bất phương trình diophante tuyến tính

✣❸■ ❍➴❈ ◗❯➮❈ ●■❆ ❍⑨ ◆❐■
❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❚Ü ◆❍■➊◆

❚❘❺◆ ❚❘×❮◆● ❙■◆❍

❇❻❚ P❍×❒◆● ❚❘➐◆❍
❉■❖P❍❆◆❚❊ ❚❯❨➌◆ ❚➑◆❍
❈❤✉②➯♥ ♥❣➔♥❤✿

P❍×❒◆● P❍⑩P ❚❖⑩◆ ❙❒ ❈❻P

▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✶✸

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙ß ❑❍❖❆ ❍➴❈

◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈
●❙✳❚❙❑❍ ◆●❯❨➍◆ ❱❿◆ ▼❾❯

❍⑨ ◆❐■ ✲ ✷✵✶✺


▼ö❝ ❧ö❝
▼ð ✤➛✉

✷

✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✹

✶✳✶

×î❝ sè ❝❤✉♥❣ ❧î♥ ♥❤➜t✳ ❚❤✉➟t t♦→♥ ❊✉❝❧✐❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹

✶✳✷

▲✐➯♥ ♣❤➙♥ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺

✶✳✸

P❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻

✶✳✸✳✶

❚➻♠ ♥❣❤✐➺♠ r✐➯♥❣ ❞ü❛ ✈➔♦ ❣✐↔♥ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻

✶✳✸✳✷

❚➻♠ ♥❣❤✐➺♠ r✐➯♥❣ ❞ü❛ ✈➔♦ t❤✉➟t t♦→♥ ❊✉❝❧✐❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻

◆❣❤✐➺♠ ♥❣✉②➯♥ ❞÷ì♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳

✼

✶✳✹

✷ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤

✽

✷✳✶

❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽

✷✳✷

❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✧❜à ❝❤➦♥✧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾

✷✳✸

◆❣❤✐➺♠ ♥❣✉②➯♥ ❞÷ì♥❣ ❝õ❛ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✳ ✳ ✳

✶✶

✷✳✸✳✶

▼ët sè ✈➼ ❞ö ❧✐➯♥ q✉❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶

✷✳✸✳✷

❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ ❞↕♥❣ ❧✐➯♥ ♣❤➙♥ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✸

✸ ▼ët sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥
✸✳✶

✶✹

◆❣❤✐➺♠ ♥❣✉②➯♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤✱
❤➺ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❧÷ñ♥❣ ❣✐→❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✳✷
✸✳✸

✶✹

P❤÷ì♥❣ tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤✱ ❤➺ ❜➜t ♣❤÷ì♥❣ tr➻♥❤
❧÷ñ♥❣ ❣✐→❝ ❝â ✤✐➲✉ ❦✐➺♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✺

❳→❝ ✤à♥❤ ♣❤➙♥ t❤ù❝ ❝❤➼♥❤ q✉② t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tr÷î❝ ✳ ✳ ✳ ✳ ✳ ✳

✶✻

❑➳t ❧✉➟♥

✶✾

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✷✵
✶



Pữỡ tr ỏ ồ ữỡ tr t
ởt tr ỳ t ớ t ừ ồ ổ q
ữỡ tr t t ồ t r ữủ ỳ t t
s s ừ số số ỳ t số số ữỡ tr t
ữ sỹ r ớ ừ số ỵ tt ữớ t ỵ tt
t t ữ ữỡ số ồ r
t ữỡ tr t t t tỹ t ữỡ tr
t t t õ ự t số õ t õ ởt t
ợ ữ ờ tr ý t ồ s ọ ờ tổ
r t ổ õ t ồ qt t
t ữỡ tr t t t ừ s ự
t ữỡ tr ợ ố ồ
s ởt t ờ t ổ ồ s tr
q tr ổ t ồ s ọ
ữủ ữỡ
ữỡ ởt số tự
ữỡ t ữỡ tr t t t
ữỡ ởt số t q

t tọ sỹ trồ ỏ t ỡ s s tợ
tớ ữợ ụ
ữ t ừ ồ trỏ tr sốt q tr ồ t
ự ú ù t t
ụ ỷ ớ ỡ t t tợ
Pỏ t ồ ỡ ồ t ổ
t t ủ t õ t t ử ừ



ỡ ổ q t ở ờ
ụ t tốt t t tr sốt tớ t ồ
t t trữớ ồ ồ ỹ ồ ố ở
ũ õ ố ữ tớ tr ở ỏ
õ tr ọ ỳ t sõt t rt
ữủ sỹ õ ỵ ừ t ổ ụ ữ ỗ
ữủ t ỡ
t ỡ
ở t
ồ tỹ

r rữớ




❈❤÷ì♥❣ ✶
▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
✶✳✶ ×î❝ sè ❝❤✉♥❣ ❧î♥ ♥❤➜t✳ ❚❤✉➟t t♦→♥ ❊✉❝❧✐❞
✣à♥❤ ♥❣❤➽❛ ✶✳✶ ✭①❡♠ ❬✶❪✮✳ ❙è ♥❣✉②➯♥ ❝ ✤÷ñ❝ ❣å✐ ❧➔ ♠ët ÷î❝ sè ❝❤✉♥❣ ❝õ❛ ❤❛✐ sè

♥❣✉②➯♥ ❛ ✈➔ ❜ ✭❦❤æ♥❣ ✤ç♥❣ t❤í✐ ❜➡♥❣ ❦❤æ♥❣✮ ♥➳✉ ❝ ❝❤✐❛ ❤➳t ❛ ✈➔ ❝ ❝❤✐❛ ❤➳t ❜✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷ ✭①❡♠ ❬✶❪✮✳ ▼ët ÷î❝ sè ❝❤✉♥❣ ❞ ❝õ❛ ❤❛✐ sè ♥❣✉②➯♥ ❛ ✈➔ ❜ ✭❦❤æ♥❣ ✤ç♥❣
t❤í✐ ❜➡♥❣ ❦❤æ♥❣✮ ✤÷ñ❝ ❣å✐ ❧➔ ÷î❝ sè ❝❤✉♥❣ ❧î♥ ♥❤➜t ❝õ❛ ❛ ✈➔ ❜ ♥➳✉ ♠å✐ ÷î❝ sè ❝❤✉♥❣
❝ ❝õ❛ ❛ ✈➔ ❜ ✤➲✉ ❧➔ ÷î❝ ❝õ❛ ❞✳

