Ta có \(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB % PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbba9q8qqaqFr0x % c9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8fr % Fve9Fve9Ff0dmeaabiqaceGabiGaaeaabaWaaeaaeaaakeaadaqada % qaaiaaikdadaahaaWcbeqaaiaadggaaaGccqGHRaWkdaWcaaqaaiaa % igdaaeaacaaIYaWaaWbaaSqabeaacaWGHbaaaaaaaOGaayjkaiaawM % caamaaCaaaleqabaGaaGOmaiaaicdacaaIXaGaaG4naaaakiabgsMi % JoaabmaabaGaaGOmamaaCaaaleqabaGaaGOmaiaaicdacaaIXaGaaG % 4naaaakiabgUcaRmaalaaabaGaaGymaaqaaiaaikdadaahaaWcbeqa % aiaaikdacaaIWaGaaGymaiaaiEdaaaaaaaGccaGLOaGaayzkaaWaaW % baaSqabeaacaWGHbaaaaaa!4F2D! {\left( {{2^a} + \frac{1}{{{2^a}}}} \right)^{2017}} \le {\left( {{2^{2017}} + \frac{1}{{{2^{2017}}}}} \right)^a}\)

\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB % PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbba9q8qqaqFr0x % c9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8fr % Fve9Fve9Ff0dmeaabiqaceGabiGaaeaabaWaaeaaeaaakeaacqGHsh % I3caaIYaGaaGimaiaaigdacaaI3aGaaeiBaiaab+gacaqGNbWaaSba % aSqaaiaaikdaaeqaaOWaaeWaaeaacaaIYaWaaWbaaSqabeaacaWGHb % aaaOGaey4kaSYaaSaaaeaacaaIXaaabaGaaGOmamaaCaaaleqabaGa % amyyaaaaaaaakiaawIcacaGLPaaacqGHKjYOcaWGHbGaaeiBaiaab+ % gacaqGNbWaaSbaaSqaaiaaikdaaeqaaOWaaeWaaeaacaaIYaWaaWba % aSqabeaacaaIYaGaaGimaiaaigdacaaI3aaaaOGaey4kaSYaaSaaae % aacaaIXaaabaGaaGOmamaaCaaaleqabaGaaGOmaiaaicdacaaIXaGa % aG4naaaaaaaakiaawIcacaGLPaaaaaa!58A0! \Rightarrow 2017{\rm{lo}}{{\rm{g}}_2}\left( {{2^a} + \frac{1}{{{2^a}}}} \right) \le a{\rm{lo}}{{\rm{g}}_2}\left( {{2^{2017}} + \frac{1}{{{2^{2017}}}}} \right)\)

\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB % PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbba9q8qqaqFr0x % c9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8fr % Fve9Fve9Ff0dmeaabiqaceGabiGaaeaabaWaaeaaeaaakeaacqGHsh % I3daWcaaqaaiaabYgacaqGVbGaae4zamaaBaaaleaacaaIYaaabeaa % kmaabmaabaGaaGOmamaaCaaaleqabaGaamyyaaaakiabgUcaRmaala % aabaGaaGymaaqaaiaaikdadaahaaWcbeqaaiaadggaaaaaaaGccaGL % OaGaayzkaaaabaGaamyyaaaacqGHKjYOdaWcaaqaaiaabYgacaqGVb % Gaae4zamaaBaaaleaacaaIYaaabeaakmaabmaabaGaaGOmamaaCaaa % leqabaGaaGOmaiaaicdacaaIXaGaaG4naaaakiabgUcaRmaalaaaba % GaaGymaaqaaiaaikdadaahaaWcbeqaaiaaikdacaaIWaGaaGymaiaa % iEdaaaaaaaGccaGLOaGaayzkaaaabaGaaGOmaiaaicdacaaIXaGaaG % 4naaaaaaa!58C0! \Rightarrow \frac{{{\rm{lo}}{{\rm{g}}_2}\left( {{2^a} + \frac{1}{{{2^a}}}} \right)}}{a} \le \frac{{{\rm{lo}}{{\rm{g}}_2}\left( {{2^{2017}} + \frac{1}{{{2^{2017}}}}} \right)}}{{2017}}\)

