33 chuyên đề khai thác tứ diện vuông cho bài toán bđt và cực trị HH

✶

✶


❑❍❆■ ❚❍⑩❈ ❚Ù ❉■➏◆ ❱❯➷◆●
❈❍❖ ❇⑨■ ❚❖⑩◆ ❇❻❚ ✣➃◆●
❚❍Ù❈ ❱⑨ ❈Ü❈ ❚❘➚
❑✐➲✉ ✣➻♥❤ ▼✐♥❤1
◆❣➔② ✶ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✶
❚â♠ t➢t ♥ë✐ ❞✉♥❣
❚ù ❞✐➺♥ ❧➔ ✤è✐ t÷ñ♥❣ ❝ì ❜↔♥ ❝õ❛ ❤➻♥❤ ❤å❝ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥✳ ❚r➯♥
t↕♣ ❝❤➼ ❚❍❚❚ sè ✸✻✼✱ t❤→♥❣ ✶ ♥➠♠ ✷✵✵✽✱ ✤➣ ❣✐î✐ t❤✐➺✉ ❜➔✐ ✈✐➳t ✧❙ü
♣❤➙♥ ❧♦↕✐ tù ❞✐➺♥ ✈➔ ù♥❣ ❞ö♥❣✧✳ ❚r♦♥❣ ❜➔✐ ❜→♦ ♥➔② ❝❤ó♥❣ tæ✐ ①✐♥ ✤÷ñ❝
tr➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ ❦❤❛✐ t❤→❝ tù ❞✐➺♥ ✈✉æ♥❣ ❝❤♦ ❜➔✐ t♦→♥ ❜➜t
✤➥♥❣ t❤ù❝ ✈➔ ❝ü❝ trà✳ ✣➸ t✐➺♥ ❝❤♦ ✈✐➺❝ t❤❡♦ ❞ã✐ ❝õ❛ ❜↕♥ ✤å❝ ❝❤ó♥❣ tæ✐
♥❤➢❝ ❧↕✐ ♠ët sè ❦➳t q✉↔ q✉❛♥ trå♥❣✳
❈❤♦ OABC ❧➔ tù ❞✐➺♥ ✈✉æ♥❣ t↕✐ O✱ OA = a, OB = b, OC = c, S =
SABC , S1 = SOAB , S2 = SOBC , S3 = SOAC ✳ ●å✐ OH ❧➔ ✤÷í♥❣ ❝❛♦
R, r, R1 ❧➛♥ ❧÷ñt ❧➔ ❜→♥ ❦➼♥❤ ♠➦t ❝➛✉ ♥❣♦↕✐ t✐➳♣✱ ♥ë✐ t✐➳♣ tù ❞✐➺♥ ✈➔
❜→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ t❛♠ ❣✐→❝ ABC; α, β, γ ❧➔ sè ✤♦ ❝→❝ ❣â❝
♥❤à ❞✐➺♥ ❝↕♥❤ AB, BC, CA✳ ❑❤✐ ✤â✿
✶✳ ❚❛♠ ❣✐→❝ ABC ❝â ❜❛ ❣â❝ ♥❤å♥❀
✷✳ OH1 = a1 + b1 + c1 ❀
✸✳ cos2 α + cos2 β + cos2 γ = 1❀
✹✳ S12 = SHAB .S, S22 = SHBC .S, S32 = SHCA.S ❀
✺✳ S12 + S22 + S32 = S 2 ✭✣à♥❤ ❧þ P②t❤❛❣♦r❡✮❀
√
✻✳ VOABC = 61 abc; Stp = 12 (ab + bc + ca + a2 b2 + b2 c2 + c2 a2 )✳
❚❛ ❝❤✐❛ t❤➔♥❤ ❝→❝ ♥❤â♠ ❜→✐ t♦→♥ s❛✉✿ ✭❇↕♥ ✤å❝ tü ✈➩ ❤➻♥❤ ❝❤♦ ❝→❝
❜➔✐ t♦→♥ tr♦♥❣ ❜➔✐ ✈✐➳t ♥➔②✮✳

2

∗1

2

2

2

●✐→♦ ✈✐➯♥ ❚❍P❚ ❝❤✉②➯♥ ❍ò♥❣ ❱÷ì♥❣ P❤ó ❚❤å

✷


✶ ❈→❝ ❜➜t ✤➥♥❣ t❤ù❝ ❧÷ñ♥❣ ❣✐→❝
❇➔✐ t➟♣ ✶✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ cosα.cosβ + cosβ.cosγ + cosγ.cosα ≤ 23 ✳
▲í✐ ❣✐↔✐

