Chủ đề này có 34 trả lời

[nxt1989]

cái này anh khuyên thế thôi .Sau này em sẽ tự nhận thức đuợc .Cái này cứ phải từ từ ,trải qua rồi mới hiểu .

[huyetdao_tama]

Khủng khiếp wá. Thế này mà cũng dám làm. Em xin bái phục .

[vuthanhtu_hd]

Trời ơi!!!Quả là con người kiên trì.Nản quá

[Toanlc_gift]

hix,cái đó biểu diễn trên maple cũng phải mấy mặt giấy đấy nhỉ
nó cồng kềnh đến mức không đủ bình tính để đặt dấu sigma vào
nhưng nếu expand như thế thì làm sao mà chứng minh $f(T) \le 0$ được

Bài viết đã được chỉnh sửa nội dung bởi Toanlc_gift: 12-08-2009 - 19:26

[Toanlc_gift]

đây là $f((a+b+c)^3)$
post vào 6 cái ảnh vậy:P






hix,đừng chém em

Bài viết đã được chỉnh sửa nội dung bởi Toanlc_gift: 12-08-2009 - 20:15

[mai quoc thang]

Cho $a;b;c \ge 0$
chứng minh rằng:
$a\sqrt{16a^4+65b^3c} +b\sqrt{16b^4+65c^3a} +c\sqrt{16c^4+65a^3b} \ge (a+b+c)^3$

Tớ có một lời giải phải nói là rất xấu theo phong cách của ngài Ji chen .

Lời giải :

Đặt :

$ m=a\sqrt{16a^4+65b^3c} \ ; \ n= b\sqrt{16b^4+65c^3a} \ ; \ t=c\sqrt{16c^4+65a^3b}$

Xét hàm :

$ F(T)=(T+m+n+t)(T+m+n-t)(T+m-n-t)(T-m-n-t)(T-m-n+t)(T-m+n+t)(T+m-n+t)(T-m+n-t) $

Ta có :

$ F \left( T \right) ={T}^{8}-64\,{b}^{6}{T}^{6}-64\,{a}^{6}{T}^{6}-260
\,{c}^{2}{a}^{3}b{T}^{6}-260\,{b}^{2}{c}^{3}a{T}^{6}-64\,{c}^{6}{T}^{6
}-260\,{a}^{2}{b}^{3}c{T}^{6}
$

$+25350\,{a}^{4}{b}^{6}{c}^{2}{T}^{4}+1024
\,{c}^{6}{b}^{6}{T}^{4}+1024\,{c}^{6}{a}^{6}{T}^{4}+4160\,{b}^{2}{c}^{
3}{a}^{7}{T}^{4}+25350\,{b}^{4}{c}^{6}{a}^{2}{T}^{4}+16900\,{b}^{5}{c}
^{4}{a}^{3}{T}^{4}+25350\,{c}^{4}{a}^{6}{b}^{2}{T}^{4} $

$ +4160\,{a}^{2}{b
}^{3}{c}^{7}{T}^{4}+4160\,{c}^{2}{a}^{9}b{T}^{4}+1024\,{a}^{6}{b}^{6}{
T}^{4}+4160\,{b}^{2}{c}^{9}a{T}^{4}+1536\,{a}^{12}{T}^{4}+1536\,{c}^{
12}{T}^{4}+1536\,{b}^{12}{T}^{4}+12480\,{c}^{8}{a}^{3}b{T}^{4}+4160\,{
c}^{2}{a}^{3}{b}^{7}{T}^{4} $

$ +12480\,{a}^{8}{b}^{3}c{T}^{4}+12480\,{b}^{
8}{c}^{3}a{T}^{4}+4160\,{a}^{2}{b}^{9}c{T}^{4}+16900\,{a}^{5}{b}^{4}{c
}^{3}{T}^{4}+16900\,{b}^{3}{c}^{5}{a}^{4}{T}^{4}-2704000\,{c}^{3}{a}^{
5}{b}^{10}{T}^{2}+133120\,{b}^{8}{c}^{9}a{T}^{2}+540800\,{b}^{9}{c}^{5
}{a}^{4}{T}^{2} $

$+1098500\,{b}^{7}{c}^{7}{a}^{4}{T}^{2}-811200\,{c}^{10}
{a}^{6}{b}^{2}{T}^{2} -1098500\,{c}^{6}{a}^{9}{b}^{3}{T}^{2}-665600\,{c
}^{2}{a}^{9}{b}^{7}{T}^{2}-2704000\,{c}^{5}{a}^{10}{b}^{3}{T}^{2}+
270400\,{b}^{4}{c}^{12}{a}^{2}{T}^{2} $

$ -199680\,{b}^{14}{c}^{3}a{T}^{2}+
133120\,{a}^{8}{b}^{3}{c}^{7}{T}^{2}-11148840\,{c}^{6}{a}^{6}{b}^{6}{T
}^{2}$

$ -16384\,{a}^{18}{T}^{2}-16384\,{c}^{18}{T}^{2}+540800\,{b}^{11}{c
}^{4}{a}^{3}{T}^{2}+1098500\,{c}^{7}{a}^{7}{b}^{4}{T}^{2}$

$ +270400a^6b^8c^4T^2 $

$+540800\,{a}^{4}{b}^{3}{c}^{11}{T}^{2}+1098500
\,{b}^{5}{c}^{8}{a}^{5}{T}^{2}+133120\,{b}^{7}{c}^{8}{a}^{3}{T}^{2}+
540800\,{a}^{11}{b}^{4}{c}^{3}{T}^{2}+270400\,{a}^{4}{b}^{6}{c}^{8}{T}
^{2}-199680\,{c}^{14}{a}^{3}b{T}^{2}$

$ -2704000\,{c}^{10}{a}^{3}{b}^{5}{T
}^{2}+133120\,{c}^{8}{a}^{9}b{T}^{2}+270400\,{c}^{4}{a}^{12}{b}^{2}{T}
^{2}+540800\,{c}^{9}{a}^{5}{b}^{4}{T}^{2}+1098500\,{c}^{5}{a}^{8}{b}^{
5}{T}^{2}+16384\,{c}^{12}{a}^{6}{T}^{2}+66560\,{a}^{2}{b}^{15}c{T}^{2}
-199680\,{a}^{14}{b}^{3}c{T}^{2}$

$ -665600\,{c}^{9}{a}^{7}{b}^{2}{T}^{2}+
66560\,{a}^{15}{c}^{2}b{T}^{2}-811200\,{b}^{10}{c}^{6}{a}^{2}{T}^{2}+
16384\,{c}^{12}{b}^{6}{T}^{2}+540800\,{a}^{9}{b}^{5}{c}^{4}{T}^{2}+
1098500\,{a}^{7}{b}^{7}{c}^{4}{T}^{2}$

$ +16384\,{a}^{6}{b}^{12}{T}^{2}-
1098500\,{b}^{6}{c}^{9}{a}^{3}{T}^{2}+270400\,{b}^{4}{c}^{6}{a}^{8}{T}
^{2}+16384\,{b}^{12}{c}^{6}{T}^{2}+16384\,{a}^{12}{b}^{6}{T}^{2}+
1098500\,{a}^{5}{b}^{8}{c}^{5}{T}^{2}-16384\,{b}^{18}{T}^{2}$

$ +66560\,{c}^{15}{b}^{2}a{T}^{2}-665600\,{c}^{7}{a}^{2}{b}^{9}{T}^{2}+66560\,{b}^
{13}{c}^{2}{a}^{3}{T}^{2}+133120\,{b}^{8}{c}^{3}{a}^{7}{T}^{2}+16384\,
{a}^{12}{c}^{6}{T}^{2}-811200\,{a}^{10}{b}^{6}{c}^{2}{T}^{2}-1098500\,
{a}^{6}{b}^{9}{c}^{3}{T}^{2} $

$ +66560\,{a}^{13}{b}^{2}{c}^{3}{T}^{2}+
66560\,{c}^{13}{a}^{2}{b}^{3}{T}^{2}+270400\,{a}^{4}{b}^{12}{c}^{2}{T}
^{2}+133120\,{a}^{8}{b}^{9}c{T}^{2}+1064960\,{b}^{20}{c}^{3}a+1064960
\,{a}^{20}{b}^{3}c+35414144\,{c}^{6}{a}^{12}{b}^{6} $

$ -71402500\,{b}^{9}{
c}^{10}{a}^{5}+35414144\,{a}^{6}{b}^{6}{c}^{12}+17576000\,{c}^{7}{a}^{
13}{b}^{4}-12979200\,{c}^{10}{a}^{6}{b}^{8}+1064960\,{c}^{20}{a}^{3}b-
12979200\,{a}^{17}{b}^{4}{c}^{3} $

$ +25958400\,{b}^{9}{c}^{11}{a}^{4}+
6489600\,{a}^{16}{b}^{6}{c}^{2}+107103750\,{b}^{10}{c}^{8}{a}^{6}+
73532420\,{c}^{8}{a}^{9}{b}^{7}+8652800\,{c}^{5}{a}^{10}{b}^{9}+
1064960\,{a}^{13}{b}^{2}{c}^{9}$

$ -12979200\,{a}^{15}{b}^{5}{c}^{4}-
12979200\,{c}^{15}{a}^{5}{b}^{4}-12979200\,{b}^{15}{c}^{5}{a}^{4}+
25958400\,{c}^{9}{a}^{11}{b}^{4}+6489600\,{c}^{14}{a}^{4}{b}^{6}+
25958400\,{a}^{9}{b}^{11}{c}^{4} $

