Solutions - Georg Mohr-Konkurrencen

7. Given a parallelogram ABCD . A circle passing through A meets the linesegments AB , AC and AD at inner points M , K , N , respectively. Provethat|AB| · |AM| + |AD| · |AN| = |AK| · |AC| .8. Let ABCD be a convex quadrilateral, and let N be the midpoint of BC .Suppose further that ̸ AND = 135 ◦ . Prove that|AB| + |CD| + 1 √2 · |BC| |AD| .9. Given a rhombus ABCD , find the locus of the points P lying inside therhombus and satisfying ̸ AP D + ̸ BP C = 180 ◦ .10. In a triangle ABC , the bisector of ̸ BAC meets the side BC at the pointD . Knowing that |BD| · |CD| = |AD| 2 and ̸ ADB = 45 ◦ , determine theangles of triangle ABC .11. The real-valued function f is defined for all positive integers. For anyintegers a > 1, b > 1 with d = gcd(a, b), we have( ( a) ( b) )f(ab) = f(d) · f + f ,d dDetermine all possible values of f(2001).12. Let a 1 , a 2 , . . . , a n be positive real numbers such thatn∑n∑a 5 i = 5. Prove that a i > 3 2 .i=1i=1n∑i=1a 3 i= 3 and13. Let a 0 , a 1 , a 2 , . . . be a sequence of real numbers satisfying a 0 = 1 anda n = a ⌊7n/9⌋ + a ⌊n/9⌋ for n = 1, 2, . . .. Prove that there exists a positiveinteger k with a k