❈→❝❤ ✶✳ ●å✐ B ❧➔ ❤➻♥❤ ❝❤✐➳✉ ✈✉æ♥❣ ❣â❝ ❝õ❛ B ❧➯♥ CA✱ t❤➳ t❤➻ OB B ✳❚❛ ❝â
1
a2 c 2
OB 2
1
1
b2 (a2 + c2 )
2
2
=
+

⇒
OB
=
⇒
tan
γ
=
=
OB 2
a2 c 2
a2 + c 2
OB 2
a2 c 2
√
b a2 + c 2
ac
⇒ tanγ =
⇒ cotγ = √
.
ac
b a2 + c 2

❚÷ì♥❣ tü t❤➻

ab
bc
cotα = √
; cotβ = √
.
c a2 + b 2

a b2 + c 2

▼➦t ❦❤→❝
cosB =

AB 2 + BC 2 − CA2
a2 + b 2 + b 2 + c 2 − a2 − c 2
=
=
2.AB.AC
2 (a2 + b2 )(b2 + c2 )

b2
(a2 + b2 )(b2 + c2 )

❙✉② r❛ cotα.cotβ = cosB ✳❱➻ ✈➟②
3
cosα.cosβ + cosβ.cosγ + cosγ.cosα ≤ .
2

❈→❝❤ ✷✳ →♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❆▼✲●▼ t❛ ❝â✿
cosα.cosβ + cosβ.cosγ + cosγ.cosα
2
2
= √ 2 b2 2 2 + √ 2 c2 2 2 + √
≤

(a +b )(b +c )
2
2

1
( b + b2b+c2 )
2 a2 +b2

+

(b +c )(c +a )
2
2
1
( c + c2c+a2 )
2 b2 +c2

a2
(a2 +b2 )(c2 +a2 )
2

+ 12 ( c2a+a2 +

a2
)
a2 +b2

= 32 .

❇➔✐ t➟♣ ✷✳ ❚➻♠ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝

M = tan2 α + tan2 β + tan2 γ + cot2 α + cot2 β + cot2 γ.
▲í✐ ❣✐↔✐

❈→❝❤ ✶✳ →♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❆▼✲●▼ t❛ ❝â
M=

a2 (b2 + c2 ) b2 (c2 + a2 ) c2 (a2 + b2 )
b2 c2
c 2 a2
a2 b2
+
+
+
+
+
b2 c 2
c 2 a2
a2 b 2
a2 (b2 + c2 ) b2 (c2 + a2 ) c2 (a2 + b2 )

✸

.


=(

a2 b 2 c 2 b 2 c 2 a2
15
b2 c 2
c2 a2
a2 b 2
3

=
.
+
+
+
+
+
)+(
+
+
)
≥
6+
b2 a2 b2 c2 a2 c2
a2 b 2 + a2 c 2 b 2 c 2 + b 2 a2 c 2 a2 + c 2 b 2
2
2

❉➜✉ ❜➡♥❣ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ a = b = c✳ ❱➟② ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ M ❧➔ 152
❦❤✐ OABC ❧➔ tù ❞✐➺♥ ✈✉æ♥❣ ❝➙♥ t↕✐ O✳
❈→❝❤ ✷✳ ✣➦t cos2α = x; cos2β = y; cos2γ = z ⇒ x + y + z = 1
⇒ tan2 α

1−x
y+z
1 − cos2 α
=
=
.
2

cos α
x
x

❚÷ì♥❣ tü ♥❤÷ t❤➳
tan2 β =

x+z
x+y
; tan2 γ =
.
y
z

❉♦ ✤â t❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ ❆▼✲●▼ t❛ ❝â
y x
x
y
z
y z
z x
+
+
+
+
+
+
+
+
x y

z y
x x
y+z z+x x+y

M=

15
3
≥ 6+ = .
2
2

❉➜✉ ❜➡♥❣ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = y = z ⇔ α = β = γ ⇔ OABC ❧➔ tù
❞✐➺♥ ✈✉æ♥❣ ❝➙♥ t↕✐ O.
❇➔✐ t➟♣ ✸✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣

cosα
cosβ

2

cosβ
+
cosγ

▲í✐ ❣✐↔✐

2

+

cosγ
cosα

2

≥

3 + tanα.tanβ + tanβ.tanγ + tanγ.tanα.