❈❤ó þ ✶✳✶✳ ◆➳✉ ❞ ❧➔ ÷î❝ sè ❝❤✉♥❣ ❧î♥ ♥❤➜t ❝õ❛ ❛ ✈➔ ❜ t❤➻ −d ❝ô♥❣ ❧➔ ÷î❝ sè ❝❤✉♥❣ ❧î♥
♥❤➜t ❝õ❛ ❛ ✈➔ ❜✳ ❱➟② t❛ q✉② ÷î❝ r➡♥❣ ÷î❝ sè ❝❤✉♥❣ ❧î♥ ♥❤➜t ❝õ❛ ❛ ✈➔ ❜ ❧➔ sè ♥❣✉②➯♥
❞÷ì♥❣✳

✣à♥❤ ♥❣❤➽❛ ✶✳✸ ✭①❡♠ ❬✶❪✮✳ ▼ët sè ♥❣✉②➯♥ ❝ ✤÷ñ❝ ❣å✐ ❧➔ ♠ët ÷î❝ sè ❝❤✉♥❣ ❝õ❛ ♥ sè
♥❣✉②➯♥ a1 , a2 , a3 , . . . , an ✭❦❤æ♥❣ ✤ç♥❣ t❤í✐ ❜➡♥❣ ❦❤æ♥❣✮ ♥➳✉ ❝ ❧➔ ÷î❝ ❝õ❛ ♠é✐ sè ✤â✳

✣à♥❤ ♥❣❤➽❛ ✶✳✹ ✭①❡♠ ❬✶❪✮✳ ▼ët ÷î❝ sè ❝❤✉♥❣ ❞ ❝õ❛ ♥ sè ♥❣✉②➯♥ a1, a2, a3, . . . , an
✭❦❤æ♥❣ ✤ç♥❣ t❤í✐ ❜➡♥❣ ❦❤æ♥❣✮ ✤÷ñ❝ ❣å✐ ❧➔ ÷î❝ sè ❝❤✉♥❣ ❧î♥ ♥❤➜t ❝õ❛ a1 , a2 , a3 , . . . , an
♥➳✉ ♠å✐ ÷î❝ sè ❝❤✉♥❣ ❝ ❝õ❛ a1 , a2 , a3 , . . . , an ✤➲✉ ❧➔ ÷î❝ ❝õ❛ ❞✳

✣à♥❤ ❧➼ ✶✳✶✳ ✭✈➲ sü tç♥ t↕✐ ÷î❝ sè ❝❤✉♥❣ ❧î♥ ♥❤➜t ❝õ❛ ♥❤✐➲✉ sè✱ ①❡♠ ❬✶❪✮ ❈❤♦ ❝→❝ sè
♥❣✉②➯♥ a1 , a2 , a3 , . . . , an ❦❤æ♥❣ ✤ç♥❣ t❤í✐ ❜➡♥❣ ❦❤æ♥❣✳ ❑❤✐ ✤â tç♥ t↕✐ ÷î❝ sè ❝❤✉♥❣ ❧î♥
♥❤➜t ❝õ❛ a1 , a2 , a3 , . . . , an ✳

❚➼♥❤ ❝❤➜t ✶✳✶ ✭①❡♠ ❬✶❪✮✳ ❈❤♦ ❛✱ ❜✱ q✱ r ❧➔ ❝→❝ sè ♥❣✉②➯♥ ✭a2 + b2 = 0✮✳ ◆➳✉ a = bq + r
✈➔ 0 ≤ r < |b| t❤➻ ✭❛✱❜✮ ❂ ✭❜✱r✮✳

❚❤✉➟t t♦→♥ ❊✉❝❧✐❞ ✭t❤✉➟t t♦→♥ t➻♠ ÷î❝ sè ❝❤✉♥❣ ❧î♥ ♥❤➜t ❝õ❛ ❤❛✐ sè ♥❣✉②➯♥ ❞÷ì♥❣ ✮✳
✹


✶✳✷ ▲✐➯♥ ♣❤➙♥ sè
✣à♥❤ ♥❣❤➽❛ ✶✳✺✳ ✭▲✐➯♥ ♣❤➙♥ sè ❤ú✉ ❤↕♥✱ ①❡♠ ❬✸❪✮

✣à♥❤ ♥❣❤➽❛ ✶✳✻✳ ✭▲✐➯♥ ♣❤➙♥ sè ✈æ ❤↕♥✱ ①❡♠ ❬✸❪✮
❚➼♥❤ ❝❤➜t ✶✳✷ ✭①❡♠ ❬✸❪✮✳ ▼é✐ sè ❤ú✉ t➾ ❧➔ ♠ët ❧✐➯♥ ♣❤➙♥ sè ❤ú✉ ❤↕♥✳
❚➼♥❤ ❝❤➜t ✶✳✸✳ ✭❚➼♥❤ ❞✉② ♥❤➜t ❝õ❛ ❧✐➯♥ ♣❤➙♥ sè ❤ú✉ ❤↕♥✱ ①❡♠ ❬✸❪✮
❚➼♥❤ ❝❤➜t ✶✳✹✳ ✭❈æ♥❣ t❤ù❝ t➼♥❤ ❣✐↔♥ ♣❤➙♥✱ ①❡♠ ❬✸❪✮
❚➼♥❤ ❝❤➜t ✶✳✺ ✭①❡♠ ❬✸❪✮✳ ●✐↔ sû {Ck } ❧➔ ❞➣② ❣✐↔♥ ♣❤➙♥ ❝õ❛ ❧✐➯♥ ♣❤➙♥ sè ❤ú✉ ❤↕♥
[a0 ; a1 , a2 , . . . , an ]✳ ❑❤✐ ✤â t❛ ❝â ❝→❝ ♠è✐ ❧✐➯♥ ❤➺ s❛✉
✐✮ Ck − Ck−1

