Cho a,b,c,d thuoc [0,1]. CMR a/(bc+cd+db+1) +b/(cd+da+ac+1) +c(da+ab+bd+1)+d/(ab+bc+ca+1)<= 3/4 +1/4abcd
Cho \(a,b,c,d\in[0;1]\)
CMR: \(\frac{a}{bc+cd+db+1}+\frac{b}{cd+da+ac+1}+\frac{c}{da+ab+bd+1}+\frac{d}{ab+bc+ca+1}\le\frac{3}{4}+\frac{1}{4abcd}\)
Cho \(a,b,c,d\in[0;1]\)
CMR: \(\frac{a}{bc+cd+db+1}+\frac{b}{cd+da+ac+1}+\frac{c}{da+ab+bd+1}+\frac{d}{ab+bc+ca+1}\le\frac{3}{4}+\frac{1}{4abcd}\)
cho a,b,c,d \(\in\left[0;1\right]\)cmr
\(\frac{a}{bc+cd+db+1}+\frac{b}{cd+da+ac+1}+\frac{c}{da+ab+bd+1}+\frac{d}{ab+bc+ca+1}\le\frac{3}{4}+\frac{1}{4abcd}\)
Đặt \(\hept{\begin{cases}x=\frac{a+b}{2}\\y=\frac{c+d}{2}\end{cases}}\)
Ta có:
\(\left(1-a\right)\left(1-b\right)\ge0\)
\(\Leftrightarrow ab+1\ge a+b\)
\(\Rightarrow ab+bc+ca+1\ge bc+ca+a+b=\left(a+b\right)\left(c+1\right)\ge\left(a+b\right)\left(c+d\right)\left(1\right)\)
Tương tự ta có:
\(bc+cd+db+1\ge\left(a+b\right)\left(b+d\right)\left(2\right)\)
\(cd+da+ac+1\ge\left(a+b\right)\left(c+d\right)\left(3\right)\)
\(da+ab+bd+1\ge\left(a+b\right)\left(c+d\right)\left(4\right)\)
Từ (1), (2), (3), (4) ta có:
\(VT\le\frac{a+b+c+d}{\left(a+b\right)\left(c+d\right)}=\frac{x+y}{2xy}\le\frac{xy+1}{2xy}\left(@\right)\)
Ta lại có:
\(VP\ge\frac{3}{4}+\frac{1}{4x^2y^2}\left(@@\right)\)
Từ \(\left(@\right),\left(@@\right)\)cái cần chứng minh trở thành.
\(\frac{xy+1}{2xy}\le\frac{3}{4}+\frac{1}{4x^2y^2}\)
\(\Leftrightarrow\left(xy-1\right)^2\ge0\)(đúng)
Vậy ta có ĐPCM.
1)Cho hình bình hành ABCD tâm o.Chứng minh:
a)AB-BC=DB
b)DA-DB+DC=VECTO KHÔNG
c)DA-DB=OD-OC
d) CO-OB=BA
e) MA+MC=MB+MD
f) MA+MB+MC+MD=4MD
g) BA+BC+OB=OD
h) AB+OD+OC=AC
2)Cho ngũ giác ABCDE.Chứng minh:
a) AB+BC+CD=AE-DE
b)AB+BC+CD+DA=VECTO KHÔNG
c) DA-CA=DB-CB
d)AC+DA+BD=AD-CD+BA
2)
a)\(\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CD}=\overrightarrow{AE}-\overrightarrow{DE}\Leftrightarrow\overrightarrow{AD}=\overrightarrow{AE}+\overrightarrow{ED}\Leftrightarrow\overrightarrow{AD}=\overrightarrow{AD}\)
b)
\(\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CD}+\overrightarrow{DA}=\overrightarrow{0}\Leftrightarrow\overrightarrow{AA}=\overrightarrow{0}\)
c)
\(\overrightarrow{DA}-\overrightarrow{CA}=\overrightarrow{DB}-\overrightarrow{CB}\Leftrightarrow\overrightarrow{DA}+\overrightarrow{AC}=\overrightarrow{DB}+\overrightarrow{BC}\Leftrightarrow\overrightarrow{DC}=\overrightarrow{DC}\)
d)\(\overrightarrow{AC}+\overrightarrow{DA}+\overrightarrow{BD}=\overrightarrow{AD}-\overrightarrow{CD}+\overrightarrow{BA}\Leftrightarrow\overrightarrow{AC}+\overrightarrow{CD}+\overrightarrow{DA}-\overrightarrow{BA}+\overrightarrow{BD}-\overrightarrow{AD}=\overrightarrow{0}\Leftrightarrow\overrightarrow{AC}+\overrightarrow{CD}+\overrightarrow{DA}+\overrightarrow{AB}+\overrightarrow{BD}+\overrightarrow{DA}=\overrightarrow{0}\Leftrightarrow\overrightarrow{AA}=\overrightarrow{0}\)
1.Cho 4 điểm A,B,C,D .Tìm các vecto:
a) u = AB+DC+BD+CA
b) v=AB+CD+BC+DA
2. Cho 4 điểm A,B,C,D . Tìm các vecto :
a) u =CA - CD - DB
b) v= AB - DC +BC - AD
Tính tổng : a . AB + BC + CD + DE b . AC + BE + CB + DA c . AB + DC + BD + CA d . BC + AD + CD + DA
1. Cho sáu điểm A,B,C,D,E,F. Chứng minh :
a) AB+BC+CD+DA=0
b) AB+DC+BD+CA=0
c) CD+BC+AB=AD
d) AB+CD=AD+CB
e) AD+BE+CF=AE+BF+CD=AF+BD+CE
Cho abcd = 1. Chứng minh rằng \(\frac{1}{1+ab+bc+ca}+\frac{1}{1+bc+cd+db}+\frac{1}{1+cd+da+ac}+\frac{1}{1+da+ab+bd}\le1\)
Xét \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge\frac{1}{\sqrt{ab}}+\frac{1}{\sqrt{bc}}+\frac{1}{\sqrt{ac}}=\sqrt{d}\left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right)\)
\(\Rightarrow ab+bc+ac\ge\frac{\sqrt{a}+\sqrt{b}+\sqrt{c}}{\sqrt{d}}\) và \(\frac{1}{1+ab+bc+ac}\le\frac{\sqrt{d}}{\sqrt{a}+\sqrt{b}+\sqrt{c}+\sqrt{d}}\)
Tương tự : \(\frac{1}{1+bc+cd+da}\le\frac{\sqrt{a}}{\sqrt{a}+\sqrt{b}+\sqrt{c}+\sqrt{d}}\)
\(\frac{1}{1+cd+da+ac}\le\frac{\sqrt{b}}{\sqrt{a}+\sqrt{b}+\sqrt{c}+\sqrt{d}}\)
\(\frac{1}{1+da+ab+bd}\le\frac{\sqrt{c}}{\sqrt{a}+\sqrt{b}+\sqrt{c}+\sqrt{d}}\)
Cộng theo vế ta được đpcm.
1)Cho hình bình hành ABCD, xác định các vectơ DA+DC,AB+DA.
2)Cho 5 điểm A, B, C, D, E. Chứng minh rằng: AC-ED+CD+EC-BC = AB
3)Cho hình vuông ABCD, tâm O cạnh bằng a.
a) Xác định vecto BA+DA+AC, AB+CA+BC, AB+AC.
b) Tính độ dài vecto DA+DC, AB-BC