$ -12979200\,{a}^{10}{b}^{12}{c}^{2}-
12979200\,{a}^{3}{b}^{17}{c}^{4}+17576000\,{c}^{8}{a}^{11}{b}^{5}+
1064960\,{c}^{15}{b}^{2}{a}^{7}-12979200\,{c}^{17}{b}^{3}{a}^{4}-
71402500\,{a}^{7}{b}^{11}{c}^{6} $

$ -17576000\,{b}^{9}{c}^{9}{a}^{6}+
17576000\,{b}^{7}{c}^{13}{a}^{4}+52728000\,{a}^{7}{b}^{7}{c}^{10}+
107103750\,{a}^{10}{b}^{8}{c}^{6}+52728000\,{a}^{5}{b}^{14}{c}^{5}-
12979200\,{c}^{8}{a}^{10}{b}^{6} $

$ -17576000\,{a}^{6}{b}^{15}{c}^{3}+
6489600\,{a}^{4}{b}^{18}{c}^{2}+17576000\,{a}^{12}{b}^{9}{c}^{3}-
17576000\,{b}^{6}{c}^{9}{a}^{9}-52728000\,{a}^{13}{b}^{7}{c}^{4}+
8652800\,{b}^{5}{c}^{10}{a}^{9} $

$ +4326400\,{a}^{3}{b}^{5}{c}^{16}+393216
\,{a}^{12}{c}^{12}-262144\,{c}^{18}{a}^{6}+393216\,{b}^{12}{c}^{12}-
262144\,{a}^{18}{c}^{6}-262144\,{b}^{18}{c}^{6}+35414144\,{a}^{6}{b}^{
12}{c}^{6}+393216\,{a}^{12}{b}^{12} $

$+17576000\,{a}^{7}{b}^{13}{c}^{4}-
262144\,{a}^{18}{b}^{6}+4326400\,{a}^{5}{b}^{16}{c}^{3}-71402500\,{c}^
{9}{a}^{10}{b}^{5}+4326400\,{c}^{4}{a}^{12}{b}^{8} $

$ +1064960\,{c}^{2}{a}
^{15}{b}^{7}+8652800\,{c}^{3}{a}^{11}{b}^{10}+1064960\,{c}^{2}{a}^{9}{
b}^{13}+6489600\,{c}^{16}{a}^{6}{b}^{2}+17576000\,{c}^{12}{a}^{9}{b}^{
3}+17850625\,{c}^{8}{a}^{12}{b}^{4}-262144\,{c}^{18}{b}^{6} $

$ -71402500\,
{c}^{7}{a}^{11}{b}^{6}-262144\,{a}^{6}{b}^{18}-52728000\,{c}^{11}{a}^{
8}{b}^{5}-17576000\,{c}^{6}{a}^{9}{b}^{9}-52728000\,{a}^{11}{b}^{8}{c}
^{5}+3194880\,{a}^{15}{c}^{8}b+17850625\,{a}^{8}{b}^{12}{c}^{4}-
3194880\,{c}^{14}{a}^{9}b $

$ +3194880\,{c}^{13}{a}^{8}{b}^{3}-71402500\,{b
}^{10}{c}^{5}{a}^{9}+3194880\,{b}^{13}{c}^{8}{a}^{3}-3194880\,{a}^{14}
{c}^{7}{b}^{3}-17576000\,{c}^{6}{a}^{15}{b}^{3}-3194880\,{b}^{14}{c}^{
9}a-12979200\,{c}^{10}{a}^{12}{b}^{2} $

$ +3194880\,{c}^{15}{b}^{8}a+
8652800\,{c}^{11}{a}^{10}{b}^{3}-3194880\,{c}^{14}{b}^{7}{a}^{3}+
3194880\,{a}^{8}{b}^{15}c-3194880\,{a}^{7}{b}^{14}{c}^{3}-52728000\,{c
}^{13}{a}^{7}{b}^{4}-3194880\,{a}^{14}{b}^{9}c $

$ +3194880\,{a}^{13}{b}^{8
}{c}^{3}-1064960\,{a}^{19}{b}^{2}{c}^{3}-1064960\,{a}^{21}{c}^{2}b-
1064960\,{b}^{21}{a}^{2}c-1064960\,{b}^{19}{c}^{2}{a}^{3}-1064960\,{c}
^{19}{a}^{2}{b}^{3}-1064960\,{c}^{21}{b}^{2}a+52728000\,{c}^{7}{a}^{7}
{b}^{10}+107103750\,{c}^{10}{a}^{8}{b}^{6}$

$ +6489600\,{c}^{4}{a}^{18}{b}
^{2}+52728000\,{c}^{5}{a}^{14}{b}^{5}+17576000\,{c}^{5}{a}^{8}{b}^{11}
+6489600\,{c}^{4}{a}^{6}{b}^{14}+73532420\,{b}^{8}{c}^{9}{a}^{7}-
52728000\,{b}^{11}{c}^{8}{a}^{5}-71402500\,{b}^{7}{c}^{11}{a}^{6}+
52728000\,{b}^{5}{c}^{14}{a}^{5}$

$-12979200\,{b}^{10}{c}^{6}{a}^{8}-
52728000\,{b}^{13}{c}^{7}{a}^{4}+6489600\,{b}^{16}{c}^{6}{a}^{2}+
17576000\,{b}^{12}{c}^{9}{a}^{3}-12979200\,{b}^{10}{c}^{12}{a}^{2}+
6489600\,{b}^{4}{c}^{6}{a}^{14}+52728000\,{b}^{7}{c}^{7}{a}^{10}+
4326400\,{c}^{5}{a}^{16}{b}^{3}$

$+4326400\,{b}^{4}{c}^{12}{a}^{8}+65536
\,{a}^{24}+65536\,{b}^{24}+65536\,{c}^{24}+17850625\,{b}^{8}{c}^{12}{a
}^{4}-17576000\,{b}^{6}{c}^{15}{a}^{3}$

$ +6489600\,{b}^{4}{c}^{18}{a}^{2}
+73532420\,{b}^{9}{c}^{7}{a}^{8}+4326400\,{b}^{12}{c}^{8}{a}^{4}+
1064960\,{a}^{2}{b}^{15}{c}^{7}+8652800\,{a}^{3}{b}^{11}{c}^{10}+
1064960\,{a}^{2}{b}^{9}{c}^{13}+8652800\,{a}^{5}{b}^{10}{c}^{9}+
17576000\,{b}^{8}{c}^{11}{a}^{5} $

Ta chứng minh : $ F((a+b+c)^3) \leq 0 $ .

Không mất tính tổng quát giả sử : $ c=min\{a;b;c\} $

Đặt : $ a=x+c \ ; \ b=y+c \ ; \ x,y \geq 0 $

Ta có :

$ F((a+b+c)^3)=-40779396864\,{y}^{2}{c}^{22}-40779396864\,{x}^{2}{c}^{22}+40779396864
\,xy{c}^{22}-4542474816\,x{y}^{2}{c}^{21}-345212036352\,{y}^{3}{c}^{21
}$

$+143031852864\,{x}^{2}y{c}^{21}-345212036352\,{x}^{3}{c}^{21}-
1389000702368\,x{y}^{3}{c}^{20}$

$-1376117913104\,{x}^{4}{c}^{20}+
950511111888\,{x}^{2}{y}^{2}{c}^{20}-1376117913104\,{y}^{4}{c}^{20}-
355980408608\,{x}^{3}y{c}^{20} $

$+3268231889920\,{x}^{3}{y}^{2}{c}^{19}-
374884969280\,{x}^{2}{y}^{3}{c}^{19}-6931813722240\,x{y}^{4}{c}^{19}-
3400454771360\,{x}^{5}{c}^{19}$

$-3400454771360\,{y}^{5}{c}^{19}-
3588270683040\,{x}^{4}y{c}^{19}+8694195294640\,{x}^{3}{y}^{3}{c}^{18}-
5761729245152\,{y}^{6}{c}^{18}$

$-11706313884304\,{x}^{5}y{c}^{18}+
3141160960560\,{x}^{4}{y}^{2}{c}^{18}-5761729245152\,{x}^{6}{c}^{18}-
12058404371040\,{x}^{2}{y}^{4}{c}^{18}$

$-18331951300624\,x{y}^{5}{c}^{18
}-7090226946264\,{x}^{5}{y}^{2}{c}^{17}-31910915937800\,{y}^{6}x{c}^{
17}+23905249549480\,{x}^{4}{y}^{3}{c}^{17}$

$+277170784280\,{x}^{3}{y}^{4
}{c}^{17}-6968375977248\,{y}^{7}{c}^{17}-6968375977248\,{x}^{7}{c}^{17
}-40052951782344\,{x}^{2}{y}^{5}{c}^{17}$

$-23021578634200\,{x}^{6}y{c}^{
17}-39365392036908\,{y}^{7}x{c}^{16}-6001362257655\,{x}^{8}{c}^{16}-
6001362257655\,{y}^{8}{c}^{16}$

$-28865583152230\,{x}^{6}{y}^{2}{c}^{16}-
75447433406390\,{y}^{6}{x}^{2}{c}^{16}+28216218821296\,{x}^{5}{y}^{3}{
c}^{16}+36554051039115\,{x}^{4}{y}^{4}{c}^{16}$

$-31086101515068\,{x}^{7}
y{c}^{16}-37290606415024\,{x}^{3}{y}^{5}{c}^{16}-30153209881820\,{x}^{
8}y{c}^{15}+10805081693192\,{x}^{4}{y}^{5}{c}^{15}$