(−1)k−1
=
✱ ✈î✐ 1 ≤ k ≤ n.
qk qk−1

✐✐✮ Ck − Ck−2 =

ak (−1)k
✱ ✈î✐ 2 ≤ k ≤ n.
qk qk−2

❚➼♥❤ ❝❤➜t ✶✳✻ ✭①❡♠ ❬✸❪✮✳ ❱î✐ ❝→❝ ❣✐↔♥ ♣❤➙♥ Ck ❝õ❛ ❧✐➯♥ ♣❤➙♥ sè ❤ú✉ ❤↕♥ [a0; a1, a2, . . . , an]
t❛ ❝â ❝→❝ ❞➣② ❜➜t ✤➥♥❣ t❤ù❝ s❛✉
✐✮ C1 > C3 > C5 > . . .
✐✐✮ C0 < C2 < C4 < . . .
✐✐✐✮ ♠é✐ ❣✐↔♥ ♣❤➙♥ ❧➫ C2j−1 ✤➲✉ ❧î♥ ❤ì♥ ♠é✐ ❣✐↔♥ ♣❤➙♥ ❝❤➤♥ C2i ✳

❚➼♥❤ ❝❤➜t ✶✳✼ ✭①❡♠ ❬✸❪✮✳ ❱î✐ ♠å✐ k = 0, 1, . . . , n t❤➻ (pk , qk ) = 1 ✭tù❝ ❧➔ pk , qk
♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉✮✳

❚➼♥❤ ❝❤➜t ✶✳✽ ✭①❡♠ ❬✸❪✮✳ ❈❤♦ a0, a1, a2, . . . ❧➔ ❞➣② ✈æ ❤↕♥ ❝→❝ sè ♥❣✉②➯♥✱ ai > 0 ✈î✐
∀i ≥ 1✳ ❱î✐ ♠é✐ ❦✱ ✤➦t Ck = [a0 ; a1 , a2 , . . . , ak ]✳ ❑❤✐ ✤â tç♥ t↕✐ ❣✐î✐ ❤↕♥

lim Ck .

k→+∞

❚➼♥❤ ❝❤➜t ✶✳✾ ✭①❡♠ ❬✸❪✮✳ ▼å✐ sè ✈æ t➾ α ✤➲✉ ❜✐➸✉ ❞✐➵♥ ✤÷ñ❝ ♠ët ❝→❝❤ ❞✉② ♥❤➜t ❞÷î✐
❞↕♥❣ ♠ët ❧✐➯♥ ♣❤➙♥ sè ✈æ ❤↕♥✳

✺


✶✳✸ P❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤
✣à♥❤ ♥❣❤➽❛ ✶✳✼ ✭①❡♠ ❬✸❪✮✳ P❤÷ì♥❣ tr➻♥❤ ❉✐♣♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ❝â ❞↕♥❣
a1 x1 + a2 x2 + . . . + an xn = c
tr♦♥❣ ✤â ❝→❝ ❤➺ sè ai , c ∈ Z,

n
i=1

a2i = 0✱ ❝→❝ ❜✐➳♥ sè xi ∈ Z, ∀i = 1, 2, . . . , n✳

✣à♥❤ ❧➼ ✶✳✷ ✭①❡♠ ❬✸❪✮✳ ❳➨t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤
Ax + By = C.

✭✶✮

✐✮ (1) ❝â ♥❣❤✐➺♠ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ d = (A, B) |C ✳
✐✐✮ ◆➳✉ (x0 , y0 ) ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ (1) t❤➻ ♠å✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

(1) ✤÷ñ❝ ❝❤♦ ❜ð✐ ❝æ♥❣ t❤ù❝


B


 x = x0 + t
d



 y = y0 − A t

, t ∈ Z.

d

◆❤➟♥ ①➨t ✶✳✶✳ ❱✐➺❝ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ (1) q✉② ✈➲ ✈✐➺❝ t➻♠
✐✮ d = (A, B).
✐✐✮ ▼ët ♥❣❤✐➺♠ r✐➯♥❣ (x0 , y0 ) ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ (1) .
✶✳✸✳✶

❚➻♠ ♥❣❤✐➺♠ r✐➯♥❣ ❞ü❛ ✈➔♦ ❣✐↔♥ ♣❤➙♥

✶✳✸✳✷

❚➻♠ ♥❣❤✐➺♠ r✐➯♥❣ ❞ü❛ ✈➔♦ t❤✉➟t t♦→♥ ❊✉❝❧✐❞

✣à♥❤ ❧➼ ✶✳✸ ✭①❡♠ ❬✸❪✮✳ ❳➨t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤
a1 x1 + a2 x2 + . . . + an xn = c.

✭✺✮

✐✮ P❤÷ì♥❣ tr➻♥❤ (5) ❝â ♥❣❤✐➺♠ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ d = (a1 , a2 , . . . , an ) |c ✳
✐✐✮ ◆➳✉ ♣❤÷ì♥❣ tr➻♥❤ (5) ❝â ♥❣❤✐➺♠ t❤➻ ♥â s➩ ❝â ✈æ sè ♥❣❤✐➺♠✳

❱➼ ❞ö ✶✳✶✳ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤
6x + 15y + 10z = 3.
✻

✭✼✮


✶✳✹ ◆❣❤✐➺♠ ♥❣✉②➯♥ ❞÷ì♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦✲
♣❤❛♥t❡ t✉②➳♥ t➼♥❤
❳➨t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♣♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤

a1 x1 + a2 x2 + . . . + an xn = c

✭✺✮

✈î✐ ❝→❝ ❤➺ sè ai , c ∈ Z+ ✱ ❝→❝ ❜✐➳♥ sè xi ∈ Z+ , ∀i = 1, 2, . . . , n✳ ❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤
✭✺✮ ❧✉æ♥ ❝â ❤ú✉ ❤↕♥ ♥❣❤✐➺♠ ♥❣✉②➯♥ ❞÷ì♥❣ x = (x1 , x2 , . . . , xn )✳ ❚ø ✤➲ ❜➔✐✱ t❛ ❝â t❤➸
❤↕♥ ❝❤➳ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ❝→❝ ❜✐➳♥ sè ❜ð✐

1 ≤ xi ≤

(c + ai ) − (a1 + a2 + . . . + an )
ai

, ∀i = 1, 2, . . . , n.