❚r÷î❝ ❤➳t t❛ ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝

x y z
+ +
y z x

2

1 1 1
+ +
x y z

≥ (x + y + z)

, ∀x, y, z > 0(∗).

❚❤➟t ✈➟②
(∗) ⇔

⇔

x
−1
y

2

+

x
y

2

+

y
z

2

+

z
x

2

+

x z y
x y z
+ + ≥3+ + +
z y x
y z x

2
2
y
z
x
−1 +
−1 +
z
x
y

✹

2

+

y
z

2

+

z
x

2

+2

x z y
+ +
z y x

≥9


❇➜t ✤➥♥❣ t❤ù❝ ❝✉è✐ ✤ó♥❣ t❤❡♦ ❆▼✲●▼ ❝❤♦ ❜❛ sè ❞÷ì♥❣✳ ⑩♣ ❞ö♥❣ (∗) ❝❤♦
x = cos2 α; y = cos2 β; z = cos2 γ t❛ ✤÷ñ❝
cosα
cosβ

2

+

cosβ
cosγ

2

+

cosγ
cosα

≥

1
1
1
+
+
2
2
cos α cos β cos2 γ

=

3 + tan2 α + tan2 β + tan2 γ

≥

3 + tanα.tanβ + tanβ.tanγ + tanγ.tanα

2

✷ ❈→❝ ❜➜t ✤➥♥❣ t❤ù❝ ❤➻♥❤ ❤å❝
❇➔✐ t➟♣ ✹✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣

R1
OH ≤ √ .
2

▲í✐ ❣✐↔✐

❈→❝❤ ✶✳ ❙û ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❆▼✲●▼✱ ✤à♥❤ ❧þ s✐♥ ✈➔ ❦➳t q✉↔ sin2A +
sin2 B + sin2 C ≤ 94 ✱ t❛ ❝â
1
1
1
9
1
=
+
+
≥
OH 2
a2 b 2 c 2
a2 + b2 + c2
a2 + b2 + c2
⇒ OH 2 ≤
9
2
(a
+
b2 ) + (b2 + c2 ) + (c2 + a2 )
2
⇒ 2OH ≤
9

▼➔ t❛ ❧↕✐ ❝â✿
(a2 + b2 ) + (b2 + c2 ) + (c2 + a2 )
AB 2 + BC 2 + CA2

=
9
9
(2R1 sinC)2 + (2R1 sinA)2 + (2R1 sinB)2
=
9
2
2
2
4R1 (sin A + sin B + sin2 C)
=
≤ R12
9

❚ø ✤â s✉② r❛

R1
OH ≤ √ .
2

✺


❈→❝❤ ✷✳ ❚❤❡♦ ✤à♥❤ ❧þ P②t❤❛❣♦r❡ t❛ ❝â
2
2
2
2
SABC
= SOAB

+ SOBC
+ SOCA
2

1
1
AB.BC.CB
1
OA2 .OB 2 + OB 2 .OC 2 + OC 2 .OA2
=
4R1
4
4
4
2
2
2
2
2
2
(OA + OB )(OB + OC )(OC + OA )
⇒
= OA2 .OB 2 + OB 2 .OC 2 + OC 2 .OA2
4R12
⇒

✣➦t OA2 = x; OB 2 = y; OC 2 = z t❤➻
R12 =

(x + y)(y + z)(z + x)

.
4(xy + yz + zx)

▲↕✐ ❝â
1
1
1
1
xy + yz + zx
xyz
2
=
+
+
=
⇒
OH
=
.
OH 2
OA2 OB 2 OC 2
xyz
xy + yz + zx

❇➜t ✤➥♥❣ t❤ù❝ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ ✤÷ì♥❣ ✈î✐
2OH 2 ≤ R12
(x + y)(y + z)(z + x)
2xyz
≤
⇔

xy + yz + zx
4(xy + yz + zx)
⇔ (x + y)(y + z)(z + x) ≥ 8xyz.

