▼ët sè ♣❤➨♣ ❝❤✉②➸♥ ✤ê✐
❜↔♦ t♦➔♥ ❝↕♥❤ ✈➔ ❣â❝ ❝õ❛ t❛♠ ❣✐→❝
❚❙✳ ❚rà♥❤ ✣➔♦ ❈❤✐➳♥
❚r÷í♥❣ ❈❛♦ ✣➥♥❣ ❙÷ P❤↕♠ ●✐❛ ▲❛✐
❚r♦♥❣ q✉→ tr➻♥❤ s→♥❣ t→❝ ❤♦➦❝ t➻♠ tá✐ ❧í✐ ❣✐↔✐ ❝❤♦ ♥❤ú♥❣ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ❝→❝
②➳✉ tè ❣â❝ ✈➔ ❝↕♥❤ ❝õ❛ t❛♠ ❣✐→❝✱ ♠ët ✈➜♥ ✤➲ tü ♥❤✐➯♥ s❛✉ ✤➙② ✤÷ñ❝ ♥↔② s✐♥❤✿ ◆❤ú♥❣
♣❤➨♣ ❜✐➳♥ ✤ê✐ ♥➔♦ ♠➔ ↔♥❤ ❝õ❛ ❜❛ ❣â❝ ✭❝↕♥❤✮ ❝õ❛ ♠ët t❛♠ ❣✐→❝ ❝ô♥❣ ❧➟♣ t❤➔♥❤ ❜❛ ❣â❝
✭❝↕♥❤✮ ❝õ❛ ♠ët t❛♠ ❣✐→❝❄ ❇➔✐ ✈✐➳t ♥➔② ♣❤➛♥ ♥➔♦ t➻♠ ❝➙✉ ❣✐↔✐ ✤→♣ ❝❤♦ ✈➜♥ ✤➲ ✤➣ ♥➯✉
✈➔ ✤✐➲✉ q✉❛♥ trå♥❣ ❧➔✱ ✤➲ ❝➟♣ ✤➳♥ ♥❤ú♥❣ →♣ ❞ö♥❣ ❝õ❛ ♥â tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ❚r✉♥❣
❤å❝ ♣❤ê t❤æ♥❣✳ ▲÷✉ þ r➡♥❣✱ tr♦♥❣ ❦❤✉æ♥ ❦❤ê ❝â ❤↕♥✱ ❜➔✐ ✈✐➳t ❝❤÷❛ ✤➲ ❝➟♣ ✤➳♥ ♥❤ú♥❣
→♣ ❞ö♥❣ s➙✉ s➢❝ ❤ì♥✱ ❧✐➯♥ q✉❛♥ ✤➳♥ ❦❤→✐ ♥✐➺♠ ✧✤ë ❣➛♥ ✤➲✉✧ ❝õ❛ ♠ët ❞➣② ❝→❝ t❛♠ ❣✐→❝
①→❝ ✤à♥❤✳
✶ P❤➨♣ ❝❤✉②➸♥ ✤ê✐ ❜↔♦ t♦➔♥ ❣â❝ ❝õ❛ t❛♠ ❣✐→❝
✶✳✶ P❤➨♣ ❝❤✉②➸♥ ✤ê✐
❚r♦♥❣ t➔✐ ❧✐➺✉ ❬✶❪✱ ❜➔✐ t♦→♥ ❝ì ❜↔♥ s❛✉ ✤➙② ✤➣ ✤÷ñ❝ ✤➲ ❝➟♣
❇➔✐ t♦→♥ ✶✳✶✳
❳→❝ ✤à♥❤ ❝→❝ ❤➔♠ sè
f(x)
❧✐➯♥ tö❝ tr♦♥❣ ✤♦↕♥
[0; π]
✱ s❛♦ ❝❤♦
f(A)
✱
f(B)
✱
f(C)
❧✉æ♥ t↕♦ t❤➔♥❤ sè ✤♦ ❝→❝ ❣â❝ ❝õ❛ ♠ët t❛♠ ❣✐→❝ ♥➔♦ ✤â ù♥❣ ✈î✐ ♠å✐ t❛♠
❣✐→❝
ABC
❝❤♦ tr÷î❝✳
●✐↔✐✳
❚r÷î❝ ❤➳t t❛ ❝â ♥❤➟♥ ①➨t r➡♥❣✱ ❤❛✐ ❤➔♠ sè
f(x) = x
✈➔
f(x) = π
3
t❤ä❛ ♠➣♥ ❜➔✐
t♦→♥✳
❚❛ ♣❤→t ❜✐➸✉ ❜➔✐ t♦→♥ ❞÷î✐ ❞↕♥❣ s❛✉✿
❳→❝ ✤à♥❤ ❝→❝ ❤➔♠ sè
f(x)
❧✐➯♥ tö❝ tr♦♥❣ ✤♦↕♥
[0; π]
✈➔
f(x)>0, f (x) + f(y) + f(π−x−y) = π, ∀x, y ∈(0; π), x +y < π.
