,
,
,
,
in some order.
be an acute triangle with orthocenter 
.
be the point such that the quadrilateral 
is a parallelogram. Let 
be the point on the line 
such that 
bisects 
.
at 
and 
.
.
, 
and 
be points on circle 
divided into three equal parts. Construct three equal circles 
,
,
tangent to 
be any point on arc 
,
,
,
to circles 
.
,
,
,
be points on circle 
equal parts. Construct 
tangent to 
,
,
to circles 
of 
(where 
)
and 
with sum 
.![\[\sum_{i=1}^n \min\{{x_i^2, x_{i+1}}\}\leq \frac{1}{2}.\]](http://latex.artofproblemsolving.com/7/f/0/7f01a8d065c8e03233fe2254946522a5762f80e2.png)
where 
.
Y by srijitbose, AnhIsGod, maplestory, SRIDEV, MSTang, phantranhuongth, hamup1, Pain rinnegan, MathSlayer4444, yiwen, kk108, GeneralCobra19, AFK16, JasperL, djy21, Icy_coffee, JustKeepRunning, Mop2018, Loserfistte, myh2910, Tang_Bushuai, Adventure10, Mango247, SBYT, and 20 other users
Hello Everyone,
This is a try to make a nice inequalities marathon in the Pre-Olympiad level
Please if you write any problem don't forget to indicate its number and if you write a solution please indicate for what problem also to prevent the confusion that happens in some marathons.
Please show your solution don't just write by AM-GM then Cauchy-Schwarz and we are done.
OK finishing the talk now we go:
Problem 1: For any positive real numbers
show that the following inequality holds ![\[ \frac{a}{b}+\frac{b}{c}+\frac{c}{a} \geq \frac{c+a}{c+b}+\frac{a+b}{a+c}+\frac{b+c}{b+a}\]](//latex.artofproblemsolving.com/4/e/9/4e9bdb317383e49c9720d3abced44a76eafd77f9.png)
This is a try to make a nice inequalities marathon in the Pre-Olympiad level
Please if you write any problem don't forget to indicate its number and if you write a solution please indicate for what problem also to prevent the confusion that happens in some marathons.
Please show your solution don't just write by AM-GM then Cauchy-Schwarz and we are done.
OK finishing the talk now we go:
Problem 1: For any positive real numbers
show that the following inequality holds ![\[ \frac{a}{b}+\frac{b}{c}+\frac{c}{a} \geq \frac{c+a}{c+b}+\frac{a+b}{a+c}+\frac{b+c}{b+a}\]](http://latex.artofproblemsolving.com/4/e/9/4e9bdb317383e49c9720d3abced44a76eafd77f9.png)
![\[ \frac {a}{b} + \frac {b}{c} + \frac {c}{a} - \frac {a + b}{c + a} - \frac {b + c}{a + b} - \frac {c + a}{b + c}\]](http://latex.artofproblemsolving.com/c/9/1/c91d74d567579c563872f91dbe53f51d347879af.png)
![\[ = \frac {abc}{(a + b)(b + c)(c + a)}\left(\frac {a^2}{b^2} + \frac {b^2}{c^2} + \frac {c^2}{a^2} - \frac {a}{b} - \frac {b}{c} - \frac {c}{a}\right)\]](http://latex.artofproblemsolving.com/9/4/4/94408e81e7a872cf5dab9d03c60d00df28d16905.png)
![\[ + \frac {abc}{(a + b)(b + c)(c + a)}\left(\frac {ab}{c^2} + \frac {bc}{a^2} + \frac {ca}{b^2} - 3\right)\]](http://latex.artofproblemsolving.com/e/5/0/e504f6306444783631006cf0c252595a7461dc9b.png)
and 
![\[ \left(\frac {a^2}{b^2} + \frac {b^2}{c^2} + \frac {c^2}{a^2}\right)^2\ge 3\left(\frac {a^2}{b^2} + \frac {b^2}{c^2} + \frac {c^2}{a^2}\right)\ge \left(\frac {a}{b} + \frac {b}{c} + \frac {c}{a}\right)^2\]](http://latex.artofproblemsolving.com/d/9/6/d969bb07aa7c8a2ad0b76e6fbf77debb0a7f9e6a.png)
for 
such that 
and 
. Then show that:
.
.
.
which is easy.
and 
in your inequality.
in my inequality . The inequality is correct . You can try to find a counterexample but you'll only lose your time . Anyway the inequality is easy . 
we get that 
.
