Problem of the Month – June 2019

Let P(x) be a non-constant polynomial with real coefficients such that all of its roots are
real numbers. Suppose that there exists a polynomial Q(x) with real coefficients such
that

(P(x))² = P(Q(x))

for all real numbers x. Determine the maximal possible number of distinct roots of P(x).

 

Correct Solutions by,

• Toshihiro Shimizu Kawasaki, Japan
• Steffen Weber Ismanıng, Germany
• Max Nilsson Lund, Sweden
• Bora Ege Duygun Bilkent University
• Vedat Deveci İstanbul
• Mehmet Yeni Tarsus, Mersin
• Halil Özkan Denizli Özel Denizli Koleji,  Denizli
• Serdar Hojayev Dashoguz, Turkmenistan
• Halil Alperen Gözeten Denizli Özel Denizli Koleji,  Denizli
• Demir Eken Bilkent University
• Bilge Köksal Bilkent University
• Ozan Kaymak Denizli Özel Denizli Koleji,  Denizli
• Roger Bengtsson Lund, Sweden
• Hakan Karakuş Antalya Yusuf Ziya Öner Fen Lisesi
• Ömer Topaloğlu Özel İzmir Fen Lisesi,  İzmir
• Mustafa Emir Çelebi İstanbul Lisesi
• İbrahim Suat Evren Denizli Özel Denizli Koleji,  Denizli
• Enes Özdemir Buca İnci Özer Tırnaklı Fen Lisesi, İzmir
• John Wright Brooklyn, New York, USA
• Jürgen Weith University of Applied Sciences, Schweinfurt, Germany

Solution: http://www.fen.bilkent.edu.tr/~cvmath/Problem/1906a.pdf