❑❤✐ ✤â✱ ❝→❝❤ ✤ì♥ ❣✐↔♥ ♥❤➜t ✤➸ t➻♠ ♥❣❤✐➺♠ ♥❣✉②➯♥ ❞÷ì♥❣ x = (x1 , x2 , . . . , xn ) ❝õ❛
♣❤÷ì♥❣ tr➻♥❤ ✭✺✮ ❧➔ t❛ ❝❤♦ ♠ët ❜✐➳♥ sè xi ♥➔♦ ✤â ❧➛♥ ❧÷ñt ❝❤↕② q✉❛ ❝→❝ ❣✐→ trà ❝â t❤➸
❝â ❝õ❛ ♥â ✈➔ t➻♠ ❝→❝ ❜✐➳♥ sè ❝á♥ ❧↕✐ tø ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦✳

❱➼ ❞ö ✶✳✷✳ ❚➻♠ ❝→❝ ♥❣❤✐➺♠ ♥❣✉②➯♥ ❞÷ì♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤
6x + 15y + 10z = 200.

✭✽✮

✣→♣ sè✿ P❤÷ì♥❣ tr➻♥❤ ✭✽✮ ❝â ❝↔ t❤↔② ✶✺ ♥❣❤✐➺♠ ♥❣✉②➯♥ ❞÷ì♥❣ (x, y, z) ❜❛♦ ❣ç♠

(5, 2, 14) , (5, 4, 11) , (5, 6, 8) , (5, 8, 5) , (5, 10, 2) , (10, 2, 11) , (10, 4, 8) , (10, 6, 5) , (10, 8, 2) ,
(15, 2, 8) , (15, 4, 5) , (15, 6, 2) , (20, 2, 5) , (20, 4, 2) , (25, 2, 2) .

✼


❈❤÷ì♥❣ ✷
❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥
t➼♥❤
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ t❛ s➩ ♥❣❤✐➯♥ ❝ù✉ ❝→❝❤ ❣✐↔✐ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡
t✉②➳♥ t➼♥❤ ✈➔ ❝→❝ t❤➼ ❞ö ♠✐♥❤ ❤å❛✳

✷✳✶ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤
✣à♥❤ ♥❣❤➽❛ ✷✳✶✳ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♣♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ❝â ❞↕♥❣
✭✾✮

a1 x1 + a2 x2 + . . . + an xn < c
✭❤♦➦❝ f (x) ≤ c, f (x) > c, f (x) ≥ c✱ ✈î✐ f (x) = a1 x1 + a2 x2 + . . . + an xn ✮
tr♦♥❣ ✤â ❝→❝ ❤➺ sè ai , c ∈ Z✱ ❝→❝ ❜✐➳♥ sè xi ∈ Z, ∀i = 1, 2, . . . , n,

❈→❝❤ ❣✐↔✐✳ ❚❛ ❝â ❜➜t ♣❤÷ì♥❣ tr➻♥❤ (9) t÷ì♥❣ ✤÷ì♥❣ ✈î✐

n

i=1

a2i = 0✳

a1 x 1 + a2 x 2 + . . . + an x n = m

✭✶✵✮

tr♦♥❣ ✤â ♠ ❧➔ t❤❛♠ sè✱ m ∈ Z, m < c✳
◆❤÷ ✈➟② ✈✐➺❝ ❣✐↔✐ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✤÷ñ❝ ✤÷❛ ✈➲ ❣✐↔✐ ♣❤÷ì♥❣
tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✭❝❤ù❛ t❤❛♠ sè✮ ♠➔ ❝❤ó♥❣ t❛ ✤➣ ❜✐➳t ❝→❝❤ ❣✐↔✐✳

❱➼ ❞ö ✷✳✶✳ ●✐↔✐ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤
342x − 123y ≥ 13.

✣→♣ sè✿ ◆❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ (11) ❧➔
x = 9k + 41t
y = 25k + 114t

, ∀t, k ∈ Z, k ≥ 5.
✽

✭✶✶✮

❱➼ ❞ö ✷✳✷✳ ●✐↔✐ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤
✭✶✷✮

6x + 9y + 18z < 5.

✣→♣ sè✿ ◆❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ (12) ❧➔
x = −k + 3u + 3t
y = −k + 4u + 2t
z = k − 3u − 2t
tr♦♥❣ ✤â k, u, t ∈ Z, k ≤ 1✳

❱➼ ❞ö ✷✳✸✳ ●✐↔✐ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤
✭✶✸✮

6x + 15y + 10z > 3.

✣→♣ sè✿ ◆❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ (13) ❧➔
x = −4m + 25u + 15t
y = − m + 6u + 4t
z = 4m − 24u − 15t
tr♦♥❣ ✤â m, u, t ∈ Z, m ≥ 4✳

❱➼ ❞ö ✷✳✹✳ ●✐↔✐ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤
✭✶✹✮

2x + 4y + 6z − 10t ≥ 1.

✣→♣ sè✿ ◆❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ (14) ❧➔



 x = k − 2u − 3v + 5w
y=
z=

 t=

u

v
w

tr♦♥❣ ✤â k, u, v, w ∈ Z, k ≥ 1✳

✷✳✷ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✧❜à ❝❤➦♥✧
✣à♥❤ ♥❣❤➽❛ ✷✳✷✳ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♣♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✧❜à ❝❤➦♥✧ ❝â ❞↕♥❣
✭✶✺✮

b ≤ a1 x 1 + a2 x 2 + . . . + an x n ≤ c
tr♦♥❣ ✤â ❝→❝ ❤➺ sè ai , b, c ∈ Z✱ ❝→❝ ❜✐➳♥ sè xi ∈ Z, ∀i = 1, 2, . . . , n✱

❈→❝❤ ❣✐↔✐✳ ❚❛ ❝â ❜➜t ♣❤÷ì♥❣ tr➻♥❤ (15) t÷ì♥❣ ✤÷ì♥❣ ✈î✐
a1 x 1 + a2 x 2 + . . . + an x n = m
✾

n
i=1

a2i = 0✳

✭✶✻✮

tr♦♥❣ ✤â ♠ ❧➔ t❤❛♠ sè✱ m ∈ Z, b ≤ m ≤ c✳
◆❤÷ ✈➟② ✈✐➺❝ ❣✐↔✐ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✤÷ñ❝ ✤÷❛ ✈➲ ❣✐↔✐ ♣❤÷ì♥❣
tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✭❝❤ù❛ t❤❛♠ sè✮ ♠➔ ❝❤ó♥❣ t❛ ✤➣ ❜✐➳t ❝→❝❤ ❣✐↔✐✳

❱➼ ❞ö ✷✳✺✳ ●✐↔✐ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✧❜à ❝❤➦♥✧
1 < 12x + 15y ≤ 10.