❇➜t ✤➥♥❣ t❤ù❝ ❝✉è✐ ✤ó♥❣ t❤❡♦ ❆▼✲●▼✳ ❱➟② t❛ ❝â OH ≤ √R2 ✳ ❉➜✉ ❜➡♥❣ ①↔②
r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ OA = OB = OC.
1

❇➔✐ t➟♣ ✺✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣

VOABC ≤

AB.BC.CA
√
12 2

✳
▲í✐ ❣✐↔✐

❚❛ ❝â

OA =

CA2 + AB 2 − BC 2
; OB =
2

✻

AB 2 + BC 2 − CA2

;
2


OC =

❙✉② r❛

BC 2 + CA2 − AB 2
2

1
OA.OB.OC
6
1 (CA2 + AB 2 − BC 2 )(AB 2 + BC 2 − CA2 )(BC 2 + CA2 − AB 2 )
=
.
6
8

VABC =

▲↕✐ ❝â
(CA2 + AB 2 − BC 2 )(AB 2 + BC 2 − CA2 ) = AB 4 − (CA2 − BC 2 )2 ≤ AB 4

❚÷ì♥❣ tü
(AB 2 + BC 2 − CA2 )(BC 2 + CA2 − AB 2 ) ≤ BC 4

❙✉② r❛

(CA2 + AB 2 − BC 2 )(AB 2 + BC 2 − CA2 ) ≤ CA4

VOABC ≤

1
6

AB.BC.CA
AB 2 BC 2 CA2
√
=
8
12 2

❇➔✐ t➟♣ ✻✳ ❈❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝

√
S1
S2
S3
3 3
√
+
+
≤
S + 2S1 S + 2S2 S + 2S3
3+2 3

▲í✐ ❣✐↔✐

✣➦t

S1
S2
S3
+
+
S + 2S1 S + 2S2 S + 2S3
2S1
2S2
2S3
⇒ 2P =
+
+
S + 2S1 S + 2S2 S + 2S3
1
1
1
= 3 − S(
+
+
)
S + 2s1 S + 2s2 S + 2s3
P =

❚❛ ❝â
1
1
1
9

+
+
≥
S + 2s1 S + 2s2 S + 2s3
3S + 2(S1 + S2 + S3
9
9
√
≥
=
2
2
2
(3 + 2 3)S
3S + 2 3(S1 + S2 + S3

✼


❙✉② r❛✿
√
√
9
6 3
3 3
√
√ ⇒P ≤
√
2P ≤ 3 − S
=

(3 + 2 3)S
3+2 3
3+2 3

✸ ❈→❝ ❜➜t ✤➥♥❣ t❤ù❝ ✤↕✐ sè
❇➔✐ t➟♣ ✼✳ ❈❤♦ x, y, z ❧➔ ❝→❝ sè t❤ü❝ ❞÷ì♥❣✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣

x
(x + y)(x + z)

y

+

(y + z)(y + x)

+

z
(z + x)(z + y)

≤

3
2

▲í✐ ❣✐↔✐

❉ü♥❣ tù ❞✐➺♥ ❖❆❇❈ ✈✉æ♥❣ t↕✐ ❖ ❝â ❝→❝ ❝↕♥❤ OA = √x❀ OB = √y❀ OC = √z
t❤➻ t❛ ❝â

cos A =
=
=

AB 2 + AC 2 − BC 2
2AB.AC
x + y + y + z − (y + z)
2 (x + y)(x + z)
x
(x + y)(x + z)

❚÷ì♥❣ tü
y

cos B =

(y + z)(y + x)

; cos C =

z
(z + x)(z + y)

❉♦ ✤â
x
(x + y)(x + z)

+

y

(y + z)(y + x)

z

+

(z + x)(z + y)

= cos A+cos B+cos C ≤

❇➔✐ t➟♣ ✽✳ ✭❑♦r❡❛✳✶✾✾✽✮ ❈❤♦ a, b, c > 0 : a + b + c = abc✳ ❈❤ù♥❣ ♠✐♥❤

r➡♥❣

√

1
1
1
3
+√
+√
≤
2
1 + a2
1 + b2
1 + c2

✽

3
2


▲í✐ ❣✐↔✐

❚❤❡♦ ❜➔✐ t♦→♥ ✼✱ ❣✐↔ sû xy + yz + zx = 1 t❤➻
√

x
+
1 + x2

y
1 + y2

+√

z
3
≤
2
1 + z2

✣➦t x = a1 ✱ y = 1b ✱ z = 1c ✱ t❛ ❝â ❜➔✐ t♦→♥ ✽✳
❇➔✐ t➟♣ ✾✳ ❈❤♦ ①✱②✱③ ❧➔ ❝→❝ sè t❤ü❝ ❞÷ì♥❣✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣

1
1
9

1
+
+
≤
(x + y)(x + z) (y + z)(y + x) (z + x)(z + y)
4(xy + yz + zx)
▲í✐ ❣✐↔✐

❉ü♥❣ tù ❞✐➺♥ ❖❆❇❈ ✈✉æ♥❣ t↕✐ ❖ ❝â ❝↕♥❤ OA = √x❀ OB = √y❀ OC = √z✳
❚❛ ❝â✿
x
xy + yz + zx
cos A =

(x + y)(x + z)

⇒ sin2 A =

(x + y)(x + z)

❉♦ ✤â s✉② r❛
xy + yz + zx
xy + yz + zx
9
xy + yz + zx
+
+
= sin2 A+sin2 B sin2 C ≤
(x + y)(x + z) (y + z)(y + x) (z + x)(z + y)
4

◆➳✉ ❣✐↔ t❤✐➳t xy + yz + zx = 1 t❤➻ t❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝ q✉❡♥ t❤✉ë❝ s❛✉✿
x2

1
1
9
1
+ 2
+ 2
≤
+1 y +1 z +1
4

❇➔✐ t➟♣ ✶✵✳ ❈❤♦ a, b, c > 0 : ab + bc + ca = (a + b)(b + c)(c + a)✳ ❈❤ù♥❣

♠✐♥❤ r➡♥❣

√
a
3 3
b
c
√
≤
+√
+√
(∗)
4
c+a

a+b
b+c

▲í✐ ❣✐↔✐

❚ø ❣✐↔ t❤✐➳t a, b, c > 0 : ab + bc + ca = (a + b)(b + c)(c + a) t❛ ❜✐➳♥ ✤ê✐ ✭✯✮
♥❤÷ s❛✉✿
(∗) ⇔ √

√
a ab + bc + ca
a+b

(a + b)(b + c)(c + a)

✾

+√

√
b ab + bc + ca
b+c

(a + b)(b + c)(c + a)

+


+√
⇔

a
(a + b)(a + c)

√
3 3
≤
4
(a + b)(b + c)(c + a)

√
c ab + bc + ca
c+a

c

+

b

ab + bc + ca
+
(b + a)(b + c)

(c + a)(c + b)

ab + bc + ca
+
(c + b)(c + a)

(b + a)(b + c)
√
3 3
ab + bc + ca
≤
(∗∗)
(a + b)(a + c)
4
√
❖ ❝â OA = √a✱OB = b✱OC = √c✳

❉ü♥❣ tù ❞✐➺♥ ❖❆❇❈ ✈✉æ♥❣ t↕✐
❆✱❇✱❈ ❧➔ ❜❛ ❣â❝ ❝õ❛ t❛♠ ❣✐→❝ ❆❇❈ ✭t❛♠ ❣✐→❝ ❆❇❈ ♥❤å♥✮✱ t❤➳ t❤➻

●å✐

a+b+a+c−b−c
AB 2 + AC 2 − BC 2
=
cos A =
AB.AC
2 (a + b)(a + c)
⇒ sin A =

√

1 − cos2 A =

ab + bc + ca
(a + b)(a + c)

❚÷ì♥❣ tü ❝ô♥❣ ❝â
cos B =

cos C =

b
(b + a)(b + c)
c
(c + a)(c + b)

; sin B =

ab + bc + ca
(b + a)(b + c)

; sin C =

ab + bc + ca
(c + a)(c + b)

❑❤✐ ✤â ✭✯✮ trð t❤➔♥❤

√
3 3
sin A cos C + sin C cos B + sin B cos A ≤
(∗ ∗ ∗)
4
√

◆❤➟♥ ①➨t r➡♥❣ ✈î✐ 0 < x < π2 t❤➻ f (x) = sin x + 12 sin 2x ≤ 3 4 3 ✳
❚❤➟t ✈➟②✿

f (x) = cos x + cos 2x, f (x) = (cos x + 1)(2 cos x − 1) ≥ 0 ⇔ 0 < x ≤ π3
√
π
3 3
⇒ f (x) ≤ f ( ) =
3
4
✣➦t P = sin A cos C + sin C cos B + sin B cos A✳ ●✐↔ sû A = min{A; B; C}✳

