{"id":11660,"date":"2019-07-10T09:07:05","date_gmt":"2019-07-10T06:07:05","guid":{"rendered":"https:\/\/w3.bilkent.edu.tr\/en\/?page_id=11660"},"modified":"2019-09-13T10:41:25","modified_gmt":"2019-09-13T07:41:25","slug":"problem-of-the-month-july-2019","status":"publish","type":"page","link":"https:\/\/w3.bilkent.edu.tr\/en\/problem-of-the-month-july-2019\/","title":{"rendered":"Problem of the Month \u2013 July, August 2019"},"content":{"rendered":"

For positive integer k<\/em>, let<\/p>\n

R<\/em>n<\/em><\/sub>\u00a0= {\u2212k<\/em>, \u2212(k<\/em> \u2212 1), . . . , \u22121, 1, . . . , k<\/em> \u2212 1, k<\/em>} for n<\/em> = 2k<\/em> and
\nR<\/em>n<\/em><\/sub>\u00a0= {\u2212k<\/em>, \u2212(k<\/em> \u2212 1), . . . , \u22121, 0, 1, . . . , k<\/em> \u2212 1, k<\/em>} for n<\/em> = 2k<\/em> + 1.<\/p>\n

A device consists of several balls and red or white ropes connecting some ball pairs. A
\nlabeling is a coloring of each ball by one of the elements of R<\/em>n<\/em><\/sub>. We say that a labeling is
\ngood if colors of any two connected balls are different. We say that a labeling is sensitive
\nif the colors of any two balls connected by white rope are different and the sum of colors
\nof any two balls connected by red rope is not equal to 0.<\/p>\n

Let n<\/em> \u2265 3 be fixed. Suppose that any device which has a good labeling by R<\/em>n<\/em><\/sub> has also a
\nsensitive labeling by R<\/em>m<\/em><\/sub>. Find the smallest possible value of m<\/em> = m<\/em>(n<\/em>).<\/p>\n

 <\/p>\n

Correct Solutions by,<\/strong><\/p>\n