![$ \Longleftrightarrow \left [d(d - 1)\right ]^2 - [d(d - 1)]+1 \ge \frac 34 \Longleftrightarrow \left [d(d - 1)-\frac {1}{2}\right ]^2\ge 0.$](http://latex.artofproblemsolving.com/9/6/c/96c11fda5081d03493e738c7717b08e69936aa46.png)
and 
![\[ \frac {a}{\left(b + c\right)^2} + \frac {b}{\left(c + a\right)^2} + \frac {c}{\left(a + b\right)^2} \geq \frac {9}{4\left(a + b + c\right)}\]](http://latex.artofproblemsolving.com/0/5/b/05b40841ce2fe57a3a0f1ab1f88f925e626c9104.png)
.
.
.
and 
prove that 
![\[ 2(a + b + c)^3 + 9abc\geq 7(ab + bc + ca)(a + b + c)\]](http://latex.artofproblemsolving.com/8/a/e/8ae50bc8e78d27ce1a3033323437468c8860665d.png)
![\[ \sum_{sym} a^3 + 6\sum_{sym} a^2b + 21abc\geq 7\sum_{sym} a^2b + 21abc\]](http://latex.artofproblemsolving.com/6/a/6/6a69df6a2d0b5eecd412372c0cd77862a3780928.png)
![\[ \sum_{sym} a^3\geq \sum_{sym} a^2b\]](http://latex.artofproblemsolving.com/7/5/1/751cffbb309554005678c3376dd8583959d97ac4.png)
and similars.
. Pove that :![\[ \sqrt {x(y + 1)} + \sqrt {y(z + 1)} + \sqrt {z(x + 1)}\le \frac {3}{2}\sqrt {(x + 1)(y + 1)(z + 1)}\]](http://latex.artofproblemsolving.com/5/6/3/5637c8dac5f0e12bc4feb420cf4e6184555a30de.png)
,
![\[ \frac{1}{a} +\frac{1}{b} +\frac{1}{c} \geq \frac{4a}{2a^2 +b^2 +c^2} +\frac{4b}{a^2 +2b^2 +c^2} +\frac{4c}{a^2 +b^2 +2c^2}\]](http://latex.artofproblemsolving.com/5/8/c/58c336949de3a040feb0fef867513e26ed8897e5.png)
we have 
.
.
.
.
we obtain the desired result .
be non-negative real numbers such that 
. Prove that:
.
and 
,
Prove that 
such that 
,
.![\[2(a^2+b^2)\le 3ab+3c^2+c(a+b)\]](http://latex.artofproblemsolving.com/f/b/4/fb4af5592cfe66739e5683f012b6a19aa266f8b7.png)
![\[3c^2+c(a+b)\ge 2(a^2+b^2)-3ab=\frac{1}{4}(a+b)^2+\frac{7}{4}(a-b)^2\ge \frac{1}{4}(a+b)^2\]](http://latex.artofproblemsolving.com/6/2/7/627d133a9f661a1d20947e280d3c26eeb9764735.png)
,
,
,
.
be positive numbers satisfying 
.![\[\frac{3}{a}+\frac{2}{b}\ge 5.\]](http://latex.artofproblemsolving.com/d/2/1/d21a3aa50a71bc5e0390a9ea4b0e6c92dd67d9df.png)
.![\[\sqrt{a^2+3}+\sqrt{b^2+8}\ge 5.\]](http://latex.artofproblemsolving.com/5/7/6/57660cc1ec4acad71f5272ff9d2e303a628938a9.png)
such as 
. Find the minimum value of![\[P=\dfrac{a^3}{a+bc}+\dfrac{b^3}{b+ac}+\dfrac{c^3}{c+ba}+\dfrac{14}{(c+1)\sqrt{c+ab}}\]](http://latex.artofproblemsolving.com/4/2/e/42e7d035acc9aa4420de6847befe6901b0f03f8a.png)
positive reals. Prove that:
![$(\frac{a+b}{\sqrt{a+2c}}+\frac{b+c}{\sqrt{b+2a}}+\frac{c+a}{\sqrt{c+2b}})^\frac{2}{3}[(a+b)(a+2c)+(b+c)(b+2a)+(c+a)(c+2b)]^\frac{1}{3}\geq{2(a+b+c)}$](http://latex.artofproblemsolving.com/c/6/3/c63442e72c43e8560d775e1cadc1b58cad7a7b1d.png)
. It is enough to show that![$\frac{[2(a+b+c)]^3}{a^2+b^2+c^2+5(ab+bc+ca)}\geq{4(a+b+c)}$](http://latex.artofproblemsolving.com/e/7/9/e79bbb1ae06df0f85a5df3c2e2821b92ae99af5d.png)
or
or
.
.
and 
Find the maximum of 
and 
then