❑❤✐ ✤â ①↔② r❛ tr÷í♥❣ ❤ñ♣ s❛✉✿

✶✵


• A ≤ B ≤ C✳

❙✉② r❛

❚❤➳ t❤➻ (sin C − sin B)(cos B − cos A) ≤ 0 ✈➔ 0 < B < π2

√
1
3 3
P ≤ sin A cos C + sin C cos A + sin B cos B = sin B + sin 2B ≤
2
4
• A ≤ C ≤ B✳

❙✉② r❛

❚❤➳ t❤➻ (sin A − sin C)(cos C − cos B) ≤ 0 ✈➔ 0 < C < π2

√
3 3
1
P ≤ sin A cos B + sin B cos A + sin C cos C = sin C + sin 2C ≤
2
4

❚❛ t❤➜② r➡♥❣ ✭✯✯✯✮ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤ ❝❤ù♥❣ tä ✭✯✯✮ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳
❱➟② ❜➜t ✤➥♥❣ t❤ù❝ ✭✯✮ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ①♦♥❣✳

✹ ❇➔✐ t➟♣ ❝õ♥❣ ❝è
❚r➯♥ ✤➙② ❧➔ ♥❤ú♥❣ ❦❤❛✐ t❤→❝ ❜❛♥ ✤➛✉ ❝õ❛ ❝❤ó♥❣ tæ✐ ✈➲ tù ❞✐➺♥ ✈✉æ♥❣ ❝❤♦ ❝→❝
❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ✈➔ ❝ü❝ trà✳ ❍② ✈å♥❣ r➡♥❣ ❝→❝ ❜↕♥ s➩ ❦❤❛✐ t❤→❝ t❤➯♠
✤÷ñ❝ ♥❤✐➲✉ ❜➔✐ t♦→♥ ♠î✐ ✈➔ ♥❤✐➲✉ ✤✐➲✉ t❤ó ✈à ✈➲ tù ❞✐➺♥ ✤➦❝ ❜✐➺t ♥➔②✳ ❈✉è✐
❝ò♥❣ ①✐♥ ♠í✐ ❝→❝ ❜↕♥ ❧➔♠ ♠ët sè ❜➔✐ t♦→♥ s❛✉✿
❇➔✐ ✶✿ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣✿ (2 + tan2 α)(2 + tan2 β)(2 + tan2 γ) ≥ 64
❇➔✐ ✷✿ ❚➻♠ ❣✐→ trà ❧î♥ ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝✿
M = (cot α. cot β + cot β. cot γ + cot γ. cot α)2 + 6(cos α. cos β. cos γ)2

❇➔✐ ✸✳ ❚➻♠ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝
P =

cos α + cos β cos β + cos γ cos γ + cos α
+
+

cos2 γ
cos2 α
cos2 β

❇➔✐ ✹✳ ❈❤ù♥❣ ♠✐♥❤✿
√
1
1 1 1
3 3
a) ≥ + + +
;
r
a b c a+b+c
b)

√
2R
≥ 3(1 + 3)
r

✶✶


❇➔✐ ✺✳ ●å✐ ▼ ❧➔ ✤✐➸♠ t❤✉ë❝ t❛♠ ❣✐→❝ ❆❇❈✳ ❚➻♠ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❜✐➸✉
t❤ù❝✿
2
2
2
AM
BM

CM
+
+
.
2
2
AO
BO
CO2

❇➔✐ ✻✳ ❳→❝ ✤à♥❤ ✈à tr➼ ✤✐➸♠ M ✤➸ ❜✐➸✉ t❤ù❝ s❛✉ ❧➔ ♥❤ä ♥❤➜t
√

3M O + M A + M B + M C.

❇➔✐ ✼✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣
S12
S22
S32
3
+
+
≤ .
2
2
2
2
2
2
S + S1

S + S2
S + S3
4

❇➔✐ ✽✳ ❈❤♦ x, y, z > 0✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣
1 1 1
+ + ≥3
x y z

3
.
xy + yz + zx

❇➔✐ ✾✳ ❈❤♦ a, b, c > 0 : a + b + c = abc✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣
√
a
b
c
3 3
√
+√
+√
≤
2
1 + a2
1 + b2
1 + c2

❇➔✐ ✶✵✳ ❈❤♦ x, y, z > 0✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣
(x + y)(x + z)

+
x

(y + z)(y + x)
+
y

✶✷

(z + x)(z + y)
≥ 6.
z