Xét hàm số \(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB % PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbba9q8qqaqFr0x % c9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8fr % Fve9Fve9Ff0dmeaabiqaceGabiGaaeaabaWaaeaaeaaakeaacaWG5b % Gaeyypa0JaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da % 9maalaaabaGaaeiBaiaab+gacaqGNbWaaSbaaSqaaiaaikdaaeqaaO % WaaeWaaeaacaaIYaWaaWbaaSqabeaacaWG4baaaOGaey4kaSYaaSaa % aeaacaaIXaaabaGaaGOmamaaCaaaleqabaGaamiEaaaaaaaakiaawI % cacaGLPaaaaeaacaWG4baaaiabg2da9maalaaabaGaaeiBaiaab+ga % caqGNbWaaSbaaSqaaiaaikdaaeqaaOWaaeWaaeaacaaI0aWaaWbaaS % qabeaacaWG4baaaOGaey4kaSIaaGymaaGaayjkaiaawMcaaiabgkHi % TiaadIhaaeaacaWG4baaaiabg2da9maalaaabaGaaeiBaiaab+gaca % qGNbWaaSbaaSqaaiaaikdaaeqaaOWaaeWaaeaacaaI0aWaaWbaaSqa % beaacaWG4baaaOGaey4kaSIaaGymaaGaayjkaiaawMcaaaqaaiaadI % haaaGaeyOeI0IaaGymaaaa!6311! y = f\left( x \right) = \frac{{{\rm{lo}}{{\rm{g}}_2}\left( {{2^x} + \frac{1}{{{2^x}}}} \right)}}{x} = \frac{{{\rm{lo}}{{\rm{g}}_2}\left( {{4^x} + 1} \right) - x}}{x} = \frac{{{\rm{lo}}{{\rm{g}}_2}\left( {{4^x} + 1} \right)}}{x} - 1\)

Ta có \(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB % PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbba9q8qqaqFr0x % c9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8fr % Fve9Fve9Ff0dmeaabiqaceGabiGaaeaabaWaaeaaeaaakeaaceWG5b % GbauaacqGH9aqpdaWcaaqaaiaaigdaaeaacaqGSbGaaeOBaiaaikda % aaWaamWaaeaadaWcaaqaamaalaaabaWaaeWaaeaacaaI0aWaaWbaaS % qabeaacaWG4baaaOGaey4kaSIaaGymaaGaayjkaiaawMcaaiaacEca % aeaacaaI0aWaaWbaaSqabeaacaWG4baaaOGaey4kaSIaaGymaaaaca % GGUaGaamiEaiabgkHiTiaabYgacaqGUbWaaeWaaeaacaaI0aWaaWba % aSqabeaacaWG4baaaOGaey4kaSIaaGymaaGaayjkaiaawMcaaaqaai % aadIhadaahaaWcbeqaaiaaikdaaaaaaaGccaGLBbGaayzxaaGaeyyp % a0ZaaSaaaeaacaaIXaaabaGaaeiBaiaab6gacaaIYaaaamaadmaaba % WaaSaaaeaacaaI0aWaaWbaaSqabeaacaWG4baaaOGaaiOlaiaabYga % caqGUbGaaGinaiaac6cacaWG4bGaeyOeI0YaaeWaaeaacaaI0aWaaW % baaSqabeaacaWG4baaaOGaey4kaSIaaGymaaGaayjkaiaawMcaaiaa % bYgacaqGUbWaaeWaaeaacaaI0aWaaWbaaSqabeaacaWG4baaaOGaey % 4kaSIaaGymaaGaayjkaiaawMcaaaqaaiaadIhadaahaaWcbeqaaiaa % ikdaaaGcdaqadaqaaiaaisdadaahaaWcbeqaaiaadIhaaaGccqGHRa % WkcaaIXaaacaGLOaGaayzkaaaaaaGaay5waiaaw2faaiabgYda8iaa % icdaaaa!76D6! y' = \frac{1}{{{\rm{ln}}2}}\left[ {\frac{{\frac{{\left( {{4^x} + 1} \right)'}}{{{4^x} + 1}}.x - {\rm{ln}}\left( {{4^x} + 1} \right)}}{{{x^2}}}} \right] = \frac{1}{{{\rm{ln}}2}}\left[ {\frac{{{4^x}.{\rm{ln}}4.x - \left( {{4^x} + 1} \right){\rm{ln}}\left( {{4^x} + 1} \right)}}{{{x^2}\left( {{4^x} + 1} \right)}}} \right] < 0\)