✭✶✼✮

✣→♣ sè✿ ◆❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ (17) ❧➔
x = −k + 5t
y = k − 4t
tr♦♥❣ ✤â k ∈ {1; 2; 3} , t ∈ Z✳

❱➼ ❞ö ✷✳✻✳ ●✐↔✐ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✧❜à ❝❤➦♥✧
12 < 6x − 18y + 54z ≤ 17.

✭✶✽✮

✣→♣ sè✿ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ (18) ✈æ ♥❣❤✐➺♠✳
❱➼ ❞ö ✷✳✼✳ ●✐↔✐ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✧❜à ❝❤➦♥✧
−1 < 4x + 10y − 20z < 20.

✭✶✾✮

✣→♣ sè✿ ◆❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ (19) ❧➔
x = 3k − 1 5u + 5t
y = k − 4u + 2t

z = k − 5u + 2t
tr♦♥❣ ✤â k, u, t ∈ Z, 0 ≤ k ≤ 9✳

❱➼ ❞ö ✷✳✽✳ ●✐↔✐ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✧❜à ❝❤➦♥✧
−2 ≤ 6x + 8y + 2z + 4t ≤ 28.

✣→♣ sè✿ ◆❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ (20) ❧➔


 x=

u
y=
v
z = k − 3u − 4v − 2w


t=
w
tr♦♥❣ ✤â k, u, v, w ∈ Z, −1 ≤ k ≤ 14✳
✶✵

✭✷✵✮


✷✳✸ ◆❣❤✐➺♠ ♥❣✉②➯♥ ❞÷ì♥❣ ❝õ❛ ❜➜t ♣❤÷ì♥❣ tr➻♥❤
❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤
❳➨t ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♣♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤

a1 x 1 + a2 x 2 + . . . + an x n ≤ c

✭✺✮

✈î✐ ❝→❝ ❤➺ sè ai , c ∈ Z+ ✱ ❝→❝ ❜✐➳♥ sè xi ∈ Z+ , ∀i = 1, 2, . . . , n✳ ❑❤✐ ✤â ❜➜t ♣❤÷ì♥❣
tr➻♥❤ ✭✺✮ ❧✉æ♥ ❝â ❤ú✉ ❤↕♥ ♥❣❤✐➺♠ ♥❣✉②➯♥ ❞÷ì♥❣ x = (x1 , x2 , . . . , xn )✳ ❚ø ✤➲ ❜➔✐✱ t❛ ❝â
t❤➸ ❤↕♥ ❝❤➳ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ❝→❝ ❜✐➳♥ sè ❜ð✐

1 ≤ xi ≤

(c + ai ) − (a1 + a2 + . . . + an )
ai

, ∀i = 1, 2, . . . , n.

❑❤✐ ✤â✱ ❝→❝❤ ✤ì♥ ❣✐↔♥ ♥❤➜t ✤➸ t➻♠ ♥❣❤✐➺♠ ♥❣✉②➯♥ ❞÷ì♥❣ x = (x1 , x2 , . . . , xn ) ❝õ❛ ❜➜t
♣❤÷ì♥❣ tr➻♥❤ ✭✺✮ ❧➔ t❛ ❝❤♦ ♠ët ❜✐➳♥ sè xi ♥➔♦ ✤â ❧➛♥ ❧÷ñt ❝❤↕② q✉❛ ❝→❝ ❣✐→ trà ❝â t❤➸
❝â ❝õ❛ ♥â ✈➔ t➻♠ ❝→❝ ❜✐➳♥ sè ❝á♥ ❧↕✐ tø ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦✳

❱➼ ❞ö ✷✳✾✳ ❚➻♠ ♥❣❤✐➺♠ ♥❣✉②➯♥ ❞÷ì♥❣ ❝õ❛ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤
3x + 4y + z + 2t ≤ 14.

✭✷✶✮

✣→♣ sè✿ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ✭✷✶✮ ❝â ❝↔ t❤↔② ✶✷ ♥❣❤✐➺♠ ♥❣✉②➯♥ ❞÷ì♥❣ (x, y, z, t) ❜❛♦ ❣ç♠
(1, 1, 1, 1) , (1, 1, 1, 2) , (1, 1, 1, 3) , (1, 1, 2, 1) , (1, 1, 2, 2) , (1, 1, 3, 1) ,
(1, 1, 3, 2) , (1, 1, 4, 1) , (1, 1, 5, 1) , (1, 2, 1, 1) , (2, 1, 1, 1) , (2, 1, 2, 1) .
✷✳✸✳✶

▼ët sè ✈➼ ❞ö ❧✐➯♥ q✉❛♥

❚✐➳♣ t❤❡♦ t❛ ①➨t ♠ët sè ✈➼ ❞ö ❧✐➯♥ q✉❛♥✳

❱➼ ❞ö ✷✳✶✵✳ ❚➻♠ ♥❣❤✐➺♠ ♥❣✉②➯♥ ❞÷ì♥❣ ❝õ❛ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤
✧❜à ❝❤➦♥✧

−1 < 4x + 10y − 20z < 20.

✣→♣ sè✿ ◆❣❤✐➺♠ ♥❣✉②➯♥ ❞÷ì♥❣ tê♥❣ q✉→t ❝õ❛ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ (22) ❧➔
x = 3k − 5l
y= k
+ 2a
z = k −l+ a
tr♦♥❣ ✤â l, a ∈ Z, l <

3k
k
, a > − , l − a < k, k = 0, 1, . . . , 9✳
5
2
✶✶

✭✷✷✮


t ởt trữớ ồ õ ồ s ọ tr õ số ồ s ọ ộ
ổ ỵ õ tữỡ ự ọ õ
q õ s ỗ tớ t õ
ộ ữủ t t q số ữủ số
ồ s ọ ũ ổ ồ t ữủ số ữ
tt s tờ số ữủ

t

số t õ tữỡ ự ợ ữỡ
(x; y; z; t) ừ tr ử t
(1; 1; 1; 1) (1; 1; 2; 1) (1; 1; 3; 1) (1; 1; 4; 1) (1; 1; 5; 1) (1; 1; 1; 2) (1; 1; 2; 2) (1; 1; 3; 2)
(2; 1; 1; 1) (2; 1; 2; 1) (1; 2; 1; 1) (1; 1; 1; 3)
t

