arrangement can be called antimagic if b1b_1, b2b_2, b3b_3, b4b_4, k1k_1, k2k_2, k3k_3, k4k_4, d1d_1, d2d_2 can be arranged into ten consecutive numbers. Determine the largest number among these ten consecutive numbers that can be obtained from an antimagic.
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