$-96584908345028\,{y}
^{7}{x}^{2}{c}^{15}-35175435038080\,{y}^{8}x{c}^{15}+7807646747492\,{x
}^{6}{y}^{3}{c}^{15}-3410350133620\,{y}^{9}{c}^{15}-3410350133620\,{x}
^{9}{c}^{15}$

$-50439484795348\,{x}^{7}{y}^{2}{c}^{15}-92817396270488\,{y
}^{6}{x}^{3}{c}^{15}+69630079888012\,{x}^{5}{y}^{4}{c}^{15}-
130810448874020\,{y}^{7}{x}^{3}{c}^{14}-56922248312170\,{x}^{8}{y}^{2}
{c}^{14}$

$+65909158471920\,{x}^{6}{y}^{4}{c}^{14}+71194715124304\,{x}^{5
}{y}^{5}{c}^{14}-794703777734\,{y}^{10}{c}^{14}-794703777734\,{x}^{10}
{c}^{14}-89088373623100\,{y}^{8}{x}^{2}{c}^{14}$

$-26123753934440\,{x}^{7
}{y}^{3}{c}^{14}-48174554126260\,{y}^{6}{x}^{4}{c}^{14}-22215280820260
\,{y}^{9}x{c}^{14}-20992933406700\,{x}^{9}y{c}^{14}-98489910922060\,{y
}^{7}{x}^{4}{c}^{13} $

$-9739722607384\,{x}^{10}y{c}^{13}+655247249392\,{y
}^{11}{c}^{13}-48254955142000\,{x}^{8}{y}^{3}{c}^{13}-126366073881940
\,{y}^{8}{x}^{3}{c}^{13}$

$+655247249392\,{x}^{11}{c}^{13}+26125041317760
\,{y}^{6}{x}^{5}{c}^{13}-45531851756140\,{x}^{9}{y}^{2}{c}^{13}-
59817646566960\,{y}^{9}{x}^{2}{c}^{13}+91280713124440\,{x}^{6}{y}^{5}{
c}^{13}-8593850024204\,{y}^{10}x{c}^{13}$

$+29047136848280\,{x}^{7}{y}^{4
}{c}^{13}-26458829532820\,{y}^{7}{x}^{5}{c}^{12}-47020363510480\,{x}^{
9}{y}^{3}{c}^{12}+63968230507640\,{x}^{7}{y}^{5}{c}^{12}$

$-26286643079152\,{x}^{10}{y}^{2}{c}^{12}-240418349520\,{y}^{11}x{c}^{12
}-106833276607195\,{y}^{8}{x}^{4}{c}^{12}-28011437835512\,{y}^{10}{x}^
{2}{c}^{12}-7840548636500\,{x}^{8}{y}^{4}{c}^{12}$

$-1908230448980\,{x}^{11}y{c}^{12}+888905253383\,{y}^{12}{c}^{12}+67054530571438\,{x}^{6}{y}
^{6}{c}^{12}+888905253383\,{x}^{12}{c}^{12}-88467848272280\,{y}^{9}{x}
^{3}{c}^{12}-7334931596240\,{y}^{11}{x}^{2}{c}^{11}$

$-10411323754040\,{x}^{11}{y}^{2}{c}^{11}+2361894748848\,{y}^{12}x{c}^{11}+1206814675688\,
{x}^{12}y{c}^{11}+58083836666004\,{x}^{7}{y}^{6}{c}^{11}-
49827013230460\,{y}^{8}{x}^{5}{c}^{11}-45158700786088\,{y}^{10}{x}^{3}
{c}^{11}$

$+23840307740944\,{x}^{6}{y}^{7}{c}^{11}+546011816620\,{x}^{13}
{c}^{11}-31215013301608\,{x}^{10}{y}^{3}{c}^{11}+24276793517940\,{x}^{
8}{y}^{5}{c}^{11}-22787922723920\,{x}^{9}{y}^{4}{c}^{11}+546011816620
\,{y}^{13}{c}^{11}$

$ -78735217905360\,{y}^{9}{x}^{4}{c}^{11}-
15987366180552\,{y}^{11}{x}^{3}{c}^{10}+1385658257020\,{x}^{13}y{c}^{
10}+887362800558\,{y}^{12}{x}^{2}{c}^{10}-42008727933836\,{y}^{10}{x}^
{4}{c}^{10}+29386455934934\,{x}^{8}{y}^{6}{c}^{10}$

$+425141134060\,{x}^{9}{y}^{5}{c}^{10}+194107227610\,{x}^{14}{c}^{10}-4767920597576\,{x}^{6
}{y}^{8}{c}^{10}+31649604172076\,{x}^{7}{y}^{7}{c}^{10}-42267014915860
\,{y}^{9}{x}^{5}{c}^{10}+1902970539260\,{y}^{13}x{c}^{10}$

$ -19261789310856\,{x}^{10}{y}^{4}{c}^{10}+194107227610\,{y}^{14}{c}^{10}
-2103047767262\,{x}^{12}{y}^{2}{c}^{10}-14928358674112\,{x}^{11}{y}^{3
}{c}^{10}-23705381781376\,{y}^{10}{x}^{5}{c}^{9}-5046511585140\,{x}^{
12}{y}^{3}{c}^{9}$

$+23830716048\,{y}^{15}{c}^{9}-11719135055000\,{x}^{6}
{y}^{9}{c}^{9}-16379088762140\,{y}^{11}{x}^{4}{c}^{9}+23830716048\,{x}
^{15}{c}^{9}+9852566232560\,{x}^{7}{y}^{8}{c}^{9}+864660823600\,{y}^{
14}x{c}^{9}$

$+718950711780\,{x}^{14}y{c}^{9}-3051453687580\,{y}^{12}{x}^
{3}{c}^{9}+18609374048160\,{x}^{8}{y}^{7}{c}^{9}+2055748110160\,{y}^{
13}{x}^{2}{c}^{9}+8571888423180\,{x}^{9}{y}^{6}{c}^{9}-5710720353776\,
{x}^{10}{y}^{5}{c}^{9}$

$+489157481700\,{x}^{13}{y}^{2}{c}^{9}-
10355975584500\,{x}^{11}{y}^{4}{c}^{9}+407823125020\,{y}^{13}{x}^{3}{c
}^{8}-7901167586150\,{x}^{6}{y}^{10}{c}^{8}+626759521940\,{x}^{14}{y}^
{2}{c}^{8}$

$+485209265040\,{x}^{7}{y}^{9}{c}^{8}-4481753621960\,{y}^{12}
{x}^{4}{c}^{8}-9533263186188\,{y}^{11}{x}^{5}{c}^{8}-3938686609225\,{x
}^{12}{y}^{4}{c}^{8}$

$ +539560240810\,{x}^{10}{y}^{6}{c}^{8}+238786738576
\,{y}^{15}x{c}^{8}-1038341117620\,{x}^{13}{y}^{3}{c}^{8}-15507566105\,
{x}^{16}{c}^{8} $

$+7528857174675\,{x}^{8}{y}^{8}{c}^{8}-4050740929028\,{x
}^{11}{y}^{5}{c}^{8}+1170135212330\,{y}^{14}{x}^{2}{c}^{8}-15507566105
\,{y}^{16}{c}^{8}$

$+223810763536\,{x}^{15}y{c}^{8}+6687472202300\,{x}^{9
}{y}^{7}{c}^{8}+568551227680\,{y}^{14}{x}^{3}{c}^{7}-715847314400\,{y}
^{13}{x}^{4}{c}^{7}-3318519957316\,{x}^{6}{y}^{11}{c}^{7}-11281109820
\,{x}^{14}{y}^{3}{c}^{7}$

$-10596866760\,{y}^{17}{c}^{7}+24202736900\,{y}
^{16}x{c}^{7}+411265914372\,{y}^{15}{x}^{2}{c}^{7}-1061526266980\,{x}^
{13}{y}^{4}{c}^{7}$

$+289916155972\,{x}^{15}{y}^{2}{c}^{7}-1674504463572
\,{x}^{12}{y}^{5}{c}^{7}-10596866760\,{x}^{17}{c}^{7}+1416634815020\,{
x}^{10}{y}^{7}{c}^{7}$

$+1899336214100\,{x}^{8}{y}^{9}{c}^{7}-
783779117776\,{x}^{11}{y}^{6}{c}^{7}-1169218051980\,{x}^{7}{y}^{10}{c}
^{7}+2984800218400\,{x}^{9}{y}^{8}{c}^{7}$

$-2795893032392\,{y}^{12}{x}^{
5}{c}^{7}+31883469180\,{x}^{16}y{c}^{7}-432218020392\,{x}^{12}{y}^{6}{
c}^{6}+237508866744\,{y}^{15}{x}^{3}{c}^{6}$

$-972801642412\,{x}^{6}{y}^{
12}{c}^{6}-184892597980\,{x}^{14}{y}^{4}{c}^{6}+80831600804\,{x}^{15}{
y}^{3}{c}^{6}+83786222520\,{x}^{16}{y}^{2}{c}^{6}$

$-472097972380\,{x}^{
13}{y}^{5}{c}^{6}-3107203010\,{x}^{18}{c}^{6}-11418784440\,{y}^{17}x{c
}^{6}+737097909890\,{x}^{10}{y}^{8}{c}^{6}+14803422700\,{y}^{14}{x}^{4
}{c}^{6}$

