Baltic Way 1999 - Georg Mohr-Konkurrencen

where [P QRS] denotes the area of the quadrilateral P QRS .15. Let ABC be a triangle with ̸ C = 60 ◦ and |AC| < |BC|. The point Dlies on the side BC and satisfies |BD| = |AC|. The side AC is extendedto the point E where |AC| = |CE|. Prove that |AB| = |DE|.16. Find the smallest positive integer k which is representable in the formk = 19 n − 5 m for some positive integers m and n.17. Does there exist a finite sequence of integers c 1 , . . . , c n such that all thenumbers a + c 1 , . . . , a + c n are primes for more than one but not infinitelymany different integers a?18. Let m be a positive integer such that m ≡ 2 (mod 4). Show that thereexists at most one factorization√m = ab where a and b are positive integerssatisfying 0 < a − b < 5 + 4 √ 4m + 1.19. Prove that there exist infinitely many even positive integers k such that forevery prime p the number p 2 + k is composite.20. Let a, b, c and d be prime numbers such that a > 3b > 6c > 12d anda 2 − b 2 + c 2 − d 2 = 1749. Determine all possible values of a 2 + b 2 + c 2 + d 2 .Solutions1. Answer: a = b = c = 3√ 2 − 1, d = 5 3√ 2 − 1.Substituting A = a + 1, B = b + 1, C = c + 1, D = d + 1, we obtainABC = 2 (1)BCD = 10 (2)CDA = 10 (3)DAB = 10 (4)Multiplying (1), (2), (3) gives C 3 (ABD) 2 = 200, which together with (4)implies C 3 = 2. Similarly we find A 3 = B 3 = 2 and D 3 = 250. Thereforethe only solution is a = b = c = 3√ 2 − 1, d = 5 3√ 2 − 1.2. Answer: 32768 is the only such integer.3