S = S(x, y, z, t) = 6x + 8y + 2z + 4t.
tữỡ ự tọ

(x; y; z; t) {(1; 1; 5; 1), (1; 1; 3; 2), (2; 1; 2; 1), (1; 2; 1; 1), (1; 1; 1; 3)} .

t r tỹ P t ộ
P ộ õ ọ r õ ữỡ
ồ tỹ tr s số t ọ r ổ ữủt q
ứ õ ữỡ số t r ọ r t

số r õ t ữỡ

P t r s ọ r số t t

t t ổ ọ õ t ộ
tọ ỗ ộ ỗ ộ õ ỗ ọ õ
ữỡ ồ t ổ tr s số t ọ r ổ ữủt
q ỗ ứ õ ữỡ số t ọ r t

số õ t ữỡ t ổ tr

ọ õ ọ õ t số t ọ r

t ỗ




✷✳✸✳✷

❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ ❞↕♥❣ ❧✐➯♥ ♣❤➙♥ sè

❱î✐ m ∈ Z ❝❤♦ tr÷î❝ t❛ ❝â
✐✮ [a0 ; a1 , a2 , . . . , ak ] > m ⇔ a0 ≥ m✱ ai ♥❣✉②➯♥ ❞÷ì♥❣✱ tò② þ✱ ∀i = 1, 2, 3, . . .
✐✐✮ [a0 ; a1 , a2 , . . . , ak ] < m ⇔ a0 ≤ m − 1✱ ai ♥❣✉②➯♥ ❞÷ì♥❣✱ tò② þ✱ ∀i = 1, 2, 3, . . .

❱➼ ❞ö ✷✳✶✶✳ ●✐↔✐ ❝→❝ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ s❛✉
[a0 ; a1 , a2 , a3 ] > 2

[b0 ; b1 , b2 ] < 3

a0 ∈ Z, a1 , a2 , a3 ∈ Z+ .

✭✷✺✮

b0 ∈ Z, b1 , b2 ∈ Z+ .

✭✷✻✮

2 < [c0 ; c1 , c2 , c3 ] < 3

c0 ∈ Z, c1 , c2 , c3 ∈ Z+ .
d0 ∈ Z, d1 , d2 , d3 , d4 ∈ Z+ .

−2 ≤ [d0 ; d1 , d2 , d3 , d4 ] ≤ 5

✣→♣ sè✿
❛✮ ◆❣❤✐➺♠ ❝õ❛ (25) ❧➔ (a0 ; a1 ; a2 ; a3 )✱ tr♦♥❣ ✤â

a0 ∈ Z, a0 ≥ 2, ai ∈ Z+ , ∀i = 1, 2, 3.
❜✮ ◆❣❤✐➺♠ ❝õ❛ (26) ❧➔ (b0 ; b1 ; b2 )✱ tr♦♥❣ ✤â

b0 ∈ Z, b0 ≤ 2, bi ∈ Z+ , ∀i = 1, 2.
❝✮ ◆❣❤✐➺♠ ❝õ❛ (27) ❧➔ (c0 ; c1 ; c2 ; c3 )✱ tr♦♥❣ ✤â

c0 = 2, ci ∈ Z+ , ∀i = 1, 2, 3.
❞✮ ◆❣❤✐➺♠ ❝õ❛ (28) ❧➔ (d0 ; d1 ; d2 ; d3 ; d4 )✱ tr♦♥❣ ✤â

d0 ∈ Z, −2 ≤ d0 ≤ 4, di ∈ Z+ , ∀i = 1, 2, 3, 4.

✶✸

✭✷✼✮

✭✷✽✮


❈❤÷ì♥❣ ✸
▼ët sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ t❛ s➩ ①➨t ♠ët sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ✈✐➺❝ ❣✐↔✐ ❜➜t
♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤✳

✸✳✶ ◆❣❤✐➺♠ ♥❣✉②➯♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣

tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤✱ ❤➺ ❜➜t ♣❤÷ì♥❣ tr➻♥❤
❧÷ñ♥❣ ❣✐→❝
❱➼ ❞ö ✸✳✶✳ ✣➳♠ sè ♥❣❤✐➺♠ ♥❣✉②➯♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
cos

π
x =0
2

tr♦♥❣ ❦❤♦↔♥❣ (−2015; 2015)✳

✣→♣ sè✿ P❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❝â ✷✵✶✻ ♥❣❤✐➺♠ ♥❣✉②➯♥ x tr♦♥❣ ❦❤♦↔♥❣ (−2015; 2015)✳
❱➼ ❞ö ✸✳✷✳ ❚➻♠ ❝→❝ ♥❣❤✐➺♠ ♥❣✉②➯♥ (x, y) ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤

π(2x + y)
1


=
 sin
6

2



 cos π(x + y) = 1
3

2

❇✐➳t x, y t❤✉ë❝ ❦❤♦↔♥❣ (−6; 10)✳

✣→♣ sè✿ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❝â ✶✹ ♥❣❤✐➺♠ ♥❣✉②➯♥ (x, y) ❜❛♦ ❣ç♠
(0, 5) , (6, 5) , (−2, −3) , (4, −3) , (−2, 9) , (4, 9) , (−4, −3) ,
(2, −3) , (−4, 9) , (8, −3) , (2, 9) , (8, 9) , (0, 1) , (6, 1)
✈î✐ x, y t❤✉ë❝ ❦❤♦↔♥❣ (−6; 10)✳

✶✹


✸✳✷ P❤÷ì♥❣ tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣
tr➻♥❤✱ ❤➺ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❧÷ñ♥❣ ❣✐→❝ ❝â ✤✐➲✉
❦✐➺♥
❈→❝❤ ❣✐↔✐✳
❇÷î❝ ✶✿ ❚➻♠ ♥❣❤✐➺♠ tê♥❣ q✉→t (x1, x2, . . . , xn) ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭❤➺ ♣❤÷ì♥❣ tr➻♥❤✱
❜➜t ♣❤÷ì♥❣ tr➻♥❤✱ ❤➺ ❜➜t ♣❤÷ì♥❣ tr➻♥❤✮ ❧÷ñ♥❣ ❣✐→❝✳

❇÷î❝ ✷✿ ❚ø ✤✐➲✉ ❦✐➺♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭❤➺ ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤✱ ❤➺ ❜➜t
♣❤÷ì♥❣ tr➻♥❤✮ ❧÷ñ♥❣ ❣✐→❝ t❛ ❤↕♥ ❝❤➳ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ❝→❝ t❤❛♠ sè tr♦♥❣ ♥❣❤✐➺♠ tê♥❣ q✉→t