$-655793824136\,{x}^{7}{y}^{11}{c}^{6}+211091427980\,{x}^{8}{y}
^{10}{c}^{6}+869440183280\,{x}^{9}{y}^{9}{c}^{6}-583468069920\,{y}^{13
}{x}^{5}{c}^{6}$

$-6829628700\,{x}^{17}y{c}^{6}+78324754304\,{x}^{11}{y}^
{7}{c}^{6}+95911483330\,{y}^{16}{x}^{2}{c}^{6}-3107203010\,{y}^{18}{c}
^{6}-91395879700\,{x}^{14}{y}^{5}{c}^{5}$

$+162589266640\,{x}^{9}{y}^{10}
{c}^{5}-55823773252\,{x}^{12}{y}^{7}{c}^{5}-78303104680\,{y}^{14}{x}^{
5}{c}^{5}-344594340\,{x}^{19}{c}^{5}$

$-5406008500\,{x}^{18}y{c}^{5}+
32813191240\,{x}^{16}{y}^{3}{c}^{5}+45210904768\,{y}^{15}{x}^{4}{c}^{5
}-127077375040\,{x}^{13}{y}^{6}{c}^{5}$

$-38780669116\,{x}^{8}{y}^{11}{c}
^{5}+104822931804\,{x}^{11}{y}^{8}{c}^{5}-344594340\,{y}^{19}{c}^{5}+
61838306320\,{y}^{16}{x}^{3}{c}^{5}$

$-12045226892\,{x}^{15}{y}^{4}{c}^{5
}-6689917100\,{y}^{18}x{c}^{5}-200044380692\,{x}^{7}{y}^{12}{c}^{5}+
15598084800\,{x}^{17}{y}^{2}{c}^{5}$

$+223839495160\,{x}^{10}{y}^{9}{c}^{
5}-203819138060\,{x}^{6}{y}^{13}{c}^{5}+13385794980\,{y}^{17}{x}^{2}{c
}^{5}+1560865030\,{x}^{18}{y}^{2}{c}^{4}$

$-24105888010\,{x}^{14}{y}^{6}{
c}^{4}-29953163730\,{x}^{6}{y}^{14}{c}^{4}-3999521728\,{y}^{15}{x}^{5}
{c}^{4}$

$+43588960498\,{x}^{10}{y}^{10}{c}^{4}+34937613760\,{x}^{11}{y}^
{9}{c}^{4}-21628269865\,{x}^{8}{y}^{12}{c}^{4}+13781514260\,{y}^{16}{x
}^{4}{c}^{4}$

$+88274106\,{y}^{20}{c}^{4}+3227891270\,{x}^{12}{y}^{8}{c}^
{4}-1853232300\,{y}^{19}x{c}^{4}-20972645460\,{x}^{13}{y}^{7}{c}^{4}+
10844622900\,{y}^{17}{x}^{3}{c}^{4}$

$-40018711720\,{x}^{7}{y}^{13}{c}^{4
}+88274106\,{x}^{20}{c}^{4}-11076565468\,{x}^{15}{y}^{5}{c}^{4}+
3365694300\,{x}^{16}{y}^{4}{c}^{4}$

$-1635221520\,{x}^{19}y{c}^{4}+
16427755380\,{x}^{9}{y}^{11}{c}^{4}+7370610540\,{x}^{17}{y}^{3}{c}^{4}
+422195940\,{y}^{18}{x}^{2}{c}^{4}$

$-2944137368\,{x}^{15}{y}^{6}{c}^{3}-
23044320\,{x}^{19}{y}^{2}{c}^{3}+6546766992\,{x}^{11}{y}^{10}{c}^{3}+
10186180\,{x}^{9}{y}^{12}{c}^{3}$

$-4490978440\,{x}^{8}{y}^{13}{c}^{3}-
1808057940\,{x}^{13}{y}^{8}{c}^{3}-241562880\,{y}^{19}{x}^{2}{c}^{3}+
2813069120\,{x}^{12}{y}^{9}{c}^{3}$

$+1026647520\,{x}^{18}{y}^{3}{c}^{3}+
5262736452\,{x}^{10}{y}^{11}{c}^{3}-5377224000\,{x}^{7}{y}^{14}{c}^{3}
-2882174948\,{x}^{6}{y}^{15}{c}^{3}$

$+1248027640\,{y}^{18}{x}^{3}{c}^{3}
-323853948\,{y}^{20}x{c}^{3}-515361720\,{x}^{16}{y}^{5}{c}^{3}+
2351959980\,{y}^{17}{x}^{4}{c}^{3}$

$+1145456760\,{x}^{17}{y}^{4}{c}^{3}+
737468560\,{y}^{16}{x}^{5}{c}^{3}-302103648\,{x}^{20}y{c}^{3}-
3810642880\,{x}^{14}{y}^{7}{c}^{3}$

$+46621620\,{x}^{21}{c}^{3}+46621620
\,{y}^{21}{c}^{3}-362756220\,{x}^{14}{y}^{8}{c}^{2}+179266140\,{y}^{17
}{x}^{5}{c}^{2}-206429766\,{x}^{16}{y}^{6}{c}^{2}+713792616\,{x}^{11}{
y}^{11}{c}^{2}$

$+84094740\,{y}^{19}{x}^{3}{c}^{2}+3077760\,{x}^{13}{y}^{
9}{c}^{2}-36504720\,{y}^{21}x{c}^{2}-51787656\,{y}^{20}{x}^{2}{c}^{2}+
82316340\,{x}^{19}{y}^{3}{c}^{2}$

$-29604456\,{x}^{20}{y}^{2}{c}^{2}+
528846458\,{x}^{12}{y}^{10}{c}^{2}-221341760\,{x}^{9}{y}^{13}{c}^{2}+
347596168\,{x}^{10}{y}^{12}{c}^{2}$

$-151217856\,{x}^{6}{y}^{16}{c}^{2}-
397558952\,{x}^{15}{y}^{7}{c}^{2}+9630900\,{y}^{22}{c}^{2}+66187200\,{
x}^{17}{y}^{5}{c}^{2}-526026990\,{x}^{8}{y}^{14}{c}^{2}+9630900\,{x}^{
22}{c}^{2}$

$-464025352\,{x}^{7}{y}^{15}{c}^{2}-35510220\,{x}^{21}y{c}^{2
}+243149230\,{y}^{18}{x}^{4}{c}^{2}+162763080\,{x}^{18}{y}^{4}{c}^{2}+
2240208\,{y}^{20}{x}^{3}c$

$+13943700\,{y}^{19}{x}^{4}c+45272568\,{x}^{12
}{y}^{11}c+16392840\,{y}^{18}{x}^{5}c-17997600\,{x}^{14}{y}^{9}c+
11657520\,{x}^{19}{y}^{4}c$

$-29944080\,{x}^{15}{y}^{8}c-5339880\,{x}^{17
}{y}^{6}c-4620060\,{y}^{21}{x}^{2}c+41882688\,{x}^{11}{y}^{12}c-
33978240\,{x}^{8}{y}^{15}c $

$+16211280\,{x}^{13}{y}^{10}c-22573176\,{x}^{
7}{y}^{16}c+1053000\,{x}^{23}c+1053000\,{y}^{23}c-21652776\,{x}^{16}{y
}^{7}c-25476240\,{x}^{9}{y}^{14}c $

$-2478600\,{y}^{22}xc+2673108\,{x}^{20
}{y}^{3}c+7255320\,{x}^{10}{y}^{13}c-1617720\,{x}^{6}{y}^{17}c+
12147300\,{x}^{18}{y}^{5}c $

$-2478600\,{x}^{22}yc-3625560\,{x}^{21}{y}^{2
}c-81000\,{y}^{23}x+329346\,{y}^{20}{x}^{4}-39960\,{y}^{21}{x}^{3}-
965520\,{x}^{15}{y}^{9}$

$-1001169\,{x}^{16}{y}^{8}+838800\,{x}^{11}{y}^{
13}-181080\,{x}^{10}{y}^{14}-364680\,{x}^{7}{y}^{17}+838800\,{x}^{13}{
y}^{11}$

$+329346\,{x}^{20}{y}^{4}+612360\,{y}^{19}{x}^{5}+50625\,{x}^{24
}-181080\,{x}^{14}{y}^{10}-153900\,{y}^{22}{x}^{2}+104580\,{y}^{18}{x}
^{6}$

$+612360\,{x}^{19}{y}^{5}-81000\,{x}^{23}y-39960\,{x}^{21}{y}^{3}-
364680\,{x}^{17}{y}^{7} $

$+104580\,{x}^{18}{y}^{6}+1703196\,{x}^{12}{y}^{12}+50625\,{y}^{24}-1001169\,{x}^{8}{y}^{16}-153900\,{x}^{22}{y}^{2}-
965520\,{x}^{9}{y}^{15} \leq 0

$

Từ đó có $ (a+b+c)^3 $ bé hơn hoặc bằng nghiệm lớn nhất của phương trình $ F(T)=0 $ là $ m+n+t $ .

Hay là :

$ a\sqrt{16a^4+65b^3c} +b\sqrt{16b^4+65c^3a} +c\sqrt{16c^4+65a^3b} \ge (a+b+c)^3 $

Đó chính là đpcm .

