2005年巴西数学奥林匹克.pdfVIP

Brazil National Olympiad 2005 Day 1 - 22 October 2005 1 A natural number is a palindrome when one obtains the same number when writing its digits in reverse order. For example, 481184, 131 and 2 are palindromes. Determine all pairs (m, n) of positive integers such that 111 . . . 1 × 111 . . . 1 is a palindrome. m ones n ones 2 Determine the smallest real number C such that the inequality C (x2005 + x2005 + · · · + x2005 ) ≥ x x x x x (x125 + x125 + · · · + x125)16 1 2 5 1 2 3 4 5 1 2 5 holds for all positive real numbers x , x , x , x , x . 1 2 3 4 5 3 A square is contained in a cube when all of its points are in the faces or in the interior of the cube. Determine the biggest 0 such that there exists a square of side contained in a cube with edge 1. This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page / Page 1 http://www.mathlinks.ro/ Brazil National Olympiad 2005 Day 2 - 23 October 2005 4 We have four charged batteries, four uncharged batteries and a radio which needs two charged batteries to work. Suppose we don’t know which batteries are charged and which ones are uncharged. Find the least number of attempts sufficient to make sure the radio will work. An attempt consists in putting two batte