✭✶✮
❈❤♦
y→0+
✱ t❛ t❤✉ ✤÷ñ❝
f(x) + f(0) + f(π−x) = π, ∀x∈(0; π)
✶
❤❛②
f(π−x) = π−f(0) −f(x),∀x∈(0; π).
❚❤❛② ✈➔♦ ✭✶✮✱ t❛ t❤✉ ✤÷ñ❝
f(x) + f(y) + (π−f(0) −f(x+y)) = π, ∀x, y ∈(0; π), x +y≤π
❤❛②
f(x) + f(y) = f(x+y) + f(0) ,∀x, y ∈[0; π], x +y < π.
✭✷✮
✣➦t
f(x) = f(0) + g(x)
✳ ❑❤✐ ✤â
g(x)
❧✐➯♥ tö❝ tr♦♥❣ ✤♦↕♥
[0; π]
✈➔ ✭✷✮ ❝â ❞↕♥❣
f(0) + g(x) + f(0) + g(y) = f(0) + g(x+y) + f(0) ,∀x, y ∈[0; π], x +y < π
⇔g(x) + g(y) = g(x+y),∀x, y ∈[0; π], x +y < π.
✭✸✮
❉♦
g(x)
❧✐➯♥ tö❝ tr♦♥❣ ✤♦↕♥
[0; π]
♥➯♥ ✭✸✮ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✱ ♠ët ❞↕♥❣
♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❝ì ❜↔♥✱ ❝â ♥❣❤✐➺♠
g(x) = αx
✳ ❙✉② r❛
f(x) = f(0) + αx
✳
✣➦t
f(0) = β
✱ t❛ ✤÷ñ❝
f(x) = αx +β
✳
❚❛ ❝➛♥ ①→❝ ✤à♥❤
α
✱
β
✤➸
f(x)>0
✱
∀x∈(0; π)
✱
x+y < π
✈➔
f(A)+f(B)+f(C) = π
❤❛②
(αx +β > 0,∀x∈(0; π) ;
αA +β+αB +β+αC +β=π.
⇔(αx +β > 0,∀x∈(0; π) ;
α(A+B+C) + 3β=π.
⇔(αx +β > 0,∀x∈(0; π) ;
απ + 3β=π.
⇔
αx +β > 0,∀x∈(0; π) ;
β=(1 −α)π
3.
❉♦ ✤â
f(x) = αx +(1 −α)π
3,∀x∈(0; π).
✭✹✮
❈❤♦
x→0+
✱ tø ✭✹✮✱ s✉② r❛
(1 −α)π
3≥0⇔α≤1.
❈❤♦
x→π−
✱ tø ✭✹✮✱ s✉② r❛
απ +(1 −α)π
3≥0
✷
❤❛②
α≥ −1
2
✳ ❱➟②
−1
2≤α≤1
✳
❱î✐
−1
2< α < 1
✱ t❤➻
f(x)
①→❝ ✤à♥❤ ❜ð✐ ✭✹✮ ❤✐➸♥ ♥❤✐➯♥ t❤ä❛ ♠➣♥ ❜➔✐ t♦→♥✳
❳➨t
α=−1
2
t❤➻
f(x) = −1
2x+π
2
t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❜➔✐ r❛✳
❚❤➟t ✈➟②✱ ✈î✐
0< x < π
t❤➻
f(x)> f (π) = 0
✳ ❙✉② r❛
f(x)>0
✱
∀x∈(0; π)
✳
❳➨t
α= 1
t❤➻
f(x) = x
❤✐➸♥ ♥❤✐➯♥ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❜➔✐ r❛✳ ❱➟② ❝→❝ ❤➔♠ sè ❝➛♥
t➻♠ ✤➲✉ ❝â ❞↕♥❣
f(x) = αx +(1 −α)π
3,−1
2≤α≤1.