P/S : dù là maple nhưng phải nói là chơi trò này mất sức quá

[L_Euler]

Hic cái gì thế này

[vuthanhtu_hd]

Hic chết mất thôi.Mấy vị huynh đề này nhàn gớm

[nguyen_ct]

em mà đưa lời giải giống anh Thắng chắc em đi luôn (độ kiên trì của anh làm em bái phục )

[Toanlc_gift]

làm sao để chứng minh cái biểu thức khủng bố kia $\le 0$ thế anh Thắng
anh trình bày thế kia là tự nhận nó $\le 0$ đấy chứ

[mai quoc thang]

làm sao để chứng minh cái biểu thức khủng bố kia $\le 0$ thế anh Thắng
anh trình bày thế kia là tự nhận nó $\le 0$ đấy chứ

Cái này dễ lắm ...... mỗi ngày cậu chứng minh 1 chút chút ......... chừng nào kiếm ra cái gì sai thì báo cho tớ ....... riêng tớ thì làm ơn tha cho tớ ..... sau cái bài của cậu và sau khi tớ post cái đống đó lên ...... tớ ko còn 1 chút sức lực nào hết .....

[L_Euler]

Cái này dễ lắm ...... mỗi ngày cậu chứng minh 1 chút chút ......... chừng nào kiếm ra cái gì sai thì báo cho tớ ....... riêng tớ thì làm ơn tha cho tớ ..... sau cái bài của cậu và sau khi tớ post cái đống đó lên ...... tớ ko còn 1 chút sức lực nào hết .....

Em thấy anh Thắng tốn công vô ích rồi Làm ra cái đống kia chẳng giải quyết được cái gì cả :-<

[Toanlc_gift]

he he,tốn công như vậy nhưng vui

[Messi_ndt]

Tớ có một lời giải phải nói là rất xấu theo phong cách của ngài Ji chen .

Lời giải :

Đặt :

$ m=a\sqrt{16a^4+65b^3c} \ ; \ n= b\sqrt{16b^4+65c^3a} \ ; \ t=c\sqrt{16c^4+65a^3b}$

Xét hàm :

$ F(T)=(T+m+n+t)(T+m+n-t)(T+m-n-t)(T-m-n-t)(T-m-n+t)(T-m+n+t)(T+m-n+t)(T-m+n-t) $

Ta có :

$ F \left( T \right) ={T}^{8}-64\,{b}^{6}{T}^{6}-64\,{a}^{6}{T}^{6}-260
\,{c}^{2}{a}^{3}b{T}^{6}-260\,{b}^{2}{c}^{3}a{T}^{6}-64\,{c}^{6}{T}^{6
}-260\,{a}^{2}{b}^{3}c{T}^{6}
$

$+25350\,{a}^{4}{b}^{6}{c}^{2}{T}^{4}+1024
\,{c}^{6}{b}^{6}{T}^{4}+1024\,{c}^{6}{a}^{6}{T}^{4}+4160\,{b}^{2}{c}^{
3}{a}^{7}{T}^{4}+25350\,{b}^{4}{c}^{6}{a}^{2}{T}^{4}+16900\,{b}^{5}{c}
^{4}{a}^{3}{T}^{4}+25350\,{c}^{4}{a}^{6}{b}^{2}{T}^{4} $

$ +4160\,{a}^{2}{b
}^{3}{c}^{7}{T}^{4}+4160\,{c}^{2}{a}^{9}b{T}^{4}+1024\,{a}^{6}{b}^{6}{
T}^{4}+4160\,{b}^{2}{c}^{9}a{T}^{4}+1536\,{a}^{12}{T}^{4}+1536\,{c}^{
12}{T}^{4}+1536\,{b}^{12}{T}^{4}+12480\,{c}^{8}{a}^{3}b{T}^{4}+4160\,{
c}^{2}{a}^{3}{b}^{7}{T}^{4} $

$ +12480\,{a}^{8}{b}^{3}c{T}^{4}+12480\,{b}^{
8}{c}^{3}a{T}^{4}+4160\,{a}^{2}{b}^{9}c{T}^{4}+16900\,{a}^{5}{b}^{4}{c
}^{3}{T}^{4}+16900\,{b}^{3}{c}^{5}{a}^{4}{T}^{4}-2704000\,{c}^{3}{a}^{
5}{b}^{10}{T}^{2}+133120\,{b}^{8}{c}^{9}a{T}^{2}+540800\,{b}^{9}{c}^{5
}{a}^{4}{T}^{2} $

$+1098500\,{b}^{7}{c}^{7}{a}^{4}{T}^{2}-811200\,{c}^{10}
{a}^{6}{b}^{2}{T}^{2} -1098500\,{c}^{6}{a}^{9}{b}^{3}{T}^{2}-665600\,{c
}^{2}{a}^{9}{b}^{7}{T}^{2}-2704000\,{c}^{5}{a}^{10}{b}^{3}{T}^{2}+
270400\,{b}^{4}{c}^{12}{a}^{2}{T}^{2} $

$ -199680\,{b}^{14}{c}^{3}a{T}^{2}+
133120\,{a}^{8}{b}^{3}{c}^{7}{T}^{2}-11148840\,{c}^{6}{a}^{6}{b}^{6}{T
}^{2}$

$ -16384\,{a}^{18}{T}^{2}-16384\,{c}^{18}{T}^{2}+540800\,{b}^{11}{c
}^{4}{a}^{3}{T}^{2}+1098500\,{c}^{7}{a}^{7}{b}^{4}{T}^{2}$

$ +270400a^6b^8c^4T^2 $

$+540800\,{a}^{4}{b}^{3}{c}^{11}{T}^{2}+1098500
\,{b}^{5}{c}^{8}{a}^{5}{T}^{2}+133120\,{b}^{7}{c}^{8}{a}^{3}{T}^{2}+
540800\,{a}^{11}{b}^{4}{c}^{3}{T}^{2}+270400\,{a}^{4}{b}^{6}{c}^{8}{T}
^{2}-199680\,{c}^{14}{a}^{3}b{T}^{2}$

$ -2704000\,{c}^{10}{a}^{3}{b}^{5}{T
}^{2}+133120\,{c}^{8}{a}^{9}b{T}^{2}+270400\,{c}^{4}{a}^{12}{b}^{2}{T}
^{2}+540800\,{c}^{9}{a}^{5}{b}^{4}{T}^{2}+1098500\,{c}^{5}{a}^{8}{b}^{
5}{T}^{2}+16384\,{c}^{12}{a}^{6}{T}^{2}+66560\,{a}^{2}{b}^{15}c{T}^{2}
-199680\,{a}^{14}{b}^{3}c{T}^{2}$

$ -665600\,{c}^{9}{a}^{7}{b}^{2}{T}^{2}+
66560\,{a}^{15}{c}^{2}b{T}^{2}-811200\,{b}^{10}{c}^{6}{a}^{2}{T}^{2}+
16384\,{c}^{12}{b}^{6}{T}^{2}+540800\,{a}^{9}{b}^{5}{c}^{4}{T}^{2}+
1098500\,{a}^{7}{b}^{7}{c}^{4}{T}^{2}$

$ +16384\,{a}^{6}{b}^{12}{T}^{2}-
1098500\,{b}^{6}{c}^{9}{a}^{3}{T}^{2}+270400\,{b}^{4}{c}^{6}{a}^{8}{T}
^{2}+16384\,{b}^{12}{c}^{6}{T}^{2}+16384\,{a}^{12}{b}^{6}{T}^{2}+
1098500\,{a}^{5}{b}^{8}{c}^{5}{T}^{2}-16384\,{b}^{18}{T}^{2}$

$ +66560\,{c}^{15}{b}^{2}a{T}^{2}-665600\,{c}^{7}{a}^{2}{b}^{9}{T}^{2}+66560\,{b}^
{13}{c}^{2}{a}^{3}{T}^{2}+133120\,{b}^{8}{c}^{3}{a}^{7}{T}^{2}+16384\,
{a}^{12}{c}^{6}{T}^{2}-811200\,{a}^{10}{b}^{6}{c}^{2}{T}^{2}-1098500\,
{a}^{6}{b}^{9}{c}^{3}{T}^{2} $

$ +66560\,{a}^{13}{b}^{2}{c}^{3}{T}^{2}+
66560\,{c}^{13}{a}^{2}{b}^{3}{T}^{2}+270400\,{a}^{4}{b}^{12}{c}^{2}{T}
^{2}+133120\,{a}^{8}{b}^{9}c{T}^{2}+1064960\,{b}^{20}{c}^{3}a+1064960
\,{a}^{20}{b}^{3}c+35414144\,{c}^{6}{a}^{12}{b}^{6} $

$ -71402500\,{b}^{9}{
c}^{10}{a}^{5}+35414144\,{a}^{6}{b}^{6}{c}^{12}+17576000\,{c}^{7}{a}^{
13}{b}^{4}-12979200\,{c}^{10}{a}^{6}{b}^{8}+1064960\,{c}^{20}{a}^{3}b-
12979200\,{a}^{17}{b}^{4}{c}^{3} $

$ +25958400\,{b}^{9}{c}^{11}{a}^{4}+
6489600\,{a}^{16}{b}^{6}{c}^{2}+107103750\,{b}^{10}{c}^{8}{a}^{6}+
73532420\,{c}^{8}{a}^{9}{b}^{7}+8652800\,{c}^{5}{a}^{10}{b}^{9}+
1064960\,{a}^{13}{b}^{2}{c}^{9}$

$ -12979200\,{a}^{15}{b}^{5}{c}^{4}-
12979200\,{c}^{15}{a}^{5}{b}^{4}-12979200\,{b}^{15}{c}^{5}{a}^{4}+
25958400\,{c}^{9}{a}^{11}{b}^{4}+6489600\,{c}^{14}{a}^{4}{b}^{6}+
25958400\,{a}^{9}{b}^{11}{c}^{4} $

