Let P1, . . . , P1024 be distinct points marked on a circle, and let a1, a2, . . . , a1024 be distinct real numbers written on these points, respectively. For each point Q on this circle different than P1, . . . , P1024, we say that Pi is Q-good if ai is the greatest number on at least one of the two arcs PiQ on this circle. Let the score of Q be the number of Q−good points on the circle. Determine the greatest integer k such that regardless of the values of a1, a2, . . . , a1024, there exists a point Q with score at least k.
Correct Solutions by,
- Toshihiro Shimizu Kawasaki, Japan
- Mehmet Sarak Bilkent University
- Yunus Orazow 40th Specialized School, Ashkabat, Turkmenistan
- Emir Kara Gazi Anadolu Lisesi, Ankara
- Mustafa Kaynak Gaziantep
- Erenalp Yuca Bilkent University
- Abdulkadir Tanrıverdi Eskişehir
- Magnus Jakobsson Lund, Sweden
- Roger Bengtsson Lund, Sweden
- Alper Balcı Ohio State University, USA
- Özgür Soysal Bilkent University
- Ananda Raidu Rajahmundry, India