\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB % PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbba9q8qqaqFr0x % c9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8fr % Fve9Fve9Ff0dmeaabiqaceGabiGaaeaabaWaaeaaeaaakeaaceWG5b % GbauaacqGH9aqpdaWcaaqaaiaaigdaaeaacaqGSbGaaeOBaiaaikda % aaWaamWaaeaadaWcaaqaaiaaisdadaahaaWcbeqaaiaadIhaaaGcca % GGUaGaaeiBaiaab6gacaaI0aWaaWbaaSqabeaacaWG4baaaOGaeyOe % I0YaaeWaaeaacaaI0aWaaWbaaSqabeaacaWG4baaaOGaey4kaSIaaG % ymaaGaayjkaiaawMcaaiaabYgacaqGUbWaaeWaaeaacaaI0aWaaWba % aSqabeaacaWG4baaaOGaey4kaSIaaGymaaGaayjkaiaawMcaaaqaai % aadIhadaahaaWcbeqaaiaaikdaaaGcdaqadaqaaiaaisdadaahaaWc % beqaaiaadIhaaaGccqGHRaWkcaaIXaaacaGLOaGaayzkaaaaaaGaay % 5waiaaw2faaiabgYda8iaaicdaaaa!5B06! y' = \frac{1}{{{\rm{ln}}2}}\left[ {\frac{{{4^x}.{\rm{ln}}{4^x} - \left( {{4^x} + 1} \right){\rm{ln}}\left( {{4^x} + 1} \right)}}{{{x^2}\left( {{4^x} + 1} \right)}}} \right] < 0\) \(;% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB % PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbba9q8qqaqFr0x % c9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8fr % Fve9Fve9Ff0dmeaabiqaceGabiGaaeaabaWaaeaaeaaakeaacqGHai % IicaWG4bGaeyOpa4JaaGimaaaa!3AE3! \forall x > 0\)

Nên y = f(x) là hàm giảm trên \(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB % PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbba9q8qqaqFr0x % c9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8fr % Fve9Fve9Ff0dmeaabaqaceGabiGaaeaabaWaaeaaeaaakeaadaqada % qaaiaaicdacaGG7aGaey4kaSIaeyOhIukacaGLOaGaayzkaaaaaa!3CA7! \left( {0; + \infty } \right)\).

Do đó \(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB % PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbba9q8qqaqFr0x % c9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8fr % Fve9Fve9Ff0dmeaabiqaceGabiGaaeaabaWaaeaaeaaakeaacaWGMb % WaaeWaaeaacaWGHbaacaGLOaGaayzkaaGaeyizImQaamOzamaabmaa % baGaaGOmaiaaicdacaaIXaGaaG4naaGaayjkaiaawMcaaaaa!41C9! f\left( a \right) \le f\left( {2017} \right)\), ( a > 0)  khi \(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB % PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbba9q8qqaqFr0x % c9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8fr % Fve9Fve9Ff0dmeaabiqaceGabiGaaeaabaWaaeaaeaaakeaacaaIWa % GaeyipaWJaamyyaiabgsMiJkaaikdacaaIWaGaaGymaiaaiEdaaaa!3E9F! 0 < a \le 2017\).