$ -12979200\,{a}^{10}{b}^{12}{c}^{2}-
12979200\,{a}^{3}{b}^{17}{c}^{4}+17576000\,{c}^{8}{a}^{11}{b}^{5}+
1064960\,{c}^{15}{b}^{2}{a}^{7}-12979200\,{c}^{17}{b}^{3}{a}^{4}-
71402500\,{a}^{7}{b}^{11}{c}^{6} $

$ -17576000\,{b}^{9}{c}^{9}{a}^{6}+
17576000\,{b}^{7}{c}^{13}{a}^{4}+52728000\,{a}^{7}{b}^{7}{c}^{10}+
107103750\,{a}^{10}{b}^{8}{c}^{6}+52728000\,{a}^{5}{b}^{14}{c}^{5}-
12979200\,{c}^{8}{a}^{10}{b}^{6} $

$ -17576000\,{a}^{6}{b}^{15}{c}^{3}+
6489600\,{a}^{4}{b}^{18}{c}^{2}+17576000\,{a}^{12}{b}^{9}{c}^{3}-
17576000\,{b}^{6}{c}^{9}{a}^{9}-52728000\,{a}^{13}{b}^{7}{c}^{4}+
8652800\,{b}^{5}{c}^{10}{a}^{9} $

$ +4326400\,{a}^{3}{b}^{5}{c}^{16}+393216
\,{a}^{12}{c}^{12}-262144\,{c}^{18}{a}^{6}+393216\,{b}^{12}{c}^{12}-
262144\,{a}^{18}{c}^{6}-262144\,{b}^{18}{c}^{6}+35414144\,{a}^{6}{b}^{
12}{c}^{6}+393216\,{a}^{12}{b}^{12} $

$+17576000\,{a}^{7}{b}^{13}{c}^{4}-
262144\,{a}^{18}{b}^{6}+4326400\,{a}^{5}{b}^{16}{c}^{3}-71402500\,{c}^
{9}{a}^{10}{b}^{5}+4326400\,{c}^{4}{a}^{12}{b}^{8} $

$ +1064960\,{c}^{2}{a}
^{15}{b}^{7}+8652800\,{c}^{3}{a}^{11}{b}^{10}+1064960\,{c}^{2}{a}^{9}{
b}^{13}+6489600\,{c}^{16}{a}^{6}{b}^{2}+17576000\,{c}^{12}{a}^{9}{b}^{
3}+17850625\,{c}^{8}{a}^{12}{b}^{4}-262144\,{c}^{18}{b}^{6} $

$ -71402500\,
{c}^{7}{a}^{11}{b}^{6}-262144\,{a}^{6}{b}^{18}-52728000\,{c}^{11}{a}^{
8}{b}^{5}-17576000\,{c}^{6}{a}^{9}{b}^{9}-52728000\,{a}^{11}{b}^{8}{c}
^{5}+3194880\,{a}^{15}{c}^{8}b+17850625\,{a}^{8}{b}^{12}{c}^{4}-
3194880\,{c}^{14}{a}^{9}b $

$ +3194880\,{c}^{13}{a}^{8}{b}^{3}-71402500\,{b
}^{10}{c}^{5}{a}^{9}+3194880\,{b}^{13}{c}^{8}{a}^{3}-3194880\,{a}^{14}
{c}^{7}{b}^{3}-17576000\,{c}^{6}{a}^{15}{b}^{3}-3194880\,{b}^{14}{c}^{
9}a-12979200\,{c}^{10}{a}^{12}{b}^{2} $

$ +3194880\,{c}^{15}{b}^{8}a+
8652800\,{c}^{11}{a}^{10}{b}^{3}-3194880\,{c}^{14}{b}^{7}{a}^{3}+
3194880\,{a}^{8}{b}^{15}c-3194880\,{a}^{7}{b}^{14}{c}^{3}-52728000\,{c
}^{13}{a}^{7}{b}^{4}-3194880\,{a}^{14}{b}^{9}c $

$ +3194880\,{a}^{13}{b}^{8
}{c}^{3}-1064960\,{a}^{19}{b}^{2}{c}^{3}-1064960\,{a}^{21}{c}^{2}b-
1064960\,{b}^{21}{a}^{2}c-1064960\,{b}^{19}{c}^{2}{a}^{3}-1064960\,{c}
^{19}{a}^{2}{b}^{3}-1064960\,{c}^{21}{b}^{2}a+52728000\,{c}^{7}{a}^{7}
{b}^{10}+107103750\,{c}^{10}{a}^{8}{b}^{6}$

$ +6489600\,{c}^{4}{a}^{18}{b}
^{2}+52728000\,{c}^{5}{a}^{14}{b}^{5}+17576000\,{c}^{5}{a}^{8}{b}^{11}
+6489600\,{c}^{4}{a}^{6}{b}^{14}+73532420\,{b}^{8}{c}^{9}{a}^{7}-
52728000\,{b}^{11}{c}^{8}{a}^{5}-71402500\,{b}^{7}{c}^{11}{a}^{6}+
52728000\,{b}^{5}{c}^{14}{a}^{5}$

$-12979200\,{b}^{10}{c}^{6}{a}^{8}-
52728000\,{b}^{13}{c}^{7}{a}^{4}+6489600\,{b}^{16}{c}^{6}{a}^{2}+
17576000\,{b}^{12}{c}^{9}{a}^{3}-12979200\,{b}^{10}{c}^{12}{a}^{2}+
6489600\,{b}^{4}{c}^{6}{a}^{14}+52728000\,{b}^{7}{c}^{7}{a}^{10}+
4326400\,{c}^{5}{a}^{16}{b}^{3}$

$+4326400\,{b}^{4}{c}^{12}{a}^{8}+65536
\,{a}^{24}+65536\,{b}^{24}+65536\,{c}^{24}+17850625\,{b}^{8}{c}^{12}{a
}^{4}-17576000\,{b}^{6}{c}^{15}{a}^{3}$

$ +6489600\,{b}^{4}{c}^{18}{a}^{2}
+73532420\,{b}^{9}{c}^{7}{a}^{8}+4326400\,{b}^{12}{c}^{8}{a}^{4}+
1064960\,{a}^{2}{b}^{15}{c}^{7}+8652800\,{a}^{3}{b}^{11}{c}^{10}+
1064960\,{a}^{2}{b}^{9}{c}^{13}+8652800\,{a}^{5}{b}^{10}{c}^{9}+
17576000\,{b}^{8}{c}^{11}{a}^{5} $

Ta chứng minh : $ F((a+b+c)^3) \leq 0 $ .

Không mất tính tổng quát giả sử : $ c=min\{a;b;c\} $

Đặt : $ a=x+c \ ; \ b=y+c \ ; \ x,y \geq 0 $

Ta có :

$ F((a+b+c)^3)=-40779396864\,{y}^{2}{c}^{22}-40779396864\,{x}^{2}{c}^{22}+40779396864
\,xy{c}^{22}-4542474816\,x{y}^{2}{c}^{21}-345212036352\,{y}^{3}{c}^{21
}$

$+143031852864\,{x}^{2}y{c}^{21}-345212036352\,{x}^{3}{c}^{21}-
1389000702368\,x{y}^{3}{c}^{20}$

$-1376117913104\,{x}^{4}{c}^{20}+
950511111888\,{x}^{2}{y}^{2}{c}^{20}-1376117913104\,{y}^{4}{c}^{20}-
355980408608\,{x}^{3}y{c}^{20} $

$+3268231889920\,{x}^{3}{y}^{2}{c}^{19}-
374884969280\,{x}^{2}{y}^{3}{c}^{19}-6931813722240\,x{y}^{4}{c}^{19}-
3400454771360\,{x}^{5}{c}^{19}$

$-3400454771360\,{y}^{5}{c}^{19}-
3588270683040\,{x}^{4}y{c}^{19}+8694195294640\,{x}^{3}{y}^{3}{c}^{18}-
5761729245152\,{y}^{6}{c}^{18}$

$-11706313884304\,{x}^{5}y{c}^{18}+
3141160960560\,{x}^{4}{y}^{2}{c}^{18}-5761729245152\,{x}^{6}{c}^{18}-
12058404371040\,{x}^{2}{y}^{4}{c}^{18}$

$-18331951300624\,x{y}^{5}{c}^{18
}-7090226946264\,{x}^{5}{y}^{2}{c}^{17}-31910915937800\,{y}^{6}x{c}^{
17}+23905249549480\,{x}^{4}{y}^{3}{c}^{17}$

$+277170784280\,{x}^{3}{y}^{4
}{c}^{17}-6968375977248\,{y}^{7}{c}^{17}-6968375977248\,{x}^{7}{c}^{17
}-40052951782344\,{x}^{2}{y}^{5}{c}^{17}$

$-23021578634200\,{x}^{6}y{c}^{
17}-39365392036908\,{y}^{7}x{c}^{16}-6001362257655\,{x}^{8}{c}^{16}-
6001362257655\,{y}^{8}{c}^{16}$

$-28865583152230\,{x}^{6}{y}^{2}{c}^{16}-
75447433406390\,{y}^{6}{x}^{2}{c}^{16}+28216218821296\,{x}^{5}{y}^{3}{
c}^{16}+36554051039115\,{x}^{4}{y}^{4}{c}^{16}$