(x1 , x2 , . . . , xn )✳

❱➼ ❞ö ✸✳✸✳ ●✐↔✐ ❤➺ ❜➜t ♣❤÷ì♥❣ tr➻♥❤
❱➼ ❞ö ✸✳✹✳ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤


1



 sin 2015x >

2



 cos 445x ≤ 1 .
2

1


 sin(2x + y) = 2


 cos(x + y) = 1
2

✈î✐ ✤✐➲✉ ❦✐➺♥ x − y ≥ 10π ✳

✣→♣ sè✿ ❈â ✹ ❤å ♥❣❤✐➺♠ (x;y) t❤ä❛ ♠➣♥π ✤➲ ❜➔✐✱ ❜❛♦ ❣ç♠

π
+ (b + t)2π


 x=
 x = − 6 + (a + t)2π
2


 y=


 x=

✱
π
+
t2π
2
π
+ (c + t)2π
2


 y=−π +
6

t2π

✱


 y = − 5π +
t2π
6

7π



 x = 6 + (d + t)2π


 y = − 3π +

✱

t2π

2

tr♦♥❣ ✤â a, b, c, d, t ∈ Z, a ≥ 6, b ≥ 5, c ≥ 5, d ≥ 4✳

❱➼ ❞ö ✸✳✺✳ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤


1


 sin(2x + y) =
2



 cos(x + y) = 1

tr➯♥ ✤♦↕♥ [−6π; 6π] ✈➔ t❤ä❛ ♠➣♥ x − y ≥ 10π ✳

2

✣→♣ sè✿ ❇➔✐ t♦→♥ ✤➣ ❝❤♦ ❝â ❤❛✐ ♥❣❤✐➺♠ (x; y) ❜❛♦ ❣ç♠
35π 11π
;−
,
6
2
✶✺

31π 11π
;−
.
6
2


✸✳✸ ❳→❝ ✤à♥❤ ♣❤➙♥ t❤ù❝ ❝❤➼♥❤ q✉② t❤ä❛ ♠➣♥ ✤✐➲✉
❦✐➺♥ ❝❤♦ tr÷î❝
✣à♥❤ ♥❣❤➽❛ ✸✳✶✳ ✭P❤➙♥ t❤ù❝ ❝❤➼♥❤
q✉② ♠ët ❜✐➳♥✱ ①❡♠❬✺❪✮ ❈❤♦ ai > 0✱ αi ∈ R ✈î✐
n
ai xαi ✈î✐ x > 0 ✤÷ñ❝ ❣å✐ ❧➔ ♣❤➙♥ t❤ù❝ ❝❤➼♥❤ q✉②

∀i = 1, 2, . . . , n✳ ❑❤✐ ✤â f (x) =

✭♠ët ❜✐➳♥ ①✮ ♥➳✉

n

i=1

ai αi = 0✳

i=1

❈❤ó þ ✸✳✶✳ P❤➙♥ t❤ù❝ ❝❤➼♥❤ q✉② f (x) ✤↕t ❣✐→ trà ♥❤ä ♥❤➜t t↕✐ x = 1.
✣à♥❤ ♥❣❤➽❛ ✸✳✷✳ ✭P❤➙♥ t❤ù❝ ❝❤➼♥❤nq✉② ❤❛✐ ❜✐➳♥✱ ①❡♠ ❬✺❪✮ ❈❤♦ ai > 0✱ αi, βi ∈ R ✈î✐
∀i = 1, 2, . . . , n✳ ❑❤✐ ✤â f (x, y) =
❝❤➼♥❤ q✉② ✭❤❛✐ ❜✐➳♥ ①✱ ②✮ ♥➳✉

n

ai xαi y βi ✈î✐ x > 0, y > 0 ✤÷ñ❝ ❣å✐ ❧➔ ♣❤➙♥ t❤ù❝

i=1

n

ai αi =
i=1

ai βi = 0✳

i=1

❈❤ó þ ✸✳✷✳ P❤➙♥ t❤ù❝ ❝❤➼♥❤ q✉② f (x, y) ✤↕t ❣✐→ trà ♥❤ä ♥❤➜t t↕✐ x = y = 1.
❚r♦♥❣ ❝→❝ ✈➼ ❞ö s❛✉ t❛ ①➨t αi , βi ∈ Z✳

❱➼ ❞ö ✸✳✻✳ ❳➨t ♣❤➙♥ t❤ù❝ ❝❤➼♥❤ q✉②
f (x) = xα1 + 2xα2 + 3xα3 + 5xα4 + 7xα5 .
❚➻♠ (α1 ; α2 ; α3 ; α4 ; α5 ) s❛♦ ❝❤♦

α1 + α2 + 2α3 + α4 − α5 > 4.

✣→♣ sè✿


α1 = −2m − a + 3b + 9c


 α2 = m − a − 4b − 8c
α3 =

a



 α4 =

b

α5 =

c

tr♦♥❣ ✤â m, a, b, c ∈ Z, m < −4✳
❈❤➥♥❣ ❤↕♥ ✈î✐ m = −5, a = 1, b = 2, c = −4 t❛ ❝â ❜➔✐ t♦→♥ s❛✉

❇➔✐ t♦→♥ ✸✳✶✳ ❈❤♦ ① ❧➔ sè t❤ü❝ ❞÷ì♥❣ tò② þ✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣
2x18 + 5x2 + 3x +

✶✻

1
7
+ 21 ≥ 18.
4
x
x

✭✷✾✮


ử t tự q
f (x, y) = x1 y 1 + 2x2 y 2 + 3x3 y 3 .
(1 ; 2 ; 3 ) (1 ; 2 ; 3 ) s tọ ỗ tớ tự s

1 + 42 33 > 0,



1 32 + 3 3.



