WholeIssue 32 1 - CMS-SMC · 2019. 8. 2. · #},%j6+ ^
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Transcript of WholeIssue 32 1 - CMS-SMC · 2019. 8. 2. · #},%j6+ ^
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1
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1
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2k
)
+
(
1
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=
(
1
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1
k + 2+ · · · +
1
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+
(
1
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cos 90◦ = 01.n:#}(�,j"�#
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cos x = sin(90◦ − x),.J�K
sin2 x = 1 − cos2 x1.n:#}u-#j&
S = sin2(90◦ − 1◦) + sin2(90◦ − 2◦) + · · · + sin2(90◦ − 89◦)
= sin2 89◦ + sin2 88◦ + · · · + sin2 1◦
= (1 − cos2 89◦) + (1 − cos2 88◦) + · · · + (1 − cos2 1◦)
= 89 − Sz
�[(Qi�/212S = 89
1.n:(Qq p (VqrdLm�erqv#%/|&)(�,j&S = 44.5
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(x + 4y)(5x + 9y) = 01
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h�$x = −9
5yz
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y = ±11Q,.J�K
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5yqrJQ&bh
x2 + xy = 12u2q�"�#-/ 81
25y2 − 9
5y2 = 12 �&)(�,-&oqr/j1 36
25y2 = 12
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f :� → � /ai p (�&)(�,j&
f(
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f(0) = mz��[(*#-J
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k = 0z A%#j&
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f(k) = m + kzV��(*#%J
f(m +k) = f(f(k)) = k +1z��[(Qi�/21
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f(m) = 2m1.n:(Qqrev#
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f(
f(n))
= n + 1987z�³|J6+�/�hQeyi�&cqUh�J�hQf'&)(Qqv/�m�$ch ^ eU#-d p h�JQ&0,%qrJ*/],C$c#-dL,%$)sV&)(�,j&&)(*#}$c#%/)i*ey&|(*hQeUK�/|�Ui�/b& ^ # p ,-i�/�#
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z A%#2&AP
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AB,.J�K
PQdo#%#j&�,j&
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&?n:hVeyqvJ�#-/RH
,%J�KCM
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x =−→OA
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z|�U&�qv/ s.J�h.n}JV&)(�,-&QP
do#%#j&0/AB
,-&R1%n:(*#%$c#
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−→OR =
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2(x + y) − z
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= (b cos A)x − (a cos B)x
+ (b cos A − a cos B)zz
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+ (b cos A + a cos B)(y − x) · z
= ρ2(a cos B − b cos A)(1 − cos 2C) + c(cos 2A − cos 2B)
= 2ρ2c(a cos B − b cos A) sin C(
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+ sin(B − A))
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(√a −
√b)2
+(√
b − √c)2
+(√
c − √a)2
(√a +
√b +
√c)2 ≤ 4
9
(
R
r− 2
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16
(
R
r− 2
) 1n:(*#%$c#
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1',%J�Kz = s − c
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R =abc
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,.J�KF = rs
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r=
abcs
4F 2=
abcs
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(x + y)(y + z)(z + x)
4xyz
z�}h.n:#�"�#-$�1
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4xyz=
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y + z
x + y=
c
a+
a
c
1
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y + z
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r≥ c