$-31086101515068\,{x}^{7}
y{c}^{16}-37290606415024\,{x}^{3}{y}^{5}{c}^{16}-30153209881820\,{x}^{
8}y{c}^{15}+10805081693192\,{x}^{4}{y}^{5}{c}^{15}$

$-96584908345028\,{y}
^{7}{x}^{2}{c}^{15}-35175435038080\,{y}^{8}x{c}^{15}+7807646747492\,{x
}^{6}{y}^{3}{c}^{15}-3410350133620\,{y}^{9}{c}^{15}-3410350133620\,{x}
^{9}{c}^{15}$

$-50439484795348\,{x}^{7}{y}^{2}{c}^{15}-92817396270488\,{y
}^{6}{x}^{3}{c}^{15}+69630079888012\,{x}^{5}{y}^{4}{c}^{15}-
130810448874020\,{y}^{7}{x}^{3}{c}^{14}-56922248312170\,{x}^{8}{y}^{2}
{c}^{14}$

$+65909158471920\,{x}^{6}{y}^{4}{c}^{14}+71194715124304\,{x}^{5
}{y}^{5}{c}^{14}-794703777734\,{y}^{10}{c}^{14}-794703777734\,{x}^{10}
{c}^{14}-89088373623100\,{y}^{8}{x}^{2}{c}^{14}$

$-26123753934440\,{x}^{7
}{y}^{3}{c}^{14}-48174554126260\,{y}^{6}{x}^{4}{c}^{14}-22215280820260
\,{y}^{9}x{c}^{14}-20992933406700\,{x}^{9}y{c}^{14}-98489910922060\,{y
}^{7}{x}^{4}{c}^{13} $

$-9739722607384\,{x}^{10}y{c}^{13}+655247249392\,{y
}^{11}{c}^{13}-48254955142000\,{x}^{8}{y}^{3}{c}^{13}-126366073881940
\,{y}^{8}{x}^{3}{c}^{13}$

$+655247249392\,{x}^{11}{c}^{13}+26125041317760
\,{y}^{6}{x}^{5}{c}^{13}-45531851756140\,{x}^{9}{y}^{2}{c}^{13}-
59817646566960\,{y}^{9}{x}^{2}{c}^{13}+91280713124440\,{x}^{6}{y}^{5}{
c}^{13}-8593850024204\,{y}^{10}x{c}^{13}$

$+29047136848280\,{x}^{7}{y}^{4
}{c}^{13}-26458829532820\,{y}^{7}{x}^{5}{c}^{12}-47020363510480\,{x}^{
9}{y}^{3}{c}^{12}+63968230507640\,{x}^{7}{y}^{5}{c}^{12}$

$-26286643079152\,{x}^{10}{y}^{2}{c}^{12}-240418349520\,{y}^{11}x{c}^{12
}-106833276607195\,{y}^{8}{x}^{4}{c}^{12}-28011437835512\,{y}^{10}{x}^
{2}{c}^{12}-7840548636500\,{x}^{8}{y}^{4}{c}^{12}$

$-1908230448980\,{x}^{11}y{c}^{12}+888905253383\,{y}^{12}{c}^{12}+67054530571438\,{x}^{6}{y}
^{6}{c}^{12}+888905253383\,{x}^{12}{c}^{12}-88467848272280\,{y}^{9}{x}
^{3}{c}^{12}-7334931596240\,{y}^{11}{x}^{2}{c}^{11}$

$-10411323754040\,{x}^{11}{y}^{2}{c}^{11}+2361894748848\,{y}^{12}x{c}^{11}+1206814675688\,
{x}^{12}y{c}^{11}+58083836666004\,{x}^{7}{y}^{6}{c}^{11}-
49827013230460\,{y}^{8}{x}^{5}{c}^{11}-45158700786088\,{y}^{10}{x}^{3}
{c}^{11}$

$+23840307740944\,{x}^{6}{y}^{7}{c}^{11}+546011816620\,{x}^{13}
{c}^{11}-31215013301608\,{x}^{10}{y}^{3}{c}^{11}+24276793517940\,{x}^{
8}{y}^{5}{c}^{11}-22787922723920\,{x}^{9}{y}^{4}{c}^{11}+546011816620
\,{y}^{13}{c}^{11}$

$ -78735217905360\,{y}^{9}{x}^{4}{c}^{11}-
15987366180552\,{y}^{11}{x}^{3}{c}^{10}+1385658257020\,{x}^{13}y{c}^{
10}+887362800558\,{y}^{12}{x}^{2}{c}^{10}-42008727933836\,{y}^{10}{x}^
{4}{c}^{10}+29386455934934\,{x}^{8}{y}^{6}{c}^{10}$

$+425141134060\,{x}^{9}{y}^{5}{c}^{10}+194107227610\,{x}^{14}{c}^{10}-4767920597576\,{x}^{6
}{y}^{8}{c}^{10}+31649604172076\,{x}^{7}{y}^{7}{c}^{10}-42267014915860
\,{y}^{9}{x}^{5}{c}^{10}+1902970539260\,{y}^{13}x{c}^{10}$

$ -19261789310856\,{x}^{10}{y}^{4}{c}^{10}+194107227610\,{y}^{14}{c}^{10}
-2103047767262\,{x}^{12}{y}^{2}{c}^{10}-14928358674112\,{x}^{11}{y}^{3
}{c}^{10}-23705381781376\,{y}^{10}{x}^{5}{c}^{9}-5046511585140\,{x}^{
12}{y}^{3}{c}^{9}$

$+23830716048\,{y}^{15}{c}^{9}-11719135055000\,{x}^{6}
{y}^{9}{c}^{9}-16379088762140\,{y}^{11}{x}^{4}{c}^{9}+23830716048\,{x}
^{15}{c}^{9}+9852566232560\,{x}^{7}{y}^{8}{c}^{9}+864660823600\,{y}^{
14}x{c}^{9}$

$+718950711780\,{x}^{14}y{c}^{9}-3051453687580\,{y}^{12}{x}^
{3}{c}^{9}+18609374048160\,{x}^{8}{y}^{7}{c}^{9}+2055748110160\,{y}^{
13}{x}^{2}{c}^{9}+8571888423180\,{x}^{9}{y}^{6}{c}^{9}-5710720353776\,
{x}^{10}{y}^{5}{c}^{9}$

$+489157481700\,{x}^{13}{y}^{2}{c}^{9}-
10355975584500\,{x}^{11}{y}^{4}{c}^{9}+407823125020\,{y}^{13}{x}^{3}{c
}^{8}-7901167586150\,{x}^{6}{y}^{10}{c}^{8}+626759521940\,{x}^{14}{y}^
{2}{c}^{8}$

$+485209265040\,{x}^{7}{y}^{9}{c}^{8}-4481753621960\,{y}^{12}
{x}^{4}{c}^{8}-9533263186188\,{y}^{11}{x}^{5}{c}^{8}-3938686609225\,{x
}^{12}{y}^{4}{c}^{8}$

$ +539560240810\,{x}^{10}{y}^{6}{c}^{8}+238786738576
\,{y}^{15}x{c}^{8}-1038341117620\,{x}^{13}{y}^{3}{c}^{8}-15507566105\,
{x}^{16}{c}^{8} $

$+7528857174675\,{x}^{8}{y}^{8}{c}^{8}-4050740929028\,{x
}^{11}{y}^{5}{c}^{8}+1170135212330\,{y}^{14}{x}^{2}{c}^{8}-15507566105
\,{y}^{16}{c}^{8}$

$+223810763536\,{x}^{15}y{c}^{8}+6687472202300\,{x}^{9
}{y}^{7}{c}^{8}+568551227680\,{y}^{14}{x}^{3}{c}^{7}-715847314400\,{y}
^{13}{x}^{4}{c}^{7}-3318519957316\,{x}^{6}{y}^{11}{c}^{7}-11281109820
\,{x}^{14}{y}^{3}{c}^{7}$

$-10596866760\,{y}^{17}{c}^{7}+24202736900\,{y}
^{16}x{c}^{7}+411265914372\,{y}^{15}{x}^{2}{c}^{7}-1061526266980\,{x}^
{13}{y}^{4}{c}^{7}$

$+289916155972\,{x}^{15}{y}^{2}{c}^{7}-1674504463572
\,{x}^{12}{y}^{5}{c}^{7}-10596866760\,{x}^{17}{c}^{7}+1416634815020\,{
x}^{10}{y}^{7}{c}^{7}$

$+1899336214100\,{x}^{8}{y}^{9}{c}^{7}-
783779117776\,{x}^{11}{y}^{6}{c}^{7}-1169218051980\,{x}^{7}{y}^{10}{c}
^{7}+2984800218400\,{x}^{9}{y}^{8}{c}^{7}$

$-2795893032392\,{y}^{12}{x}^{
5}{c}^{7}+31883469180\,{x}^{16}y{c}^{7}-432218020392\,{x}^{12}{y}^{6}{
c}^{6}+237508866744\,{y}^{15}{x}^{3}{c}^{6}$

$-972801642412\,{x}^{6}{y}^{
12}{c}^{6}-184892597980\,{x}^{14}{y}^{4}{c}^{6}+80831600804\,{x}^{15}{
y}^{3}{c}^{6}+83786222520\,{x}^{16}{y}^{2}{c}^{6}$

$-472097972380\,{x}^{
13}{y}^{5}{c}^{6}-3107203010\,{x}^{18}{c}^{6}-11418784440\,{y}^{17}x{c
}^{6}+737097909890\,{x}^{10}{y}^{8}{c}^{6}+14803422700\,{y}^{14}{x}^{4
}{c}^{6}$