◆❤÷ ✈➟②✱ ❧í✐ ❣✐↔✐ tr➯♥ ✤➙② ✤➣ ✈➨t ❤➳t t➜t ❝↔ ❝→❝ ♥❣❤✐➺♠✱ ❧➔ ❝→❝ ❤➔♠ sè
f(x)
✱ t❤ä❛
♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ❜➔✐ t♦→♥✳
❇➙② ❣✐í✱ t❛ t✐➳♣ tö❝ t➻♠ ❦✐➳♠ ♥❤ú♥❣ →♣ ❞ö♥❣ ❝ö t❤➸ ❝õ❛ ❜➔✐ t♦→♥ tr➯♥ ✈➔ ①➨t ♥❤ú♥❣
tr÷í♥❣ ❤ñ♣ ❦❤→❝ ♠➔ ❜➔✐ t♦→♥ ❝❤÷❛ ✤➲ ❝➟♣✳
❚ø ❇➔✐ t♦→♥ ✶✳✶✱ t❛ ❝â
▼➺♥❤ ✤➲ ✶✳✶✳
❱î✐
−1
2≤α≤1
✱ ♥➳✉
A
✱
B
✱
C
❧➔ ❜❛ ❣â❝ ❝õ❛ ♠ët t❛♠ ❣✐→❝✱ t❤➻
A1
✱
B1
✱
C1
①→❝ ✤à♥❤ ♥❤÷ s❛✉
A1=αA +(1 −α)π
3, B1=αB +(1 −α)π
3, C1=αC +(1 −α)π
3,
❝ô♥❣ ❧➔ ❜❛ ❣â❝ ❝õ❛ ♠ët t❛♠ ❣✐→❝✳
▼➺♥❤ ✤➲ ✶✳✷✳
❱î✐
α < −1
2
✱ ♥➳✉
A
✱
B
✱
C
❧➔ ❜❛ ❣â❝ ❝õ❛ ♠ët t❛♠ ❣✐→❝ t❤ä❛ ♠➣♥
max {A, B, C}<(α−1) π
3α
✱ t❤➻
A1
✱
B1
✱
C1
①→❝ ✤à♥❤ ♥❤÷ s❛✉
A1=αA +(1 −α)π
3, B1=αB +(1 −α)π
3, C1=αC +(1 −α)π
3,
❝ô♥❣ ❧➔ ❜❛ ❣â❝ ❝õ❛ ♠ët t❛♠ ❣✐→❝✳
❈❤ù♥❣ ♠✐♥❤✳
❚❤➟t ✈➟②✱ ✈î✐
α < −1
2
✱ t❛ ❝â
max {A, B, C}<(α−1) π
3α⇒A < (α−1) π
3α
⇒3αA + (1 −α)π > 0⇒αA +(1 −α)π
3>0⇒A1>0.
❚÷ì♥❣ tü
B1>0
✈➔
C1>0
✳ ❍ì♥ ♥ú❛✱
A1+B1+C1=π
✱ ♥➯♥ t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣
♠✐♥❤✳
✸
▼➺♥❤ ✤➲ ✶✳✸✳
❱î✐
α > 1
✱ ♥➳✉
A
✱
B
✱
C
❧➔ ❜❛ ❣â❝ ❝õ❛ ♠ët t❛♠ ❣✐→❝ t❤ä❛ ♠➣♥
min {A, B, C}>(α−1) π
3α
✱ t❤➻
A1
✱
B1
✱
C1
①→❝ ✤à♥❤ ♥❤÷ s❛✉
A1=αA +(1 −α)π
3, B1=αB +(1 −α)π
3, C1=αC +(1 −α)π
3,
❝ô♥❣ ❧➔ ❜❛ ❣â❝ ❝õ❛ ♠ët t❛♠ ❣✐→❝✳
❈❤ù♥❣ ♠✐♥❤✳
❚❤➟t ✈➟②✱ ✈î✐
α > 1
✱ t❛ ❝â
min {A, B, C}>(α−1) π
3α⇒A > (α−1) π
3α
⇒3αA + (1 −α)π > 0⇒αA +(1 −α)π
3>0⇒A1>0.