a+
a
c
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R
r− 2 ≥ c2 + a2
ac− 2 =
(a − c)2
ac=
(√
a +√
c)2(√
a − √c)2
ac
≥ 4√
ac(√
a − √c)2
ac
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a ≥ b ≥ cz��-qrJ p #
(√
a −√
c)2 =(
(√
a −√
b) + (√
b −√
c))2
≥ (√
a −√
b)2 + (√
b −√
c)21
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a +√
b +√
c)2 ≥ (√
a + 2√
c)2 ≥ 8√
ac1.n:#}(�,j"�#
D =(√
a −√
b)2 + (√
b − √c)2 + (
√c − √
a)2
(√
a +√
b +√
c)2
≤ 2(√
a − √c)2
8√
ac=
4√
ac(√
a − √c)2
16ac
z w=� z� $ch�d w � z ,.J�KQw=� z 1�n:# h ^ &c,-qvJ R
r−2 ≥ 16D
1j,.J�K�(*#-J p #D ≤ 1
16
(
R
r− 2
) z
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D ≤ 4
27
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r− 2
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ha
wa
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wb
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wc
≥ 1 +4r
R
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h2a
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b
w2b
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c
w2c
≥ 1 +4r
R
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wb
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wc
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b
w2b
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c
w2c
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2+ cos2
C − A
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2
= 32
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12
A)
sin(
12
B)
sin(
12
C)
= 1 +r
R
�
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n:#:h ^ &c,-qvJ
32
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≥ 32
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= 32
+ 12
[
8(
1 + 4 sin(
12
A)
sin(
12
B)
sin(
12
C))
− 9]
= 32
+ 4 (1 + r/R) − 92
= 1 + 4r/R1
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v+
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5
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v+
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w=
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1�n:#�dCi�/)&�(�,2"�#v = 4
z���#%J p #�1 1
w=
N − 1
2N + 2
,%J�K
w =2N + 2
N − 1= 2 +
4
N − 1
zO�U&}fGhQeyeUh.n}/:&)(�,j&N − 1
K�q "�qvK�#-/4z���(*#%$c#jfGh�$c#�1
N = 3h�$
N = 5z��]h.n
N = 4ev#%,.K*/ &ch
w = 3 < v14, p h�JQ&b$0,6KQq p &cqvh�J�z4��(Qi�/j1
N = 3,.J�K
w = 41Qn:(Qq p (V+QqU#jeUK�/
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v = 4h�$
v = 3z
³|/ , ^ h-"�#�1v = 4
eU#-,6K�/|&bhw =
12N + 12
5N − 7
1QqrdLm�e�+QqvJ�u�&)(�,j&5N − 7
dki�/b&TK�q "�qvK�#12N + 12
z���(*#%J5N − 7
dCi�/)&�KQq�"QqUK*#5(12N + 12) − 12(5N − 7) = 144
z�-qvJ p #
N + 1qr/�,VdCi*ey&cqrm*ev#VhQf ^ h6&)(
3,%J�K
41�n:#�/�#.#�&)(�,-&
N ≥ 11z��U&Oqr/]#-,./)qre +
/�#%#%J�&)(�,j& &)(*#kh�J�e�+�m4h�/a/aq ^ qyerq�& +oqv/5N − 7 = 48
,.J�KN = 11
z � i�&O&)(Qqv/�u2q "�#%/w = 3 < v
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eU#-,6K�/ &chN = 5
h�$1114,.J�Kk&ch
&)(*#}/�hQeyi�&cqUh�J*/(3, 3, 6, 5)
,%J�K(3, 3, 4, 11)
z
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v = 6151
41*h�$
3,.J�KVeU#-,6K�/|&bh
9doh�$c#�/�hQeyi�&cqUh�J*/-z
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qvJQ&c#.u-#-$0/u1v1w1x1[/ai p (o&)(�,-&
u1v1w1[,%J�K
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N + 1w ^ i�&�J�h�J�#
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1
u+
1
v+
1
w+
1
x=
N
N + 1
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f(k) = a(k + 1) − 21
n:(*#%$c#a(k)
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N + 1
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ABC^ #V,:&b$cqr,.J�u2ev#�14,%J�KoeU#2&
