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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