Shortlist 7
Posted: Sun Mar 28, 2010 7:44 pm
Rezolvati ecuatia:
\( (2^x+3^x+4^x)(5^x+6^x+7^x+8^x)=(4^x+5^x)(8^x+9^x+10^x) \).
Nicolae Papacu, Shortlist 2003
Rezolvare:facem calculele,dupa care facem impartire prin \( 28^x \), si ne da:
\( \frac{10}{28}^x+\frac{12}{28}^x+\frac{14}{28}^x+\frac{16}{28}^x+\frac{15}{28}^x+\frac{18}{28}^x+\frac{21}{28}^x+\frac{20}{28}^x+2*\frac{24}{28}^x+1=\frac{36}{28}^x+2*\frac{40}{28}^x+\frac{45}{28}^x+\frac{50}{28}^x \). Deci, in stanga este o functie strict descrescatoare, iar in dreapta o functie strict crescatoare, deci este cel mult o solutie unica. Care este aceea???.
\( (2^x+3^x+4^x)(5^x+6^x+7^x+8^x)=(4^x+5^x)(8^x+9^x+10^x) \).
Nicolae Papacu, Shortlist 2003
Rezolvare:facem calculele,dupa care facem impartire prin \( 28^x \), si ne da:
\( \frac{10}{28}^x+\frac{12}{28}^x+\frac{14}{28}^x+\frac{16}{28}^x+\frac{15}{28}^x+\frac{18}{28}^x+\frac{21}{28}^x+\frac{20}{28}^x+2*\frac{24}{28}^x+1=\frac{36}{28}^x+2*\frac{40}{28}^x+\frac{45}{28}^x+\frac{50}{28}^x \). Deci, in stanga este o functie strict descrescatoare, iar in dreapta o functie strict crescatoare, deci este cel mult o solutie unica. Care este aceea???.
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. Dar se poate gasi matematic, fara a folosi vreun program de calcul? 