A1 ∈ BC1B1 ∈ CA
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/ai p (&)(�,-&BA1
A1C=
CB1
B1A=
AC1
C1B= k > 0
z
� z���$ch%"�#]&)(�,j&w&)(*#}/�#%ujdo#-JQ&c/AA1
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rk
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T ��*z[Tk] =
k2 + k + 1
(k + 1)2[ABC]
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K*#%J�h6&c#%/|&)(*#},%$c#%,:hQf4&b$cqr,.J�u2ev#T �
{[zRk ≥ k
√k P (ABC)
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rk >k2 + k + 1
(k + 1)2r1.n:(*#%$c#
rqr/|&)(*#�qvJ*$0,.K�qyi�/�hQf 4ABC
z
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a = BC1b = CA
1c = AB
1�,%J�Ks = 1
2(a + b + c)
z��-qrJ p #
BA1
A1C=
CB1
B1A=
AC1
C1B= k
1
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ak
k + 1
,.J�KA1C =
a
k + 1
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CB11B1A
1AC1
1Q,.J�KC1B
z � +�&)(*#BA-,2n hQf'�[h�/aqrJ�#%/21.n:#}h ^ &0,%qrJ
AA21 = c2 +
a2k2
(k + 1)2− k
k + 1· 2ac cos B
=c2(k + 1)2 + a2k2 − (k2 + k)(a2 + c2 − b2)
(k + 1)2
=b2(k2 + k) + c2(k + 1) − a2k
(k + 1)2z
� z�A%#2&A2 ∈ BC
1B2 ∈ CA
1Q,.J�KC2 ∈ AB
/)i p (V&)(�,j&
BA2
A2C=
CB2
B2A=
AC2
C2B= k + 1
z
�"�
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a
k + 2
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b(k + 1)
k + 2
z � +�&)(*#BA-,2n hQf'�[h�/aqrJ�#%/21
A2B22 =
a2
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b2(k + 1)2
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(k + 2)2· 2ab cos C
=a2 + b2(k + 1)2 − (k + 1)(a2 + b2 − c2)
(k + 2)2
=b2(k2 + k) + c2(k + 1) − a2k
(k + 2)2=
(k + 1)2
(k + 2)2· AA2
1
z
�}#-J p #Q1AA1 =
k + 2
k + 1· A2B2
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1BB1
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r
r
A
A1 A2
A′
B
B1
B2
C
C1
C2
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ABqrJQ&b#-$0/�# p &c/
AA1,j&�,Om�hQqvJQ&
A′ z � +�/aqv�dkqyerq�&ci4K�#Q1A′A1 = AA1/k
,.J�KCA′ = c/k
zl�EJ 4ACA′ 1Ln:#�(�,2"�#AA′ < AC + CA′ z�{*�Qi*q�"Q,-eU#-JQ&ce�+�1 AA1 +
AA1
k< b +
c
k
1�h�$AA1 <
bk + c
k + 1
z�}#-J p #Q1
P (Tk) = AA1+BB1+CC1 <bk + c + ck + a + ak + b
k + 1= P (ABC)
z
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bk + c
k + 1
z-��(Qqv/qvJ�#%��i�,-erq�& +:qv/|#.�Qi*q�"Q,-eU#-JQ&�&bh
b2(k2 + k)+ c2(k +1) − a2k < (bk + c)21Qh�$_w ^ +
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z��z���#}(�,2"�#
[CA2B2] =A2C · B2C · sin C
2=
ab(k + 1) sin C
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k + 1
(k + 2)2[ABC]
z�-qvJ p #]&)(*#t/)qUK*#%/ThQf 4A2B2C2
,.$c#]qrJ�&)(*#t$0,j&cqUhk + 2 : k + 1
&bh}&)(*#t/aqvK�#-/OhQfTk1
n:#:K*#.KQi p #t&)(�,j&
[Tk] =(k + 2)2
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(k + 2)2
(k + 1)2
(
[ABC] − 3k + 1
(k + 2)2[ABC]
)
=(k2 + k + 1)
(k + 1)2[ABC]
z
��{
{'z���#}(�,2"�#AA1 =
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k + 1
≥√
2bck + b2k + c2k − a2k
k + 1=
√k√
(b + c)2 − a2
k + 1
=
√k√
(b + c + a)(b + c − a)
k + 1=
2√
k√
s(s − a)
k + 1
z
�-qvJ p #Rk =
AA1 · BB1 · CC1
4[Tk]
1Qq�&�fGhQeyeUh.n}/�&)(�,-&
Rk ≥ 8k√
k√
s3(s − a)(s − b)(s − c)
(k + 1)3 · 4 · k2 + k + 1
(k + 1)2[ABC]
=2k
√k · s · [ABC]
(k + 1)(k2 + k + 1)[ABC]=
k√
k P (ABC)
(k + 1)(k2 + k + 1)
1
n:(*#%$c#�n:#L(�,2"�#Ci�/�#%K_��#%$ch�J[´ / � h�$cdki*er,[ABC] =
√
s(s − a)(s − b)(s − c)z
{*�Qi�,%eyqy&E+k(*hQeUK�/Tqyf�,.J�Kkh�J�e�+�qrfbk = c
1ck = a
1Q,.J�Kak = b �
&)(�,j&�qr/j16qyf�,.J�Kkh�J�e +qrf
abck3 = abc1.n:(Qq p (CdL#%,%J*/T&)(�,j&
k = 1,%J�K
a = b = cz
� z'�-qrJ p #rk =
2[Tk]
P (Tk)
,.J�Kr =
2[ABC]
P ([ABC])
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