$-655793824136\,{x}^{7}{y}^{11}{c}^{6}+211091427980\,{x}^{8}{y}
^{10}{c}^{6}+869440183280\,{x}^{9}{y}^{9}{c}^{6}-583468069920\,{y}^{13
}{x}^{5}{c}^{6}$

$-6829628700\,{x}^{17}y{c}^{6}+78324754304\,{x}^{11}{y}^
{7}{c}^{6}+95911483330\,{y}^{16}{x}^{2}{c}^{6}-3107203010\,{y}^{18}{c}
^{6}-91395879700\,{x}^{14}{y}^{5}{c}^{5}$

$+162589266640\,{x}^{9}{y}^{10}
{c}^{5}-55823773252\,{x}^{12}{y}^{7}{c}^{5}-78303104680\,{y}^{14}{x}^{
5}{c}^{5}-344594340\,{x}^{19}{c}^{5}$

$-5406008500\,{x}^{18}y{c}^{5}+
32813191240\,{x}^{16}{y}^{3}{c}^{5}+45210904768\,{y}^{15}{x}^{4}{c}^{5
}-127077375040\,{x}^{13}{y}^{6}{c}^{5}$

$-38780669116\,{x}^{8}{y}^{11}{c}
^{5}+104822931804\,{x}^{11}{y}^{8}{c}^{5}-344594340\,{y}^{19}{c}^{5}+
61838306320\,{y}^{16}{x}^{3}{c}^{5}$

$-12045226892\,{x}^{15}{y}^{4}{c}^{5
}-6689917100\,{y}^{18}x{c}^{5}-200044380692\,{x}^{7}{y}^{12}{c}^{5}+
15598084800\,{x}^{17}{y}^{2}{c}^{5}$

$+223839495160\,{x}^{10}{y}^{9}{c}^{
5}-203819138060\,{x}^{6}{y}^{13}{c}^{5}+13385794980\,{y}^{17}{x}^{2}{c
}^{5}+1560865030\,{x}^{18}{y}^{2}{c}^{4}$

$-24105888010\,{x}^{14}{y}^{6}{
c}^{4}-29953163730\,{x}^{6}{y}^{14}{c}^{4}-3999521728\,{y}^{15}{x}^{5}
{c}^{4}$

$+43588960498\,{x}^{10}{y}^{10}{c}^{4}+34937613760\,{x}^{11}{y}^
{9}{c}^{4}-21628269865\,{x}^{8}{y}^{12}{c}^{4}+13781514260\,{y}^{16}{x
}^{4}{c}^{4}$

$+88274106\,{y}^{20}{c}^{4}+3227891270\,{x}^{12}{y}^{8}{c}^
{4}-1853232300\,{y}^{19}x{c}^{4}-20972645460\,{x}^{13}{y}^{7}{c}^{4}+
10844622900\,{y}^{17}{x}^{3}{c}^{4}$

$-40018711720\,{x}^{7}{y}^{13}{c}^{4
}+88274106\,{x}^{20}{c}^{4}-11076565468\,{x}^{15}{y}^{5}{c}^{4}+
3365694300\,{x}^{16}{y}^{4}{c}^{4}$

$-1635221520\,{x}^{19}y{c}^{4}+
16427755380\,{x}^{9}{y}^{11}{c}^{4}+7370610540\,{x}^{17}{y}^{3}{c}^{4}
+422195940\,{y}^{18}{x}^{2}{c}^{4}$

$-2944137368\,{x}^{15}{y}^{6}{c}^{3}-
23044320\,{x}^{19}{y}^{2}{c}^{3}+6546766992\,{x}^{11}{y}^{10}{c}^{3}+
10186180\,{x}^{9}{y}^{12}{c}^{3}$

$-4490978440\,{x}^{8}{y}^{13}{c}^{3}-
1808057940\,{x}^{13}{y}^{8}{c}^{3}-241562880\,{y}^{19}{x}^{2}{c}^{3}+
2813069120\,{x}^{12}{y}^{9}{c}^{3}$

$+1026647520\,{x}^{18}{y}^{3}{c}^{3}+
5262736452\,{x}^{10}{y}^{11}{c}^{3}-5377224000\,{x}^{7}{y}^{14}{c}^{3}
-2882174948\,{x}^{6}{y}^{15}{c}^{3}$

$+1248027640\,{y}^{18}{x}^{3}{c}^{3}
-323853948\,{y}^{20}x{c}^{3}-515361720\,{x}^{16}{y}^{5}{c}^{3}+
2351959980\,{y}^{17}{x}^{4}{c}^{3}$

$+1145456760\,{x}^{17}{y}^{4}{c}^{3}+
737468560\,{y}^{16}{x}^{5}{c}^{3}-302103648\,{x}^{20}y{c}^{3}-
3810642880\,{x}^{14}{y}^{7}{c}^{3}$

$+46621620\,{x}^{21}{c}^{3}+46621620
\,{y}^{21}{c}^{3}-362756220\,{x}^{14}{y}^{8}{c}^{2}+179266140\,{y}^{17
}{x}^{5}{c}^{2}-206429766\,{x}^{16}{y}^{6}{c}^{2}+713792616\,{x}^{11}{
y}^{11}{c}^{2}$

$+84094740\,{y}^{19}{x}^{3}{c}^{2}+3077760\,{x}^{13}{y}^{
9}{c}^{2}-36504720\,{y}^{21}x{c}^{2}-51787656\,{y}^{20}{x}^{2}{c}^{2}+
82316340\,{x}^{19}{y}^{3}{c}^{2}$

$-29604456\,{x}^{20}{y}^{2}{c}^{2}+
528846458\,{x}^{12}{y}^{10}{c}^{2}-221341760\,{x}^{9}{y}^{13}{c}^{2}+
347596168\,{x}^{10}{y}^{12}{c}^{2}$

$-151217856\,{x}^{6}{y}^{16}{c}^{2}-
397558952\,{x}^{15}{y}^{7}{c}^{2}+9630900\,{y}^{22}{c}^{2}+66187200\,{
x}^{17}{y}^{5}{c}^{2}-526026990\,{x}^{8}{y}^{14}{c}^{2}+9630900\,{x}^{
22}{c}^{2}$

$-464025352\,{x}^{7}{y}^{15}{c}^{2}-35510220\,{x}^{21}y{c}^{2
}+243149230\,{y}^{18}{x}^{4}{c}^{2}+162763080\,{x}^{18}{y}^{4}{c}^{2}+
2240208\,{y}^{20}{x}^{3}c$

$+13943700\,{y}^{19}{x}^{4}c+45272568\,{x}^{12
}{y}^{11}c+16392840\,{y}^{18}{x}^{5}c-17997600\,{x}^{14}{y}^{9}c+
11657520\,{x}^{19}{y}^{4}c$

$-29944080\,{x}^{15}{y}^{8}c-5339880\,{x}^{17
}{y}^{6}c-4620060\,{y}^{21}{x}^{2}c+41882688\,{x}^{11}{y}^{12}c-
33978240\,{x}^{8}{y}^{15}c $

$+16211280\,{x}^{13}{y}^{10}c-22573176\,{x}^{
7}{y}^{16}c+1053000\,{x}^{23}c+1053000\,{y}^{23}c-21652776\,{x}^{16}{y
}^{7}c-25476240\,{x}^{9}{y}^{14}c $

$-2478600\,{y}^{22}xc+2673108\,{x}^{20
}{y}^{3}c+7255320\,{x}^{10}{y}^{13}c-1617720\,{x}^{6}{y}^{17}c+
12147300\,{x}^{18}{y}^{5}c $

$-2478600\,{x}^{22}yc-3625560\,{x}^{21}{y}^{2
}c-81000\,{y}^{23}x+329346\,{y}^{20}{x}^{4}-39960\,{y}^{21}{x}^{3}-
965520\,{x}^{15}{y}^{9}$

$-1001169\,{x}^{16}{y}^{8}+838800\,{x}^{11}{y}^{
13}-181080\,{x}^{10}{y}^{14}-364680\,{x}^{7}{y}^{17}+838800\,{x}^{13}{
y}^{11}$

$+329346\,{x}^{20}{y}^{4}+612360\,{y}^{19}{x}^{5}+50625\,{x}^{24
}-181080\,{x}^{14}{y}^{10}-153900\,{y}^{22}{x}^{2}+104580\,{y}^{18}{x}
^{6}$

$+612360\,{x}^{19}{y}^{5}-81000\,{x}^{23}y-39960\,{x}^{21}{y}^{3}-
364680\,{x}^{17}{y}^{7} $

$+104580\,{x}^{18}{y}^{6}+1703196\,{x}^{12}{y}^{12}+50625\,{y}^{24}-1001169\,{x}^{8}{y}^{16}-153900\,{x}^{22}{y}^{2}-
965520\,{x}^{9}{y}^{15} \leq 0

$

Từ đó có $ (a+b+c)^3 $ bé hơn hoặc bằng nghiệm lớn nhất của phương trình $ F(T)=0 $ là $ m+n+t $ .

Hay là :

$ a\sqrt{16a^4+65b^3c} +b\sqrt{16b^4+65c^3a} +c\sqrt{16c^4+65a^3b} \ge (a+b+c)^3 $

Đó chính là đpcm .

P/S : dù là maple nhưng phải nói là chơi trò này mất sức quá

Ôi mẹ ơi, Ngài Ji chen.

[CD13]

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