\(1=a+b\ge2\sqrt{ab}\Rightarrow ab\le\frac{1}{4}\)
\(A=a^2b^2+\frac{1}{a^2b^2}+2\)
\(A=a^2b^2+\frac{1}{256a^2b^2}+\frac{255}{256\left(ab\right)^2}+2\)
\(A\ge2\sqrt{\frac{a^2b^2}{256a^2b^2}}+\frac{255}{256.\left(\frac{1}{4}\right)^2}+2=\frac{289}{16}\)
\(A_{min}=\frac{289}{16}\) khi \(a=b=\frac{1}{2}\)
Cho 0<a,b,c<1 và ab+bc+ca=1 tìm \(P_{min}=\frac{a^2\left(1-2b\right)}{b}+\frac{b^2\left(1-2c\right)}{c}++\frac{c^2\left(1-2a\right)}{a}\)
1. \(\left\{{}\begin{matrix}a,b,c>0\\a+b+c=1\end{matrix}\right.\). Cmr: \(\frac{ab}{\sqrt{\left(1-c\right)^2\left(1+c\right)}}+\frac{bc}{\sqrt{\left(1-a\right)^2\left(1+a\right)}}+\frac{ca}{\sqrt{\left(1-b\right)^3\left(1+b\right)}}\le\frac{3\sqrt{2}}{8}\)
2. \(\left\{{}\begin{matrix}a,b,c>0\\a+b+c\le1\end{matrix}\right.\). Cmr: \(\frac{1}{a^2+b^2+c^2}+\frac{1}{ab\left(a+b\right)}+\frac{1}{bc\left(b+c\right)}+\frac{1}{ac\left(a+c\right)}\ge\frac{87}{2}\)
3. \(\left\{{}\begin{matrix}a,b,c>0\\ab+bc+ca=2abc\end{matrix}\right.\). Cmr: \(\frac{1}{a\left(2a-1\right)^2}+\frac{1}{b\left(2b-1\right)^2}+\frac{1}{c\left(2c-1\right)^2}\ge\frac{1}{2}\)
4. \(\left\{{}\begin{matrix}x,y,z>0\\x+y+z=2015\end{matrix}\right.\). Tìm min \(A=\frac{x^4+y^4}{x^3+y^3}+\frac{y^4+z^4}{y^3+z^3}+\frac{z^4+x^4}{z^2+x^2}\)
Mn giúp mk với ạ! Thanks nhiều
1. a) Tìm \(n\in N\)*, \(n>2008\) sao cho \(2^{2008}+2^{2012}+2^{2013}+2^{2014}+2^{2016}+2^n\) là số chính phương
b) tìm x,y > 0 thỏa mãn \(x^2+y^2=2\left(x+y\right)\left(\sqrt{x}+\sqrt{y}-2\right)\)
2. a) \(\left\{{}\begin{matrix}a\ge0\\a+b\ge1\end{matrix}\right.\). Min \(A=\frac{8a^2+b}{4a}+b^2\)
b) \(\left\{{}\begin{matrix}a,b\ge0\\\left(a-b\right)^2=a+b+2\end{matrix}\right.\). Cmr: \(\left(1+\frac{a^3}{\left(b+1\right)^3}\right)\left(1+\frac{b^3}{\left(b+1\right)^3}\right)\le9\)
c) \(x,y>0;\left(x+\sqrt{1+x^2}\right)\left(y+\sqrt{1+y^2}\right)=2020\). Min P = x + y
d) \(x,y,z>0;\sqrt{x^2+y^2}+\sqrt{y^2+z^2}+\sqrt{z^2+x^2}=6\). Min \(P=\frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}\)
e) \(\left\{{}\begin{matrix}x,y,z>0\\x+y+z+4xyz=4\end{matrix}\right.\) Cmr: \(\left(1+xy+\frac{y}{z}\right)\left(1+yz+\frac{z}{x}\right)\left(1+zx+\frac{x}{y}\right)\ge27\)
f) \(\left\{{}\begin{matrix}x,y,z\ge1\\3x^2+4y^2+5z^2=52\end{matrix}\right.\). Min P = x + y + z
g) \(x,y>0\). Min \(P=\frac{2}{\sqrt{\left(2x+y\right)^3+1}-1}+\frac{2}{\sqrt{\left(x+2y\right)^3+1}-1}+\frac{\left(2x+y\right)\left(x+2y\right)}{4}-\frac{8}{3\left(x+y\right)}\)
Cho a, b là 2 sô dương thỏa mãn a + b = 1. Tìm Min T = \(\frac{1}{a^2}+\left(a+\frac{1}{b}\right)^2+\left(b+\frac{1}{a}\right)^2+\frac{1}{b^2}\)
1. a) \(\left\{{}\begin{matrix}x,y,z>0\\xyz=1\end{matrix}\right.\). Tìm max \(P=\frac{1}{\sqrt{x^5-x^2+3xy+6}}+\frac{1}{\sqrt{y^5-y^2+3yz+6}}+\frac{1}{\sqrt{z^5-z^2+zx+6}}\)
b) \(\left\{{}\begin{matrix}x,y,z>0\\xyz=8\end{matrix}\right.\). Min \(P=\frac{x^2}{\sqrt{\left(1+x^3\right)\left(1+y^3\right)}}+\frac{y^2}{\sqrt{\left(1+y^3\right)\left(1+z^3\right)}}+\frac{z^2}{\sqrt{\left(1+z^3\right)\left(1+x^3\right)}}\)
