✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ◆●❯❨➍◆ ❙❒◆ ❍⑨ P❍×❒◆● ❚❘➐◆❍✱ ❇❻❚ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ❈❒ ❇❷◆ ❚❘➊◆ ❚❾P ❙➮ ❚Ü ◆❍■➊◆ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙ß ❚❖⑩◆ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ✲ ◆❿▼ ✷✵✶✹ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ◆●❯❨➍◆ ❙❒◆ ❍⑨ P❍×❒◆● ❚❘➐◆❍✱ ❇❻❚ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ❈❒ ❇❷◆ ❚❘➊◆ ❚❾P ❙➮ ❚Ü ◆❍■➊◆ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙ß ❚❖⑩◆ ❍➴❈ ❈❤✉②➯♥ ♥❣❤➔♥❤✿ Pì PP P số ữớ ữợ ❞➝♥ ❦❤♦❛ ❤å❝ ●❙✳ ❚❙❑❍✳ ◆●❯❨➍◆ ❱❿◆ ▼❾❯ ❚❍⑩■ ◆●❯❨➊◆ ✲ ◆❿▼ ✷✵✶✹ ▼ö❝ ❧ö❝ ▼ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ❈❤÷ì♥❣ ✶✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ tr➯♥ t➟♣ sè tü ♥❤✐➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❞↕♥❣ ❈❛✉❝❤② tr➯♥ t➟♣ sè tü ♥❤✐➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ợ ữỡ tr ❈❛✉❝❤② ❧✐➯♥ tö❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ợ ữỡ tr ❤➔♠ ❞↕♥❣ ❈❛✉❝❤② tr➯♥ t➟♣ sè tü ♥❤✐➯♥ ✹ ✶✳✶✳✸✳ ❈→❝ ✈➼ ❞ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❞↕♥❣ ❏❡♥s❡♥ tr➯♥ t➟♣ sè tü ♥❤✐➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✷✳✶✳ ❈→❝ ❞↕♥❣ t♦→♥ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❏❡♥s❡♥ tr➯♥ t➟♣ sè tü ♥❤✐➯♥ ✶✸ ✶✳✷✳✷✳ ❈→❝ ✈➼ ❞ö ♠✐♥❤ ❤å❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✸✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❉✬❆❧❡♠❜❡rt tr➯♥ t➟♣ sè tü ♥❤✐➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✸✳✶✳ ❈→❝ ❞↕♥❣ t♦→♥ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❉✬❆❧❡♠❜❡rt tr➯♥ t➟♣ sè tü ♥❤✐➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✸✳✷✳ ❈→❝ ✈➼ ❞ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✶✳ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② tr➯♥ t➟♣ sè tü ♥❤✐➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✷✳ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❏❡♥s❡♥ tr➯♥ t➟♣ sè tü ♥❤✐➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✸✳ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❉✬❆❧❡♠❜❡rt tr➯♥ t➟♣ sè tü ♥❤✐➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✶✳ ❇➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ❞➣② sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✷✳ ❇➜t ✤➥♥❣ t❤ù❝ ❤➔♠ ❝❤✉②➸♥ ✤ê✐ ❝→❝ ✤↕✐ ❧÷đ♥❣ tr✉♥❣ ❜➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✸✳✸✳ ❱➼ ❞ö →♣ ❞ö♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ❈❤÷ì♥❣ ✷✳ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ tr➯♥ t➟♣ sè tü ♥❤✐➯♥ ✳ ✳ ✳ ✳ ✷✸ ❈❤÷ì♥❣ ✸✳ ▼ët sè ❞↕♥❣ ❦❤→❝ ❝õ❛ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ tr➯♥ t➟♣ sè tü ♥❤✐➯♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺ ✐ ▼ð ✤➛✉ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✈➔ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ♥ë✐ ❞✉♥❣ ❝❤✉②➯♥ ✤➲ q✉❛♥ trå♥❣ t❤✉ë❝ ❝❤÷ì♥❣ tr➻♥❤ ❝❤✉②➯♥ t♦→♥ tr♦♥❣ ❝→❝ tr÷í♥❣ tr✉♥❣ ❤å❝ ♣❤ê t❤ỉ♥❣ ❝❤✉②➯♥✳ ❈→❝ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✈➔ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ t❤÷í♥❣ ❧➔ ♥❤ú♥❣ ❜➔✐ t♦→♥ ❦❤â✱ t❤÷í♥❣ ❣➦♣ tr♦♥❣ ❝→❝ ❦➻ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ ❝➜♣ q✉è❝ ❣✐❛✱ ❦❤✉ ✈ü❝✱ ❖❧②♠♣✐❝ s✐♥❤ ✈✐➯♥ ✈➔ q✉è❝ t➳✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ tr✉♥❣ ❤å❝ ♣❤ê t❤ỉ♥❣ ❝❤✉②➯♥ r➜t ♣❤♦♥❣ ♣❤ó ✈➔ ✤❛ ❞↕♥❣✱ t❤÷í♥❣ ❦❤â ♣❤➙♥ ❧♦↕✐ ❝❤✐ t✐➳t t❤❡♦ ❞↕♥❣ ❜➔✐ ✈➔ ❝→❝ ❝❤✉②➯♥ ✤➲ r✐➯♥❣ ❜✐➺t✳ ❚✉② ♥❤✐➯♥✱ ❝❤♦ ✤➳♥ ♥❛② ✈➜♥ ✤➲ ✈➲ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤✉②➯♥ s➙✉ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✈➔ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❞ị♥❣ ❝❤♦ ❤➺ tr✉♥❣ ❤å❝ ♣❤ê t❤ỉ♥❣ ❝❤✉②➯♥ ✈✐➳t ❜➡♥❣ t✐➳♥❣ ✈✐➺t ❝á♥ ❦❤→ ➼t ä✐ ❝❤õ ②➳✉ ❧➔ ❝→❝ ❝æ♥❣ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ❝æ♥❣ ❜è ❜➡♥❣ t✐➳♥❣ ❛♥❤ ð ♠ù❝ ✤ë t♦→♥ ❝❛♦ ❝➜♣ ✈➔ ✤✐ s ỵ tt ữỡ tr t ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✈✐➳t ð ❜ë ♠ỉ♥ ❣✐↔✐ t➼❝❤ ❤➔♠ ❞ò♥❣ ❝❤♦ s✐♥❤ ✈✐➯♥ ✤↕✐ ❤å❝✱ ❝→❝ t➔✐ ❧✐➺✉ ✈✐➳t t ữợ t tr ữỡ t trt ♥➯♥ ✈✐➺❝ t➻♠ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤♦ t♦→♥ ♣❤ê t❤æ♥❣ ✈✐➳t ❜➡♥❣ t✐➳♥❣ ✈✐➺t ❝á♥ r➜t ❦❤â ❦❤➠♥✳ ❈→❝ ❜➔✐ t➟♣ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✈➔ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ tr➯♥ ❝→❝ t➟♣ ✤➣ ❦❤â ✤è✐ ✈ỵ✐ ❝→❝ ❤å❝ s✐♥❤ tr✉♥❣ ❤å❝ ♣❤ê t❤æ♥❣ ❝❤✉②➯♥ t♦→♥ ♥â✐ ❝❤✉♥❣ ♥➯♥ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✈➔ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ tr➯♥ t➟♣ sè tü ♥❤✐➯♥ ❧↕✐ ❝➔♥❣ ❦❤â ❦❤➠♥ ❤ì♥ ✈➻ ❝❤ó♥❣ ✤÷đ❝ ①➨t tr➯♥ t➟♣ rí✐ r↕❝✳ ❈❤➼♥❤ ✈➻ ♥❤ú♥❣ ❦❤â ❦❤➠♥ ✤➣ ✤➲ ❝➟♣ ð tr➯♥ ♥➯♥ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② t→❝ ❣✐↔ ❝è ❣➢♥❣ ✤÷❛ ❝→❝ ❜➔✐ t➟♣ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❝ì ❜↔♥ tr➯♥ t➟♣ sè tü ♥❤✐➯♥ ✈➲ ♥❤ú♥❣ ❞↕♥❣ t♦→♥ ❝ö t❤➸ ✈➔ ❞➵ ♥❤➟♥ ❜✐➳t ❤ì♥✳ ◆❤ú♥❣ ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ tr♦♥❣ ❜➔✐ ✈✐➳t ❝ư t❤➸ ♥❤÷ s❛✉✿ ❈❤÷ì♥❣ ✶✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❞↕♥❣ ❈❛✉❝❤② tr➯♥ t➟♣ sè tü ♥❤✐➯♥✳ ✶✳✶✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② tr➯♥ t➟♣ sè tü ♥❤✐➯♥✳ ✶ ✶✳✷✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❏❡♥s❡♥ tr➯♥ t➟♣ sè tü ♥❤✐➯♥✳ ✶✳✸✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❉✬❆❧❡♠❜❡rt tr➯♥ t➟♣ sè tü ♥❤✐➯♥✳ ✶✳✹✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✤❛ ➞♥ tr➯♥ t➟♣ sè tü ♥❤✐➯♥✳ ❈❤÷ì♥❣ ✷✳ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② tr➯♥ t➟♣ sè tü ♥❤✐➯♥✳ ✷✳✶✳ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② tr➯♥ t➟♣ sè tü ♥❤✐➯♥✳ ✷✳✷✳ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❏❡♥s❡♥ tr➯♥ t➟♣ sè tü ♥❤✐➯♥✳ ✷✳✸✳ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❉✬❆❧❡♠❜❡rt tr➯♥ t➟♣ sè tü ♥❤✐➯♥✳ ❈❤÷ì♥❣ ✸✳ ▼ët sè ❞↕♥❣ ❦❤→❝ ❝õ❛ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ tr➯♥ t➟♣ sè tü ♥❤✐➯♥✳ ✸✳✶✳ ❇➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ❞➣② sè✳ ✸✳✷✳ ❇➜t ✤➥♥❣ t❤ù❝ ❤➔♠ ❝❤✉②➸♥ ✤ê✐ ❝→❝ ✤↕✐ ❧÷đ♥❣ tr✉♥❣ ❜➻♥❤✳ ✸✳✸✳ ❱➼ ❞ö →♣ ❞ö♥❣ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤è✐ ✈ỵ✐ ●❙✳❚❙❑❍ ◆❣✉②➵♥ ❱➠♥ ▼➟✉✱ ♥❣÷í✐ t❤➛② ✤➣ trü❝ t✐➳♣ t➟♥ t➻♥❤ ❝❤➾ ❜↔♦ ✈➔ ữợ t tr t ỳ ❦✐♥❤ ♥❣❤✐➺♠ ✈➲ ♠➦t ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❧➔♠ ❧✉➟♥ ✈➠♥✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❝→❝ t❤➛② ✭❝æ✮ ❣✐→♦ tr♦♥❣ ❦❤♦❛ ❚♦→♥ ✲ ❚✐♥✱ ♣❤á♥❣ ✣➔♦ t↕♦ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✱ ✣↕✐ ❤å❝ ❚❤→✐ ♥❣✉②➯♥✱❚r÷í♥❣ P ỏ số ỗ ✤➣ ❣✐ó♣ ✤ï t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tỉ✐ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ✦ ❚❤→✐ ◆❣✉②➯♥✱ ✷✵✶✹ ◆❣✉②➵♥ ❙ì♥ ❍➔ ✷ ❈❤÷ì♥❣ ✶ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ tr➯♥ t➟♣ sè tü ♥❤✐➯♥ ✶✳✶✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❞↕♥❣ ❈❛✉❝❤② tr➯♥ t➟♣ sè tü ♥❤✐➯♥ ✶✳✶✳✶✳ ❈→❝ ợ ữỡ tr tử ✶✳✶ ✳ ❍➔♠ sè f : R → R ❧✐➯♥ tö❝ tr➯♥ R ✈➔ t❤ä❛ ♠➣♥ ✭❈❛✉❝❤②✱ ❬✶❪✮ ✤✐➲✉ ❦✐➺♥ f (x + y) = f (x) + f (y), ∀x, y ∈ R, ❧➔ ❤➔♠ sè ❞↕♥❣ f (x) = ax, a R tũ ỵ ✭❉✬❆❧❡♠❜❡rt✱ ❬✶❪✮ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✳ ❍➔♠ sè f : R → R ❧✐➯♥ tö❝ tr➯♥ R ✈➔ t❤ä❛ f (x + y) = f (x)f (y), ∀x, y ∈ R, ❧➔ ♠ët tr♦♥❣ ❝→❝ ❤➔♠ f (x) ≡ 0, f (x) ≡ ✈➔ f (x) = ax , = a R+ tũ ỵ ✳ ✭❉↕♥❣ ❧♦❣❛r✐t✱ ❬✶❪✮ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❍➔♠ sè f : R+ → R ❧✐➯♥ tö❝ tr➯♥ R+ ✈➔ f (xy) = f (x) + f (y), ∀x, y ∈ R+ , ❧➔ ❤➔♠ f (x) = a ln x a R tũ ỵ ❧ô② t❤ø❛✱ ❬✶❪✮ ✈➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✳ ❍➔♠ sè f : R+ → R ❧✐➯♥ tö❝ tr➯♥ R+ f (xy) = f (x)f (y), ∀x, y ∈ R+ , ✸ ❧➔ ♠ët tr♦♥❣ ❝→❝ ❤➔♠ f (x) ≡ 0, f (x) ≡ ✈➔ ❤➔♠ f (x) = xm = m R tũ ỵ ợ ữỡ tr tr t số tü ♥❤✐➯♥ ❧➔ • ❚➻♠ ❤➔♠ f : X → Y t❤ä❛ ♠➣♥ N, N∗ ❀ Y ❝â t❤➸ ❧➔ N, N∗ , Z, R✮✳ • ❚➻♠ sè ❤↕♥❣ tê♥❣ q✉→t ❝õ❛ ❞➣② ✤✐➲✉ ❦✐➺♥ ♥➔♦ ✤â ✭tr♦♥❣ ✤â X õ t số trữợ ❞ö ✶✳✶✳ ❬✣➲ ✤➲ ♥❣❤à ■▼❖ ✶✾✽✽❪ ❳→❝ ✤à♥❤ ❤➔♠ sè f :N→N t❤ä❛ ♠➣♥ f (f (n) + f (m)) = n + m, ∀m, n ∈ N ▲í✐ sỷ tỗ t số f (x) tọ ♠➣♥ ②➯✉ ❝➛✉ ❜➔✐ r❛✳ ❚❛ t❤➜② f (x) ❧➔ ✤ì♥ →♥❤✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ n, m ∈ N ✈➔ f (n) = f (m)✱ t❛ ❝â f (f (n) + f (m)) = n + m = f (f (n) + f (n)) = n + n→ n = m✳ ❱ỵ✐ ♠å✐ n ∈ N, n ≥ 1✱ t❛ ❝â f (f (n) + f (n)) = 2n = (n − 1) + (n + 1) = f (f (n − 1) + f (n + 1)) ✤✐➲✉ ❦✐➺♥✿ f (n) + f (n) = f (n − 1) + f (n + 1) ✭❞♦ f ❧➔ ✤ì♥ →♥❤✮✳ ❚❤❡♦ ♥❤➟♥ ①➨t ❜❛♥ ✤➛✉ t❤➻ f ❧➔ ❤➔♠ sè t✉②➳♥ t➼♥❤✱ tù❝ ❧➔ f ❝â ❞↕♥❣ f (n) = an + b✳ ❚❤û ❧↕✐ t❛ ♣❤↔✐ ❝â a [(an + b) + (am + b)] + b = n + m ✈ỵ✐ ♠å✐ n, m ∈ N, tø ✤â ✤÷đ❝ a = 1, b = 0✳ ❱➟② f (n) = n ❧➔ ❤➔♠ sè ❝➛♥ t➻♠✳ ♥➯♥ ❱➼ ❞ö f : N N ỗ f (mn) = f (m) · f (n) , ∀m, n ∈ N ❬P✉t♥❛♠ ✶✾✻✸❪ ❳→❝ ✤à♥❤ t➜t ❝↔ ❝→❝ ❤➔♠ sè f (2) = sỷ tỗ t ❤➔♠ sè f (x) t❤ä❛ ♠➣♥ ②➯✉ ❝➛✉ ❜➔✐ r❛✳ f (x) số ỗ f (0) < f (1) < f (2) = ♥➯♥ f (0) = 0, f (1) = 1✳ ✣➦t f (3) = + k, k ∈ N; f (6) = f (2) · f (3) t❤➻ f (6) = + 2k ◆❤÷ ✈➟② f (5) ≤ + 2k ♥➯♥ f (10) = f (2) · f (5) ≤ 10 + 4k ✳ ▲➟♣ ❧✉➟♥ t÷ì♥❣ tü ✤÷đ❝ f (9) ≤ + 4k ♥➯♥ f (18) ≤ 18 + 8k, s✉② r❛ f (15) ≤ 15 + 8k ✳ ❜✐➳♥✱ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥✿ ✹ ▼➦t ❦❤→❝ f (3) = + k; f (5) ≥ + k ♥➯♥ f (15) = f (3) f (5) ≥ (3 + k)(5 + k) ❚â♠ ❧↕✐ t❛ ✤÷đ❝✿ ❱➟② (3 + k)(5 + k) ≤ 15 + 8k ⇔ k ≤ ⇔ k = 0✳ f (3) = 3✳ n ∈ N∗ ✳ ❚❤➟t ✈➟②✱ ❤✐➸♥ ♥❤✐➯♥✱ ❦❤➥♥❣ ✤à♥❤ ✤ó♥❣ ✈ỵ✐ n = n n ●✐↔ sû ❦❤➥♥❣ ✤à♥❤ ✤ó♥❣ tỵ✐ ♥✱ tù❝ ❧➔✿ f (2 + 1) = + 1, ❦❤✐ ✤â f 2n+1 + = f (2) f (2n + 1) = (2n + 1) = 2n+1 + 2✳ f số ỗ ỡ →♥❤ ♥➯♥ t➟♣ f (2n + 2); f (2n + 3); ; f (2n+1 + 2) ỗ 2n +1 sè ✤æ✐ ♠ët ❦❤→❝ ♥❤❛✉✱ n s➢♣ ①➳♣ t tự tỹ t t ỗ + sè ✤æ✐ ♠ët ❦❤→❝ n n n+1 ♥❤❛✉ + 2; + 3; ; +2 ✳ n n n ◆❤÷ ✈➟②✱ t❛ ❝â f (2 + i) = + i, ✈ỵ✐ ♠å✐ i ∈ {2; 3; ; + 2} tù❝ ❧➔ f (2n + 1) = 2n + 1✳ ◆â✐ ❝→❝❤ ❦❤→❝✱ ❦❤➥♥❣ ✤à♥❤ ✤ó♥❣ tỵ✐ n + ỵ q ✤ó♥❣ ✈ỵ✐ ♠å✐ n ∈ N ✳ ▲➟♣ ❧✉➟♥ ❤♦➔♥ t♦➔♥ t÷ì♥❣ tü t❛ ❝ơ♥❣ ✤÷đ❝ f (n) = n ✈ỵ✐ ♠å✐ n ∈ N✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ f (2n + 1) = 2n + 1, ✈ỵ✐ ♠å✐ ❉➵ t❤➜② ❤➔♠ sè ♥➔② t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦✳ ❱➟② f (n) = n ❱➼ ❞ö ✶✳✸✳ ❧➔ ❤➔♠ sè ❝➛♥ t➻♠✳ ❚➻♠ t➜t ❝↔ ❝→❝ ❤➔♠ sè f :N→R t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ f (0) = 1; f (1) = 2; f (n + 1) f (n − 1) = f (n) , ∀n ∈ N∗ sỷ tỗ t số f (n) t❤ä❛ ♠➣♥ ②➯✉ ❝➛✉ ❜➔✐ r❛✳ f (n) > 0, ∀n ∈ N✳ ▲➜② ❧æ❣❛r✐t ❝õ❛ ❝→❝ ❜✐➸✉ t❤ù❝ tr➯♥ t❛ ✤÷đ❝✿ ln f (0) = 0, ln f (1) = ln ✈➔ ln f (n + 1) + ln f (n − 1) = ln f (n) , ∀n ∈ N✳ ✣➦t xn = ln f (n) , (n ∈ N) t❛ ✤÷đ❝ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ t✉②➳♥ t➼♥❤✿ x0 = 0; x1 = ln 2; xn+2 − 3xn+1 + 2xn = 0, (n ∈ N)✳ P❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣✿ λ − 3λ + = ⇔ λ = ❤♦➦❝ λ = 2✳ n n n ◆❣❤✐➺♠ tê♥❣ q✉→t✿ xn = A · + B · = A + B · ✳ ❇➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ q✉② ♥↕♣ t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ✺ ❚ø ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤➣ ❝❤♦ t❛ ✤÷đ❝✿ A+B =0 A = − ln A + 2B = ln ⇔ B = ln n 2n −1 ❙✉② r❛ xn = (2 − 1) · ln = ln f (n) n −1 ❚ø ✤â t❛ ✤÷đ❝ f (n) = ✳ ❉➵ t❤➜② ❤➔♠ sè ♥➔② t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦✳ f (n) = 22 ❱➟② ❤➔♠ sè ❝➛♥ t➻♠ ❧➔ ❱➼ ❞ö ✶✳✹✳ ❳→❝ ✤à♥❤ ❤➔♠ sè n −1 ✱ f :N→R (n ∈ N)✳ t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠✿ 8f (n) + 1, ∀n ∈ N f (0) = 2, f (n + 1) = 3f (n) + sỷ tỗ t số f (n) t❤ä❛ ♠➣♥ ②➯✉ ❝➛✉ ❜➔✐ r❛✳ ❚ø ❣✐↔ t❤✐➳t t❛ ❝â✿ f (n + 1) − 3f (n) = 8f (n) + (≥ > 0, ∀n ∈ N), ♥➯♥ (f (n + 1) + 3f (n))2 = 8f (n) + ❙✉② r❛ f (n + 1) + f (n) = 6f (n) f (n + 1) + ❚❤❛② n ❜ð✐ n−1 ✭✶✳✶✮ t❛ ✤÷đ❝ f (n) + f (n − 1) = 6f (n − 1) · f (n) + ✭✶✳✷✮ ❚rø tø♥❣ ✈➳ ❝õ❛ ✭✶✳✷✮ ❝❤♦ ✭✶✳✶✮✱ t❛ ✤÷đ❝ f (n + 1) − f (n − 1) = 6f (n) (f (n + 1) − f (n − 1)) ❚ø ❣✐↔ t❤✐➳t t❛ ❝á♥ ❝â f (n) > ✈ỵ✐ ♠å✐ n ✭❝❤ù♥❣ ♠✐♥❤ ❜➡♥❣ q✉② ♥↕♣✮✳ ◆❣♦➔✐ r❛ f (n + 1) > 3f (n) = 9f (n − 1) + 8f (n − 1) + > f (n − 1) ♥➯♥ f (n + 1) − f (n − 1) > ♥➯♥ f (n + 1) + f (n − 1) = 6f (n)✳ ❱➟② t❛ ✤÷đ❝ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ √ f (0) = 2, f (1) = + 33, f (n + 2) − 6f (n + 1) + f (n) = 0, ∀n ∈ N ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② t❛ ✤÷đ❝✿ f (n) = ❉➵ t❤➜② ❤➔♠ sè (8 + √ f (n) 66)(3 + √ n 8) + (8 − √ tr➯♥ ❧➔ ❤➔♠ sè ❝➛♥ t➻♠✳ ✻ 66)(3 − √ n 8) ❱➼ ❞ö ✶✳✺✳ ❚➻♠ ❤➔♠ sè f :N→N t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ f (1) > 0; f (m2 + n2 ) = f (m) + f (n), ∀m, n ∈ N f (n) t❤ä❛ ♠➣♥ ②➯✉ ❝➛✉ ❜➔✐ r❛✳ ❚❛ ❝â f (1) = f (12 + 02 ) = f (1) + f (0) ♥➯♥ f (0) = 0; f (1) = 2 ∗ ❚❤❡♦ ❜➔✐ r❛✱ ❝❤♦ m = ✤÷đ❝ f (n ) = f (n), ∀n ∈ N ✳ 2 ❚❛ ❝â f (2) = f (1 + 1) = 2f (1) = 2, f (4) = f (22 ) = f (2) = 22 = 4, f (5) = f (22 + 12 ) = f (2) + f (1) = + = 5, f (25) = f (52 ) = 25 = f (32 + 42 ) = f (3) + f (4) = f (3) + 4, ♥➯♥ f (3) = 3, 2 = f (3) + f (1) = 32 + 12 = 100, f (100) = f (102 ) = f 32 + 12 f (100) = f (62 + 82 ) = f (6) + f (8) = f (6) + f (22 + 22 ) = f (6) + f (2) + f (2) = f (6) + (4 + 4)2 = f (6) + 64 ⇒ f (6) = ▲í✐ ❣✐↔✐✳ ●✐↔ sỷ tỗ t số s ự f (n) = n, ∀n ∈ N∗ ✭✶✳✸✮ n = ●✐↔ sû ✭✶✳✸✮ ✤ó♥❣ ✈ỵ✐ ♠å✐ m < n, (n ≥ 6)✳ ❑❤✐ ✤â✱ ♥➳✉ n = 2k + ✈➔ 2 2 ❞♦ (2k + 1) + (k − 2) = (2k − 1) + (k + 2) ♥➯♥ t❛ ❝â f ((2k + 1)2 + (k − 2)2 ) = f (2k + 1) + f (k − 2) f ((2k − 1)2 + (k + 2)2 ) = f (2k − 1) + f (k + 2) 2 2 ❙✉② r❛ f (2k + 1) + f (k − 2) = f (2k − 1) + f (k + 2) ▼➔ < k − < k + < 2k − < 2k + = n ♥➯♥ t❤❡♦ ❣✐↔ t❤✐➳t q✉② 2 f (k − 2) = (k − 2) ♥↕♣✱ t❛ ❝â✿ f (2k − 1) = (2k − 1)2 f (k + 2) = (k + 2)2 2 2 ❙✉② r❛ f (2k + 1) = (2k − 1) + (k + 2) − (k − 2) = (2k + 1) ✳ ❱➟② t❛ ❝â f (n) = f (2k + 1) = 2k + = n✳ ❚÷ì♥❣ tü✱ ❦❤✐ n = 2k + sû ❞ö♥❣ ✤➥♥❣ t❤ù❝ ❚❤➟t ✈➟②✱ t❤❡♦ tr➯♥ ✤➣ ✤ó♥❣ ✤➳♥ (2k + 2)2 + (k − 4)2 = (2k − 2)2 + (k + 4)2 , ✼ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ♥❤➟♥ ①➨t tr➯♥ ❜➡♥❣ q✉② ♥↕♣✿ • ✭✸✳✸✶✮ ✤ó♥❣ ✈ỵ✐ n ≤ 5✳ • ●✐↔ sû ✭✸✳✸✶✮ ✤ó♥❣ ✈ỵ✐ n = m, (m ≥ 6)✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ✭✸✳✸✶✮ ✤ó♥❣ ✈ỵ✐ n = m + 1✳ ❚r÷í♥❣ ❤đ♣ ✶✿ m + ❧➔ sè ❝❤➤♥✱ ✤➦t m + = 2q ✈ỵ✐ q = (ak ak−1 a1 )2 ✈➔ m + = 2q = (ak ak−1 a1 0)2 ✳ ❚❛ ✤÷đ❝ f (m + 1) = f (2q) = 2f (q) − = 2(ak−1 2k−1 + · · · + a1 21 + ak 20 ) − = (ak−1 a1 01)2 = (ak−1 a1 0ak )2 m + ❧➔ sè ❧➫✱ ✤➦t m + = 2q + ✈ỵ✐ q = (ak ak−1 a1 )2 m + = 2q + = (ak ak−1 a1 1)2 ✳ ❚r÷í♥❣ ❤đ♣ ✷✿ ✈➔ ❚❛ ✤÷đ❝✿ f (m + 1) = f (2q + 1) = 2f (q) + = 2(ak−1 2k−1 + · · · + a1 21 + ak 20 ) + = (ak−1 a1 11)2 = (ak−1 a1 1ak )2 n = m + 1✳ ❉♦ ✤â ♥➳✉ n ❝â ❜✐➸✉ ❞✐➵♥ tr♦♥❣ ❤➺ ♥❤à ♣❤➙♥ ❧➔✿ n = (ak ak−1 a1 )2 ✈ỵ✐ ak = t❤➻ f ((ak ak−1 a1 )2 ) = (ak−1 a1 ak )2 = ak−1 2k−1 + · · · + a1 21 + ak 20 ✳ ❚❤û ❧↕✐ t❛ t❤➜② ❤➔♠ sè f t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❜➔✐ t♦→♥✳ ❱➟② ✭✸✳✸✶✮ ✤ó♥❣ ✈ỵ✐ ❱➼ ❞ư ✸✳✸✷✳ ❚➻♠ t➜t ❝↔ ❝→❝ ❤➔♠ sè f ✿N → N t❤ä❛ ♠➣♥✿ f (f (f (n))) + f (f (n)) = 3f (n) − n, ∀n ∈ N ▲í✐ ❣✐↔✐✳ sỷ tỗ t số tỹ t ①➨t ❞➣② sè (an ) f t❤ä❛ ♠➣♥ ②➯✉ ❝➛✉ t ợ ữ s a0 = n an 0, ∀n ∈ N P❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣✿ ✻✷ ✈➔ an+1 = f (an )✳ n ❧➔ sè ❑❤✐ ✤â✱ √ λ3 + λ2 − 3λ + = ⇔ λ ∈ 1, −1, ± ✳ ❉♦ ✤â✿ ❱➻ √ n √ n an = C + A(−1 − 2) + B(−1 + 2) , ∀n ∈ N √ √ −1 + < < −1 − ♥➯♥ lim (−1 + n→+∞ ✈➔ lim (−1 − √ n→+∞ 2n+1 2) √ ✭✸✳✸✷✮ n 2) = 0, = −∞, lim (−1 − n→+∞ √ 2n 2) = +∞ A > t❛ ❝❤♦ n ❧➫ ✈➔ ✤û ❧ỵ♥ s➩ ❝â an < 0✱ ✈ỉ ❧➼✱ ♥➳✉ n ❝❤➤♥ ✈➔ ✤õ ❧ỵ♥ s➩ ❝â an < 0✱ ✈ỉ ❧➼✳ ❉♦ ✤â A = 0✳ ❱➟②✿ √ n an = C + B(−1 + 2) , ∀n ∈ N ✭✸✳✸✸✮ ❉♦ ✤â tø ✭✸✳✸✷✮✱ ♥➳✉ Aa t❛ ❝â f (n) = b✳ f t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❜➔✐ t♦→♥✳ ❚❛ ❝â✿ f (n+2)−f (n+1) = f (n+1)−f (n)+f (f (n−1)) ≥ f (n+1)−f (n)✳ ❙✉② r❛✿ f (n + 1) − f (n) ❧➔ ♠ët số t õ tỗ t n0 s ✈ỵ✐ ♠å✐ n ≥ n0 t❤➻✿ f (n + 1) f (n) sỷ tỗ t số n1 s❛♦ ❝❤♦ f (n1 + 1) − f (n1 ) ≥ 1✳ ❉♦ ✤â f (n) t➠♥❣ t❤ü❝ sü ợ n > n1 r tỗ t n2 > n1 + s❛♦ ❝❤♦ f (n2 − 1) > n1 ứ õ sỷ tỗ t↕✐ ❤➔♠ sè f (n2 + 2) − f (n2 + 1) = f (n2 + 1) − f (n2 ) + f (f (n2 − 1)) ≥ f (n2 + 1) − f (n2 ) + ≥ 2, (❱➻ f (f (n2 − 1)) > f (n1 ) ≥ 0) ❚ø ♠ët c f (n + 1) − f (n) ❧➔ ❤➔♠ sè t➠♥❣✱ ♥â ❝â ♥❣❤➽❛ ❧➔ ♥➔♦ ✤â✳ ✻✺ f (n) ≥ 2n − c ✈ỵ✐ ❙✉② r❛✱ ❱➻ t❤➳✱ f (n) ≥ n + ✈ỵ✐ n ✈ỵ✐ n ✤õ ❧ỵ♥ t❤➻ ✤õ ❧ỵ♥✳ f (n + 2) − f (n + 1) = f (n + 1) − f (n) + f (f (n − 1)) ≥ f (f (n − 1)) ≥ f (n + 3) > f (n + 2), ✭✈æ ❙✉② r❛✿ f (n + 1) = f (n) = f (a) = b, ∀n ≥ a✳ ❱➼ ❞ö ✸✳✸✼ ✭✣✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✮✳ ✭✣➲ ♥❣❤à t❤✐ ❖❧②♠♣✐❝ t♦→♥ q✉è❝ t➳ tỗ t ởt ởt số f : N∗ → N∗ ✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ s❛♦ ❝❤♦✿ f (m + f (n)) = n + f (m + 95), ∀m, n ∈ N∗ ▲í✐ ❣✐↔✐✳ ❧➼✮✳ sỷ tỗ t số tọ ❝➛✉ ❜➔✐ t♦→♥✳ f ❧➔ ✤ì♥ →♥❤✳ ❚❤➟t ✈➟②✱ ❣✐↔ sû f (n1 ) = f (n2 )✱ t❛ ❝â✿ f (f (n1 ) + f (1)) = f (f (n2 ) + f (1)) ⇒ n1 + f (f (1) + 95) = n2 + f (f (1) + 95) ⇒ n1 = n2 ✳ ∗ ❱➟② f ❧➔ ✤ì♥ →♥❤✳ ❱ỵ✐ ♠å✐ n ∈ N ✱ t❛ ❝â✿ f (f (n)+f (1)) = n+f (f (1)+95) = n+1+f (95+95) = f (95+f (n+1))✳ ∗ ❙✉② r❛ f (n + 1) + 95 = f (n) + f (1)✳ ❚ø ✤â ∀n ∈ N t❤➻ f (n + 1) − f (n) = f (1) − 95 = a✱ ✈ỵ✐ a = f (1) − 95✳ ❙✉② r❛✿ f (n) − f (n − 1) = · · · = f (2) − f (1) = f (1) − 95 = a✱ ∗ ❞➝♥ ✤➳♥✿ f (n) − 95 = na ⇒ f (n) = na + 95, ∀n ∈ N ✳ ∗ ❱ỵ✐ f (n) = na + 95, ∀n ∈ N ✳ ❚❤❛② ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✭✸✳✹✵✮ t❛ ❝â✿ f (m + f (n)) = n + f (m + 95) ⇔ na2 + (m + 95)a + 95 = n + (m + 95)a + 95 ⇒ a2 = ⇒ a = ✭✈➻ ♥➳✉ a = −1 t❤➻ f (n) ∈ / N∗ ❦❤✐ n ≥ 95✮✳ ∗ ❱➟② f (n) = n + 95, ∀n ∈ N ✳ ❚❤û ❧↕✐ t❤➜② ❤➔♠ sè ✈ø❛ t➻♠ t❤ä❛ ♠➣♥ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ❤➔♠ sè ②➯✉ ❝➛✉ ❜➔✐ r❛✳ ❱➼ ❞ö ✸✳✸✽✳ ❚➻♠ t➜t ❝↔ ❝→❝ ❤➔♠ sè f : N∗ → N∗ t❤ä❛ ♠➣♥✿ f (f (f (n))) + f (f (n)) + f (n) = 3n, ∀n ∈ N∗ ✻✻ ✭✸✳✹✶✮ f t❤ä❛ ♠➣♥ ❝→❝ ②➯✉ ❝➛✉ ✤➲ ❜➔✐✳ ∗ ∗ ❚ø ✭✸✳✹✶✮ ❝❤♦ n = ✈➔ sû ❞ö♥❣ f : N → N t❛ ✤÷đ❝✿ f (f (f (1))) + f (f (1)) + f (1) = ⇒ f (f (f (1))) = f (f (1)) = f (1) = 1✳ ●✐↔ sû f (k) = k, ∀k ≤ n✱ t❛ ❝❤ù♥❣ ♠✐♥❤ f (n + 1) = n + 1✳ ●✐↔ sû f (n1 ) = f (n2 ) tứ s r sỷ tỗ t↕✐ ❤➔♠ sè 3n1 = f (f (f (n1 ) + f (f (n1 )) + f (n1 ) = f (f (f (n2 ))) + f (f (n2 )) + f (n2 ) = 3n2 n1 = n2 ✱ s✉② r❛ f ❧➔ ✤ì♥ →♥❤✳ ❚❛ ❝â f (m) > n ✈ỵ✐ m > n ✭✈➻ ♥➳✉ f (m) n t t tt q tỗ t↕✐ k ≤ n < m s❛♦ ❝❤♦ f (m) = f (k)✱ ♠➙✉ t❤✉➝♥ ✈ỵ✐ f ❧➔ ✤ì♥ →♥❤✮✳ ❙û ❞ư♥❣ ✤✐➲✉ ♥➔② t❛ ✤÷đ❝✿ ❉♦ ✤â f (n + 1) > n ⇒ f (n + 1) ≥ n + ⇒ f (f (n + 1)) ≥ n + ⇒ f (f (f (n + 1))) ≥ n + ⇒ f (f (f (n + 1))) + f (f (n + 1)) + f (n + 1) ≥ 3(n + 1) ✭✸✳✹✷✮ = ð ✭✸✳✹✷✮ ①↔② r❛✱ ❞♦ ✤â✿ ⇒ f (f (f (n + 1))) = f (f (n + 1)) = f (n + 1) = n + 1✳ ∗ ❱➟② t❤❡♦ ♥❣✉②➯♥ ❧➼ q✉② ♥↕♣ s✉② r❛ f (n) = n, ∀n ∈ N ✳ ❚❤û ❧↕✐ t❤➜② ❤➔♠ s✉② r❛ ❞➜✉ sè t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✤➛✉ ❜➔✐✳ ❱➼ ❞ö ✸✳✸✾ ✳ ✭❖❧②♠♣✐❝ ❚♦→♥ ❈❍ ❙❡❝ ✷✵✵✼✮ ❳➨t t➜t ❝↔ ❝→❝ ❤➔♠ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥✿ f (mf (n)) = nf (m), ∀m, n ∈ N∗ ✳ f (2007)✳ ▲í✐ ❣✐↔✐✳ ●å✐ S ❧➔ t➟♣ ❝→❝ ❤➔♠ sè f t❤ä❛ ♠➣♥ ✤✐➲✉ ●✐↔ sû f ∈ S ✱ ✤➦t a = f (1)✳ ❈❤å♥ m = t❛ ✤÷đ❝✿ f : N∗ → N∗ ❍➣② t➻♠ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❦✐➺♥ ❜➔✐ t♦→♥✳ f (1.f (n)) = n.f (1) = na ⇒ f (f (n)) = na ❈❤å♥ n=1 ✭✸✳✹✸✮ t❛ ✤÷đ❝✿ f (m.f (1)) = 1.f (m) ⇒ f (ma) = f (m) f ❧➔ ✤ì♥ →♥❤✳ ❚❤➟t ✈➟②✱ ❣✐↔ f (f (n1 )) = f (f (n2 )) ⇒ n1 a = n2 a ⇒ n1 = n2 ✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ✻✼ sû f (n1 ) = f (n2 )✱ ✭✸✳✹✹✮ t❛ ✤÷đ❝✿ f ❧➔ ✤ì♥ →♥❤✳ ❚ø ✭✸✳✹✹✮ t❛ ❝â f (na) = f (n) ⇒ na = n, ∀n ∈ N∗ ✱ s✉② r❛ a = ❤❛② f (1) = 1✳ ❚❛ ❝â✿ f (f (m).f (n)) = n.f (f (m)) = nm = f (f (m.n))✳ ❙✉② r❛ ❱➟② f (m.n) = f (m).f (n) ✭✸✳✹✺✮ p ❧➔ sè ♥❣✉②➯♥ tè ❜➜t ❦➻✱ ❣✐↔ sû f (p) = u.v ✱ ✈ỵ✐ u, v ∈ N∗ ✳ ❚❛ ❝â✿ p = f (f (p)) = f (u.v) = f (u).f (v)✳ ❉♦ ✤â f (u) = ❤♦➦❝ f (v) = 1✳ ●✐↔ sû f (u) = 1✱ t❛ ✤÷đ❝ u = f (f (u)) = f (1) = 1✳ ❙✉② r❛ f (u) ❧➔ sè ♥❣✉②➯♥ tè✳ ❑❤✐ ✤â f (n) ❧➔ ❤➔♠ ❝❤✉②➸♥ ❝→❝ sè ♥❣✉②➯♥ tè ❦❤→❝ ♥❤❛✉ t❤➔♥❤ ❱ỵ✐ ❝→❝ sè ♥❣✉②➯♥ tè ❦❤→❝ ♥❤❛✉✳ 2007 = 32 223; f (2007) = f (3).f (223)✳ ❉♦ ✤â✱ ✤➸ ♥❤➟♥ ✤÷đ❝ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ f (2007) t❛ ♣❤↔✐ ❝❤å♥ ❤➔♠ f (n) s❛♦ ❝❤♦ f (3), f (223) ❧➔ ❝→❝ sè ♥❣✉②➯♥ tè ♥❤ä ♥❤➜t✱ ❦❤→❝ ♥❤❛✉✳ ❍✐➸♥ ♥❤✐➯♥✱ ♥➳✉ t❛ ❝❤å♥ ✤÷đ❝ ❤➔♠ f¯(n) s❛♦ ❝❤♦✿ f¯(3) = 2, f¯(2) = 3, f¯(223) = 5, f¯(5) = 223, t❤➻ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ f (2007) = = 20 ✭✈➻ f (n) ≥ f¯(n)✮✳ ∗ ∗ ❚❛ ①➙② ❞ü♥❣ ❤➔♠ f : N → N ♥❤÷ s❛✉✿ f (1) = 1, f (2) = 3, f (3) = 2, f (5) = 223, f (223) = f (p) = p, ∀p ∈ P \ {2; 3; 5; 223} k1 k2 v`a n = p1 p2 pkmm th`i f (n) = f k1 (p1 )f k2 (p2 ) f km (pm ) ❑❤✐ ✤â f (n) t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥✿ f (1) = 1; f (f (n)) = n; f (mn) = f (m).f (n); ∀m, n ∈ N∗ ✳ ❉♦ ✤â f (n) ∈ S ✳ ❱➟② ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ f (2007) = 20✳ ❚❛ ❧↕✐ ❝â✿ ❱➼ ❞ö ✸✳✹✵✳ ❚➻♠ t➜t ❝↔ ❝→❝ ❤➔♠ f :N→N t❤ä❛ ♠➣♥✿ f (f (m) + f (n) + f (p)) = m + n + p, m, n, p N sỷ tỗ t↕✐ ❤➔♠ sè f (n) t❤ä❛ ♠➣♥ ②➯✉ ❝➛✉ ❜➔✐ t♦→♥✳ f (n) ❧➔ ✤ì♥ →♥❤✳ ❚❤➟t ✈➟②✱ ❣✐↔ sû f (n1 ) = f (n2 )✳ f (f (n1 ) + f (1) + f (1)) = f (f (n2 ) + f (1) + f (1)) ⇔ n1 + + = n2 + + ⇔ n1 = n2 ✳ ❑❤✐ ✤â✱ ✻✽ ❚❛ ❝â✿ ✭✸✳✹✻✮ ❉♦ ✤â f ❧➔ ✤ì♥ →♥❤✳ ❚ø ✭✸✳✹✻✮ ❝❤♦ n=p t❛ ✤÷đ❝✿ f (f (m) + 2f (n)) = m + 2n, ∀m, n ∈ N m + 2n = (m + 2) + 2(n − 1) ♥➯♥ tø ✭✸✳✹✼✮ f (f (m) + 2f (n)) = f (f (m + 2) + 2f (n − 1))✳ ▼➔ f ❧➔ ✤ì♥ →♥❤ ♥➯♥✿ ❱➻ s✉② r❛✿ f (m) + 2f (n) = f (m + 2) + 2f (n − 1) ❚ø ✭✸✳✹✽✮ t❤❛② n ❜ð✐ m+2 ✭✸✳✹✼✮ ✭✸✳✹✽✮ t❛ ✤÷đ❝✿ f (m) + 2f (m + 2) = f (m + 2) + 2f (m + 1) ⇔ f (m + 2) − f (m + 1) = f (m + 1) − f (m) ✭✸✳✹✾✮ {f (m)}m ❧➔ ❝➜♣ sè ❝ë♥❣✱ ❞♦ ✤â f (m) = am + b, ∀m ∈ N ✭a, b ❧➔ ❝→❝ ❤➡♥❣ sè tü ♥❤✐➯♥✮✳ a p = m t❛ ✤÷đ❝ ❚ø ✭✸✳✹✻✮ ❝❤♦✿ n = m v` ❚ø ✭✸✳✹✾✮ s✉② r❛ ❞➣② f (3f (m)) = 3m, ∀m ∈ N ✭✸✳✺✵✮ f (m) = am + b, ∀m ∈ N ✈➔♦ ✭✸✳✺✵✮ t❛ ✤÷đ❝✿ a(3am + 3b) + b = 3m, ∀m ∈ N ⇔ 3a2 m + 3ab + b = 3m, ∀m ∈ N N 3a2 = a,b∈ ⇔ a=1 ⇔ b=0 3ab + b = ❱➟② f (m) = m, ∀m ∈ N✳ ❚❤û ❧↕✐ t❤➜② t❤ä❛ ♠➣♥✳ ❚❤❛② ❱➼ ❞ö ✸✳✹✶✳ ❚➻♠ t➜t ❝↔ ❝→❝ ❤➔♠ sè f : N∗ → N∗ t❤ä❛ ♠➣♥✿ f (1) + f (2) + · · · + f (n) = [f (1) + f (2) + · · · + f (n)]2 , ∀n = 1, 2, ✭✸✳✺✶✮ n = t❛ ✤÷đ❝✿ f (1) = f (1) ⇔ f (1) = ✭❞♦ f : N∗ → N∗ ✮✳ ▲í✐ ❣✐↔✐✳ ❚ø ✭✸✳✺✶✮ ❧➜② ❚ø ✭✸✳✺✶✮ t❛ ❝â✿ f (1)+f (2)+· · ·+f (n)+f (n+1) = [f (1) + f (2) + · · · + f (n) + f (n + 1)]2 = [f (1) + f (2) + · · · + f (n)]2 +2f (n+1) [f (1) + f (2) + · · · + f (n)]+f (n+1) ✭✸✳✺✷✮ ✻✾ ❚ø ✭✸✳✺✶✮ ✈➔ ✭✸✳✺✷✮ s✉② r❛✿ f (n + 1) = [f (1) + f (2) + · · · + f (n)]2 + 2f (n + 1)[f (1) + f (2) + · · · + f (n)] + f (n + 1) ⇔ f (n + 1) = [f (1) + f (2) + · · · + f (n)] + f (n + 1) ✭✸✳✺✸✮ ⇒ f (n + 2) = [f (1) + f (2) + · · · + f (n) + f (n + 1)] + f (n + 2) ✭✸✳✺✹✮ ▲➜② ✭✸✳✺✹✮ trø ✭✸✳✺✸✮ t❛ ✤÷đ❝✿ f (n + 2) − f (n + 1) = f (n + 2) + f (n + 1) ⇔ [f (n + 2) + f (n + 1)] [f (n + 2) − f (n + 1) − 1] = ⇔ f (n + 2) − f (n + 1) − = ⇔ f (n + 2) = f (n + 1) + ✭✸✳✺✺✮ ❚ø ✭✸✳✺✺✮ t❛ ❝â✿ f (1) = f (2) = f (1) + f (3) = f (2) + f (n) = f (n − 1) + f (n) = n✱ ∀n = 1, 2, n(n + 1) ▼➦t + + ··· + n = ❈ë♥❣ ❧↕✐ t❛ ✤÷đ❝ ❚❛ ❝â ❦❤→❝ ❜➡♥❣ q✉② ♥↕♣ t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝✿ n(n + 1) + + ··· + n = , ∀n = 1, 2, ❱➻ t❤➳ ♥➯♥✿ 13 + 23 + · · · + n3 = (1 + + · · · + n)2 , ∀n = 1, 2, ❉♦ ✤â ❤➔♠ sè f (n) = n, ∀n = 1, 2, t❤ä❛ ♠➣♥ ❝→❝ ②➯✉ ❝➛✉ 3 ❱➼ ❞ö ✸✳✹✷✳ ❚➻♠ ❞➣② sè {xn }+∞ n=1 ❜✐➳t✿ x1 = α, xn+1 = ax2n + b, ∀n ∈ N∗ , ab = −2 ▲í✐ ❣✐↔✐✳ ✣➦t xn = −byn ✱ a ✳ b = −abyn2 − ⇒ yn+1 = 2yn2 − ❦❤✐ ✤â −byn+1 = ab2 yn2 + b ⇒ yn+1 y1 = − ✼✵ ✤➲ ❜➔✐✳ ✣➦t xn = −byn t❛ t➻♠ ✤÷đ❝ ✿ ❝ỉ♥❣ t❤ù❝ ❧÷đ♥❣ ❣✐→❝ ✤÷❛ ❞➣② sè ✤➣ ❝❤♦ ✈➲ ❞➣② sè {yn }+∞ n=1 cos 2α = 2cos2 α − t❤ä❛ ♠➣♥✿ yn+1 = 2yn2 − 1, ∀n = 1, 2, ✣➦t xn = pyn ✳ ✭✸✳✺✻✮ ❑❤✐ ✤â✿ b pyn+1 = ap2 yn2 + b ⇒ yn+1 = apyn2 + p ap = b ⇒ ❚ø ✭✸✳✺✻✮ ✈➔ ✭✸✳✺✼✮ s✉② r❛ t❛ ❝➛♥ t➻♠ p s❛♦ ❝❤♦✿ = −1 p ✭t❤ä❛ ♠➣♥ ab = −2✮✳ ❱➟② t❛ s➩ ✤➦txn = yn ⇔ xn = −byn ✭❞♦ ab = −2✮✳ a ❱➼ ❞ö ✸✳✹✸✳ ❚➻♠ sè ❤↕♥❣ tê♥❣ q✉→t ❝õ❛ ❞➣② sè x1 = ▲í✐ ❣✐↔✐✳ 2+ √ ✈➔ (xn ) ✭✸✳✺✼✮ a p = −b p= ❝❤♦ ❜ð✐ xn+1 = x4n − 4x2n + 2, ∀n = 1, 2, ✣➦t✿ xn = 2yn ✳ ❑❤✐ ✤â 2yn+1 = 16yn4 − 16yn2 + 2, ∀n = 1, 2, ⇔ yn+1 = 8yn4 − 8yn2 + 1, ∀n = 1, 2, ❚❛ ❝â✿ √ π 2+ = cos π ✳ y1 = = 2 12 ❚❤❡♦ ❝æ♥❣ t❤ù❝ cos 4α = 8cos α − 8cos α + ♥➯♥ t❛ ❝â✿ √ 1+ = + cos π π π ) − 8cos2 (4 ) + = cos(✹2 ) 12 12 12 n−1 π ●✐↔ sû yn = cos ✳ ❑❤✐ ✤â✿ 12 π π π yn+1 = 8cos4 (4n−1 ) − 8cos2 (4n−1 ) + = cos(✹n ) 12 12 12 n−1 π ❱➟② t❤❡♦ ♥❣✉②➯♥ ❧➼ q✉② ♥↕♣ s✉② r❛ yn = cos , ∀n = 1, 2, 12 n−1 π ❉♦ ✤â xn = cos , ∀n = 1, 2, 12 y3 = 8cos4 (4 ✼✶ ❱➼ ❞ö ✸✳✹✹ sè (un ) ✳ ✭✣➲ t❤✐ P ỗ ♥❤÷ s❛✉✿ u1 = ✈➔ un+1 = u4n u4n − 8u2n +8 , ∀n ∈ N∗ ✳ ❚➻♠ sè ❤↕♥❣ tê♥❣ q✉→t ❝õ❛ ❞➣② sè ✤➣ ❝❤♦✳ = ✳ ❑❤✐ ✤â v1 = ✳ ❚ø ❣✐↔ t❤✐➳t s✉② r❛ un 8 = − + ⇒ vn+1 = 8vn4 − 8vn2 + 1, ∀n = 1, 2, un+1 un un 1 a=2 ❳➨t sè t❤ü❝ a s❛♦ ❝❤♦✿ = a+ ⇔ 2a2 − 5a + = ⇔ a = 2−1 ✳ a 1 ❱➟② ♥➳✉ ✤➦t a = th` a+ , a = ✳ ❙û ❞ö♥❣ ❝→❝ ♣❤➨♣ ❜✐➳♥ = i a ❍❉●✳ ✣➦t ✤ê✐ ♥❤÷ ❝→❝ ✈➼ ❞ư tr➯♥ t❛ ♥❤➟♥ ✤÷đ❝ ❦➳t q✉↔ ❜➔✐ t♦→♥ ❧➔✿ 4n−1 1 4n−1 n−1 a + 4n−1 = + 2−4 , ∀n = 1, 2, 2 a ✤â un = , ∀n = 1, 2, n−1 24 + 2−4n−1 = ❉♦ ❱➼ ❞ö ✸✳✹✺✳ ❈❤♦ ❞➣② sè (un ) ♥❤÷ s❛✉✿ x1 = , xn+1 = 16x5n − 20x3n + 5xn , ∀n = 1, 2, ❚➻♠ sè ❤↕♥❣ tê♥❣ q✉→t ❝õ❛ ❞➣② sè ✤➣ ❝❤♦✳ cos 5α = 16cos5 α − 20cos3 α + cos α✳ π π π 5π 5π ❱➟② x1 = cos ✈➔ x2 = 16cos − 20cos3 + cos = cos x3 3 3 5π 5π 5π π = 16cos5 − 20cos3 + cos = cos ✳ 3 3 5n−1 π ●✐↔ sû xn = cos ✱ ❦❤✐ ✤â n−1 n−1 π 5n−1 π 5n π π 55 35 xn+1 = 16cos − 20cos + cos = cos ✳ 3 3 5n−1 π ❚❤❡♦ ♥❣✉②➯♥ ❧➼ q✉② ♥↕♣✱ t❛ s✉② r❛ xn = cos , ∀n = 1, 2, ▲í✐ ❣✐↔✐✳ ❚❛ ❝â ❱➼ ❞ư ✸✳✹✻✳ ❈❤♦ ❞➣② sè x1 = α ∈ R ✈➔ (xn ) ♥❤÷ s❛✉✿ xn+1 = x2n − 14xn + 56, ∀n = 1, 2, ❚➻♠ sè ❤↕♥❣ tê♥❣ q✉→t ❝õ❛ ❞➣② sè ✤➣ ❝❤♦✳ ▲í✐ ❣✐↔✐✳ ❚❛ ❝â✿ ✼✷ xn − = x2n−1 − 14xn−1 + 49 = (xn−1 − 7)2 = (xn−2 − 7)2 n−1 n−1 = (xn−3 − 7)2 = · · · = (x1 − 7)2 = (α − 7)2 ✳ ❱➟② sè ❤↕♥❣ tê♥❣ q✉→t ❝õ❛ ❞➣② sè ✤➣ ❝❤♦ ❧➔✿ n−1 xn = + (α − 7)2 ❱➼ ❞ö ✸✳✹✼✳ , ∀n ∈ N∗ ✳ ❈❤♦ ❞➣② sè u1 = α ∈ R (un ) ✈➔ ♥❤÷ s❛✉✿ un+1 = 3u2n + 4un + , ∀n = 1, 2, ❚➻♠ sè ❤↕♥❣ tê♥❣ q✉→t ❝õ❛ ❞➣② sè ✤➣ ❝❤♦✳ xn ✳ ❑❤✐ ✤â ❞➣② = x2n + 4xn + 2, ∀n ∈ N∗ ✳ ▲í✐ ❣✐↔✐✳ ✣➦t ✈➔ xn+1 un = sè (xn ) t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ x1 = 3α ❚❛ ❝â xn+2 = x2n−1 + 4xn−1 + = (xn−1 + 2)2 = (xn−2 + 2)2 = n−1 = (x1 + 2)2 ❱➟② xn = (3α + 2)2 n−1 − 2✳ ❙è ❤↕♥❣ tê♥❣ q✉→t ❝õ❛ ❞➣② sè ✤➣ ❝❤♦ ❧➔✿ n−1 ❱➼ ❞ö ✸✳✹✽✳ (3α + 2)2 un = ❈❤♦ ❞➣② sè (un ) u1 = α ∈ R −2 , ∀n ∈ N∗ ♥❤÷ s❛✉✿ ✈➔ un+1 = −5u2n − 4un − ❚➻♠ sè ❤↕♥❣ tê♥❣ q✉→t ❝õ❛ ❞➣② sè ✤➣ ❝❤♦✳ un = − xn ✳ ❑❤✐ ✤â✱ 1 − xn+1 = − xn + xn − ⇔ xn+1 = x2n − 4xn + ⇔ xn+1 − = (xn − 2)2 ✳ 5 5 22 2n−1 ❱➟② xn − = (xn−1 − 2) = (xn−2 − 2) = · · · = (x1 − 2) ✳ ▲í✐ ❣✐↔✐✳ ✣➦t ❙è ❤↕♥❣ tê♥❣ q✉→t ❝õ❛ ❞➣② sè ✤➣ ❝❤♦ ❧➔ un = − n−1 + (5α + 2)2 , ∀n ∈ N∗ ✼✸ ❑➳t ❧✉➟♥ ▲✉➟♥ ✈➠♥ ✏P❤÷ì♥❣ tr➻♥❤ ✈➔ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ tr➯♥ t➟♣ sè tü ♥❤✐➯♥✑ ✤➣ tr➻♥❤ ❜➔② ♥❤ú♥❣ ✈➜♥ ✤➲ s❛✉✿ ✶✳ ❚r➻♥❤ ❜➔② ❝❤✐ t✐➳t ♠ët sè ❞↕♥❣ t♦→♥ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❞↕♥❣ ❈❛✉❝❤②✱ ❉✬❆❧❡♠❜❡rt✱ ❏❡♥s❡♥ tr➯♥ t➟♣ sè tü ♥❤✐➯♥ ✈➔ ❝→❝ ✈➼ ❞ö ♠✐♥❤ ❤å❛✳ ✷✳ ❚✐➳♣ t❤❡♦✱ ①➨t ♠ët sè ❧ỵ♣ ❜➔✐ t♦→♥ ✈➲ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❞↕♥❣ ❈❛✉❝❤②✱ ❉✬❆❧❡♠❜❡rt✱ ❏❡♥s❡♥ tr➯♥ t➟♣ sè tü ♥❤✐➯♥ ✈➔ ❝→❝ ✈➼ ❞ö ♠✐♥❤ ❤å❛✳ ✸✳ ❈✉è✐ ❝ị♥❣✱ ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ❝→❝ ù♥❣ ❞ư♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✱ ❉✬❆❧❡♠❜❡rt✱ ❏❡♥s❡♥ tr➯♥ t➟♣ sè tü ♥❤✐➯♥ ✈➔♦ ✈✐➺❝ ❣✐↔✐ ❝→❝ ❜➔✐ t➟♣ ✈➲ ❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ❞➣② sè✱ ❜➜t ✤➥♥❣ t❤ù❝ ❤➔♠ ❝❤✉②➸♥ ✤ê✐ ❝→❝ ✤↕✐ ❧÷đ♥❣ tr✉♥❣ ❜➻♥❤✱ ❝→❝ ✤➲ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ tr✉♥❣ ❤å❝ ♣❤ê t❤æ♥❣ ❱✐➺t ◆❛♠✱ ❖▲②♠♣✐❝ ❦❤✉ ✈ü❝ ✈➔ q✉è❝ t➳✳✳✳ ▼➦❝ ❞ò t→❝ ❣✐↔ ✤➣ ❤➳t sù❝ ❝è ❣➢♥❣ ✈➔ ♥❣❤✐➯♠ tó❝ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ♥❤÷♥❣ ❞♦ t❤í✐ ❣✐❛♥ ❝â ❤↕♥ ✈➔ ❦❤↔ ♥➠♥❣ ❝â ❤↕♥ ♥➯♥ ❝❤➢❝ ❝❤➢♥ ❧✉➟♥ ✈➠♥ ♥➔② ❝á♥ ❝â ♥❤✐➲✉ t❤✐➳✉ sât✳ ❚→❝ ❣✐↔ r➜t ữủ ỵ õ õ qỵ t ổ ỗ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ✼✹ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ◆❣✉②➵♥ ❚➔✐ ❈❤✉♥❣✱ ▲➯ ❍♦➔♥❤ P❤á✱ P❤÷ì♥❣ tr➻♥❤ ❤➔♠✱ ◆❳❇ ✣❍◗●❍◆✱ ✷✵✵✻✳ ❬✷❪ ◆❣✉②➵♥ ❱➠♥ ▼➟✉✱ ❬✸❪ ◆❣✉②➵♥ ❱➠♥ ▼➟✉✱ P❤÷ì♥❣ tr➻♥❤ ❤➔♠✱ ◆❳❇ ●✐→♦ ❉ư❝✱ ✶✾✾✼✳ ✣❛ t❤ù❝ ✤↕✐ sè ✈➔ ♣❤➙♥ t❤ù❝ ❤ú✉ t✛✱ ◆❳❇ ●✐→♦ ❉ö❝✱ ✷✵✵✺✳ ❬✹❪ ◆❣✉②➵♥ ❱➠♥ ▼➟✉✱ ◆❣✉②➵♥ ❱➠♥ ❚✐➳♥✱ ❣✐ä✐ t P rồ ỗ ❞÷ï♥❣ ❤å❝ s✐♥❤ ◆❳❇ ●✐→♦ ❉ư❝✱ ✷✵✵✾✳ ❇➔✐ t♦→♥ ❤➔♠ sè q✉❛ ❝→❝ ❧ý t❤✐ ♦❧②♠♣✐❝✱ ◆❳❇ ●✐→♦ ❉ö❝✱ ✷✵✵✹✳ ❬✻❪ ❈→❝ ❜➔✐ t❤✐ ❖❧②♠♣✐❝ ❚♦→♥ tr✉♥❣ ❤å❝ ♣❤ê t❤æ♥❣ ❱✐➺t ◆❛♠ ✭✶✾✾✵✲✷✵✵✻✮✱ ◆❳❇ ●✐→♦ ❉ö❝✱ ✷✵✵✽✳ ✼✺ ❳⑩❈ ◆❍❾◆ ❈❍➓◆❍ ❙Û❆ ▲❯❾◆ ❱❿◆ ❳→❝ ♥❤➟♥ ❧✉➟♥ ✈➠♥ t❤↕❝ s➽ ❝õ❛ ❤å❝ ✈✐➯♥ ❝❛♦ ❤å❝ ◆❣✉②➵♥ ❙ì♥ ❍➔ ✳ ❚➯♥ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ P❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❝ì ❜↔♥ tr➯♥ t➟♣ sè tü ♥❤✐➯♥ ❈❤✉②➯♥ ♥❣➔♥❤✿ P❤÷ì♥❣ ♣❤→♣ t♦→♥ ❝➜♣ ▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✶✸ ❇↔♦ ✈➺ ♥❣➔② ✶✷✳✶✵✳✷✵✶✹ ✣➣ ❝❤➾♥❤ sû❛ t❤❡♦ ♥❤÷ ❦➳t ❧✉➟♥ ❝õ❛ ❍ë✐ ỗ t trữớ ữợ ◆❣✉②➵♥ ❱➠♥ ▼➟✉ ✼✻ ... + 1) = (2n + 1) = 2n+1 + f số ỗ ✈➔ ❧➔ ✤ì♥ →♥❤ ♥➯♥ t➟♣ f (2n + 2); f (2n + 3); ; f (2n+1 + 2) ỗ 2n +1 số ổ ởt n s➢♣ ①➳♣ t❤❡♦ t❤ù tü t➠♥❣ ❞➛♥✱ ❧➔ ↔♥❤ t ỗ + số ổ ởt n n n+1 ♥❤❛✉ + 2; +... (n) = n, ∀n ∈ N∗ ●✐↔ sỷ tỗ t số tọ t♦→♥✳ d ❧➔ ♣❤➛♥ tû ♥❤ä ♥❤➜t tr♦♥❣ ♠✐➲♥ ❣✐→ trà ❝õ❛ ❤➔♠ sè f ✱ d = {f (n) : n ∈ N∗ } t❤❡♦ ♥❣✉②➯♥ ❧➼ s➢♣ tự tỹ tốt d tỗ t tự ❧➔ ❞✉② ♥❤➜t✳ m ∈ N∗ s❛♦ ❝❤♦... ❞➣② ✤✐➲✉ ❦✐➺♥ ♥➔♦ ✤â tr õ X õ t số trữợ ✈➼ ❞ö ❱➼ ❞ö ✶✳✶✳ ❬✣➲ ✤➲ ♥❣❤à ■▼❖ ✶✾✽✽❪ ❳→❝ ✤à♥❤ ❤➔♠ sè f :N→N t❤ä❛ ♠➣♥ f (f (n) + f (m)) = n + m, ∀m, n N sỷ tỗ t số f (x) t❤ä❛ ♠➣♥ ②➯✉ ❝➛✉ ❜➔✐ r❛✳