số
(1 ; 2 ; 3 ) = (9m 7a; 3m + 2a; m + a) ,
(1 ; 2 ; 3 ) = (4n + 11b; n + 2b; 2n 5b)
tr õ m, n, a, b Z, m 1, n 3
ợ m = 1, n = 3, a = 1, b = 2 t õ tự q

f (x, y) =

y 10
3x2
5
+
2x
y
+
x16
y4

õ tr ọ t

min f (x, y) = f (1, 1) = 6.
t x =

b
a
, y = t t õ t s
2
2

t a, b số tỹ ữỡ tũ ỵ tr ọ t ừ tự
M=

64b10 a5 b 12a2
+
+ 4 .
a16

32
b

ử tự q

f (x) = ax2 + bx4 +

c
x2

ợ a, b, c số ữỡ ở số (a, b, c) s tr ọ t ừ f (x)
ổ ữủt q

số õ ở số (a, b, c) tọ t ỗ
(1, 1, 3) , (2, 1, 4) , (1, 2, 5) , (3, 1, 5) , (2, 2, 6) , (1, 3, 7) , (4, 1, 6) .
ợ t õ tự q

f (x) = x2 + x4 +

x 3 t ữủ t s


3
.
x2


❇➔✐ t♦→♥ ✸✳✸✳ ❈❤♦ ① ❧➔ sè t❤ü❝ ❞÷ì♥❣ t❤❛② ✤ê✐✳ ❚➻♠ ❣✐→ trà ❧î♥ ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝
f (x) =

5
1
− 4 − 9x2 .
2
x
x

❱➼ ❞ö ✸✳✾✳ ❈❤♦ ❤➔♠ ♣❤➙♥ t❤ù❝ ❝❤➼♥❤ q✉②
f (x, y) = ax2 y + b

x5
y3
y
+c 4 +d 3
x
y
x

✈î✐ a, b, c, d ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣✳ ❚➻♠ ❜ë sè (a, b, c, d) s❛♦ ❝❤♦ 2a + b + 3c + 2d < 40✳

✣→♣ sè✿ ❈â ✷ ❜ë sè (a, b, c, d) t❤ä❛ ♠➣♥ ②➯✉ ❝➛✉ ❜➔✐ t♦→♥✱ ❣ç♠
(1, 2, 3, 5) , (4, 1, 4, 9) .

❈❤➥♥❣ ❤↕♥ ✈î✐ (a, b, c, d) = (1, 2, 3, 5) t❛ ❝â ♣❤➙♥ t❤ù❝ ❝❤➼♥❤ q✉②

f (x, y) = x2 y +

2y 3 3x5 5y
+ 4 + 3
x

y
x

❝â ❣✐→ trà ♥❤ä ♥❤➜t

min f (x, y) = f (1, 1) = 11.
◆➳✉ t❤❛② x = a, y =

b
t❤➻ t❛ t❤✉ ✤÷ñ❝ ❜➔✐ t♦→♥ s❛✉
2

❇➔✐ t♦→♥ ✸✳✹✳ ❈❤♦ a, b ❧➔ ❝→❝ sè t❤ü❝ ❞÷ì♥❣✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣
2a2 b +

b3 192a5 10b
+
+ 3 ≥ 44.
a
b4
a

✶✽


❑➳t ❧✉➟♥
❙❛✉ ♠ët t❤í✐ ❣✐❛♥ ❤å❝ t➟♣ t↕✐ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ❚ü ♥❤✐➯♥ ✲ ✣↕✐ ❤å❝ ◗✉è❝
❣✐❛ ❍➔ ◆ë✐✱ ✤÷ñ❝ ❝→❝ t❤➛② ❝æ t➟♥ t➻♥❤ ❣✐↔♥❣ ❞↕② ✈➔ ❞÷î✐ sü ❤÷î♥❣ ❞➝♥ ❝õ❛ ●❙✳❚❙❑❍
◆❣✉②➵♥ ❱➠♥ ▼➟✉✱ t→❝ ❣✐↔ ✤➣ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ✈î✐ ✤➲ t➔✐ ✧❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦✲
♣❤❛♥t❡ t✉②➳♥ t➼♥❤✧✳ ▲✉➟♥ ✈➠♥ ✤➣ ✤↕t ✤÷ñ❝ ♠ët sè ❦➳t q✉↔ s❛✉✿

✶✳ ❚r➻♥❤ ❜➔② ✤÷ñ❝ ♠ët ❝→❝❤ ❝â ❤➺ t❤è♥❣ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ❧➔♠ ❝ì sð ❝❤♦ ✈✐➺❝
❣✐↔✐ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✭♠➔ t❤ü❝ ❝❤➜t ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡
t✉②➳♥ t➼♥❤ ❝â ❝❤ù❛ t❤❛♠ sè✮✳
✷✳ ✣÷❛ r❛ ✤÷ñ❝ ❤❛✐ ❝→❝❤ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤✳ ▲➜② ✤â ❧➔♠ ❝ì sð
✤➸ ✤÷❛ r❛ ❝→❝❤ ❣✐↔✐ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ❝ô♥❣ ♥❤÷ ❜➜t ♣❤÷ì♥❣ tr➻♥❤
❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✧❜à ❝❤➦♥✧✳
✸✳ ❚➻♠ tá✐✱ ✤÷❛ r❛ ♠ët sè ❞↕♥❣ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ✈✐➺❝ ❣✐↔✐ ❜➜t ♣❤÷ì♥❣ tr➻♥❤
❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤✱ ❝â t❤➸ ❞ò♥❣ ❝❤♦ ✈✐➺❝ æ♥ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ r➜t ❤ú✉ ➼❝❤✳
▼➦❝ ❞ò tr♦♥❣ q✉→ tr➻♥❤ ❧➔♠ ❧✉➟♥ ✈➠♥✱ t→❝ ❣✐↔ ✤➣ r➜t ❝è ❣➢♥❣ s♦♥❣ ❝❤➢❝ ❝❤➢♥ ❧✉➟♥
✈➠♥ ✈➝♥ ❦❤â tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❚→❝ ❣✐↔ r➜t ♠♦♥❣ ♥❤➟♥ ✤÷ñ❝ ♥❤ú♥❣ þ ❦✐➳♥
❣â♣ þ ❝õ❛ ❝→❝ t❤➛② ❝æ ✈➔ ❜↕♥ ❜➧ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷ñ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳
❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦

✶✾


t
ụ ữỡ tr t t t t s
ồ ồ ồ ỹ ồ ố ở
P

t ỡ ừ số ồ ử

số ồ ỗ ữù ồ s ọ t tr
ồ Pữỡ tr ử tr

P

ố ồ ử



t tự ử ử

r ụ ũ

ởt số số ồ ồ ồ ử
ũ ồ ụ ừ
ử



số ồ