c) \(x,y,z>0.\) Min \(P=\sqrt{\frac{x^3}{x^3+\left(y+z\right)^3}}+\sqrt{\frac{y^3}{y^3+\left(z+x\right)^3}}+\sqrt{\frac{z^3}{z^3+\left(x+y\right)^3}}\)
d) \(a,b,c>0;a^2+b^2+c^2+abc=4.Cmr:2a+b+c\le\frac{9}{2}\)
e) \(\left\{{}\begin{matrix}a,b,c>0\\a+b+c=3\end{matrix}\right.\). Cmr: \(\frac{a}{b^3+ab}+\frac{b}{c^3+bc}+\frac{c}{a^3+ca}\ge\frac{3}{2}\)
f) \(\left\{{}\begin{matrix}a,b,c>0\\ab+bc+ca+abc=4\end{matrix}\right.\) Cmr: \(\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\le3\)
g) \(\left\{{}\begin{matrix}a,b,c>0\\ab+bc+ca+abc=2\end{matrix}\right.\) Max : \(Q=\frac{a+1}{a^2+2a+2}+\frac{b+1}{b^2+2b+2}+\frac{c+1}{c^2+2c+2}\)
1. cho \(0< a\le b\le c\) . Cmr: \(\frac{2a^2}{b^2+c^2}+\frac{2b^2}{c^2+a^2}+\frac{2c^2}{a^2+b^2}\le\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\)
2. cho \(a,b,c\ge0\). cmr: \(a^2+b^2+c^2+3\sqrt[3]{\left(abc\right)^2}\ge2\left(ab+bc+ca\right)\)
3. \(a,b,c>0.\) Cmr: \(\sqrt{\left(a^2b+b^2c+c^2a\right)\left(ab^2+bc^2+ca^2\right)}\ge abc+\sqrt[3]{\left(a^3+abc\right)\left(b^3+abc\right)\left(c^3+abc\right)}\)
4. \(a,b,c>0\). Tìm Min \(P=\left(\frac{a}{a+b}\right)^4+\left(\frac{b}{b+c}\right)^4+\left(\frac{c}{c+a}\right)^4\)
Cho a,b >0 và \(a+b\le3\). Tìm min
\(K=\dfrac{1}{a^2+b^2-2\left(a+b\right)+2}+\dfrac{1}{ab-\left(a+b\right)+1}+4\left(ab-a-b\right)\)
1. Tìm tất cả các số tự nhiên n thỏa mãn 2n+1,3n+1 là các số chính phương và 2n+9 là số nguyên tố
2. Tìm tất cả các cặp số nguyên dương (m,n) để \(2^m\cdot5^n+25\) là số chính phương
3. a) cho a,b,c thỏa mãn \(2\left(a^2+ab+b^2\right)=3\left(3-c^2\right)\). Tìm max, min \(P=a+b+c\)
b) \(\left\{{}\begin{matrix}a,b,c>0\\a+b+c=1\end{matrix}\right.\). Cmr: \(6\left(ab+bc+ca\right)+a\left(a-b\right)^2+b\left(b-c\right)^2+c\left(c-a\right)^2\le2\)
c) \(\left\{{}\begin{matrix}x,y,z>0\\x+y+z=3\end{matrix}\right.\). Tìm min \(P=\frac{1}{2xy^2+1}+\frac{1}{2yz^2+1}+\frac{1}{2zx^2+1}\)
d) \(\left\{{}\begin{matrix}a,b,c\ge0\\a+b+c=3\end{matrix}\right.\). Tìm max \(P=a\sqrt[3]{b^3+1}+b\sqrt[3]{c^3+1}+c\sqrt[3]{a^3+1}\)
e) \(\left\{{}\begin{matrix}-1\le a,b,c\le1\\0\le x,y,z\le1\end{matrix}\right.\). Max \(P=\left(\frac{1-a}{1-bz}\right)\left(\frac{1-b}{1-cx}\right)\left(\frac{1-c}{1-ay}\right)\)
f) \(\left\{{}\begin{matrix}a,b>0\\a+2b\le3\end{matrix}\right.\). Max \(P=\frac{1}{\sqrt{a+3}}+\frac{1}{\sqrt{b+3}}\)
g) \(\left\{{}\begin{matrix}x,y,z>0\\xyz=x+y+z+2\end{matrix}\right.\). Max \(P=\frac{1}{\sqrt{x^2+2}}+\frac{1}{\sqrt{y^2+2}}+\frac{1}{\sqrt{z^2+2}}\)
h) \(a,b,c>0\). Tìm min \(P=\frac{1}{\left(a+b\right)^2}+\frac{1}{\left(a+c\right)^2}+2\sqrt{a^2+bc}\)
Cho a,b,c dương và abc=1
CMR: \(\frac{a^4}{2\left(b+c\right)^2}+\frac{b^4}{2\left(a+c\right)^2}+\frac{c^4}{2\left(a+b\right)^2}+\frac{1}{c^2\left(a+c\right)\left(a+b\right)}+\frac{1}{b^2\left(a+b\right)\left(b+c\right)}+\frac{1}{a^2\left(a+c\right)\left(a+b\right)}\ge\frac{1}{8}\)