❚÷ì♥❣ tü
B1>0
✈➔
C1>0
✳ ❍ì♥ ♥ú❛✱
A1+B1+C1=π
✱ ♥➯♥ t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣
♠✐♥❤✳
❉÷î✐ ✤➙② ❧➔ ♠ët sè tr÷í♥❣ ❤ñ♣ r✐➯♥❣✱ ♠✐♥❤ ❤å❛ ❝❤♦ ❝→❝ ♠➺♥❤ ✤➲ tr➯♥✳
✲ ❚ø ▼➺♥❤ ✤➲ ✶✳✶✱ ✈î✐
α=−1
2
✱ t❛ ❝â
❍➺ q✉↔ ✶✳✶✳
◆➳✉
A
✱
B
✱
C
❧➔ ❜❛ ❣â❝ ❝õ❛ ♠ët t❛♠ ❣✐→❝✱ t❤➻
A1
✱
B1
✱
C1
①→❝ ✤à♥❤
♥❤÷ s❛✉
A1=π−A
2, B1=π−B
2, C1=π−C
2
❤❛②
A1=B+C
2, B1=C+A
2, C1=A+B
2
❝ô♥❣ ❧➔ ❜❛ ❣â❝ ❝õ❛ ♠ët t❛♠ ❣✐→❝✳
✲ ❚ø ▼➺♥❤ ✤➲ ✶✳✶✱ ✈î✐
α=1
2
✱ t❛ ❝â
❍➺ q✉↔ ✶✳✷✳
◆➳✉
A
✱
B
✱
C
❧➔ ❜❛ ❣â❝ ❝õ❛ ♠ët t❛♠ ❣✐→❝✱ t❤➻
A1
✱
B1
✱
C1
①→❝ ✤à♥❤
♥❤÷ s❛✉
A1=π+ 3A
6, B1=π+ 3B
6, C1=π+ 3C
6
❤❛②
A1=4A+B+C
6, B1=4B+C+A
6, C1=4C+A+B
6
❝ô♥❣ ❧➔ ❜❛ ❣â❝ ❝õ❛ ♠ët t❛♠ ❣✐→❝✳
✹
✲ ❚ø ▼➺♥❤ ✤➲ ✶✳✷✱ ✈î✐
α=−2
3
✱ t❛ ❝â
❍➺ q✉↔ ✶✳✸✳
◆➳✉
A
✱
B
✱
C
❧➔ ❜❛ ❣â❝ ❝õ❛ ♠ët t❛♠ ❣✐→❝ t❤ä❛ ♠➣♥
max {A, B, C}<5π
6
✱
t❤➻
A1
✱
B1
✱
C1
①→❝ ✤à♥❤ ♥❤÷ s❛✉
A1=5π−6A
9, B1=5π−6B
9, C1=5π−6C
9
❤❛②
A1=5B+ 5C−A
9, B1=5C+ 5A−B
9, C1=5A+ 5B−C
9
❝ô♥❣ ❧➔ ❜❛ ❣â❝ ❝õ❛ ♠ët t❛♠ ❣✐→❝✳
✲ ❚ø ▼➺♥❤ ✤➲ ✶✳✷✱ ✈î✐
α=−4
5
✱ t❛ ❝â
❍➺ q✉↔ ✶✳✹✳
◆➳✉
A
✱
B
✱
C
❧➔ ❜❛ ❣â❝ ❝õ❛ ♠ët t❛♠ ❣✐→❝ t❤ä❛ ♠➣♥
max {A, B, C}<3π
4
✱
t❤➻
A1
✱
B1
✱
C1
①→❝ ✤à♥❤ ♥❤÷ s❛✉
A1=3π−4A
5, B1=3π−4B
5, C1=3π−4C
5
❤❛②
A1=3B+ 3C−A
5, B1=3C+ 3A−B
5, C1=3A+ 3B−C
5
❝ô♥❣ ❧➔ ❜❛ ❣â❝ ❝õ❛ ♠ët t❛♠ ❣✐→❝✳
✲ ❚ø ▼➺♥❤ ✤➲ ✶✳✷✱ ✈î✐
α=−1
✱ t❛ ❝â
❍➺ q✉↔ ✶✳✺✳
◆➳✉
A
✱
B
✱
C
❧➔ ❜❛ ❣â❝ ❝õ❛ ♠ët t❛♠ ❣✐→❝ t❤ä❛ ♠➣♥
max {A, B, C}<2π
3
✱
t❤➻
A1
✱
B1
✱
C1
①→❝ ✤à♥❤ ♥❤÷ s❛✉
A1=2π
3−A, B1=2π
3−B, C1=2π
3−C
❤❛②
A1=2B+ 2C−A
3, B1=2C+ 2A−B
3, C1=2A+ 2B−C
3
❝ô♥❣ ❧➔ ❜❛ ❣â❝ ❝õ❛ ♠ët t❛♠ ❣✐→❝✳
✲ ❚ø ▼➺♥❤ ✤➲ ✶✳✷✱ ✈î✐
α=−2
✱ t❛ ❝â
❍➺ q✉↔ ✶✳✻✳
◆➳✉
A
✱
B
✱
C
❧➔ ❜❛ ❣â❝ ❝õ❛ ♠ët t❛♠ ❣✐→❝ t❤ä❛ ♠➣♥
max {A, B, C}<π
2
✱
tù❝ ❧➔ t❛♠ ❣✐→❝
ABC
♥❤å♥✱ t❤➻
A1
✱
B1
✱
C1
①→❝ ✤à♥❤ ♥❤÷ s❛✉
A1=π−2A, B1=π−2B, C1=π−2C
✺