AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
satisfied 
Find minimum and maximum of 
such that:
with all 
is real number
be the set of integers. Determine all functions 
such that, for all integers 
and 
, 
Proposed by Liam Baker, South Africa
be a positive integer. Find all real solutions 
to the system:![\[a_1^2 + a_1 - 1 = a_2\]](http://latex.artofproblemsolving.com/a/4/e/a4e26b83c929b82800576bac11e12de0c3aa1cf6.png)
![\[ a_2^2 + a_2 - 1 = a_3\]](http://latex.artofproblemsolving.com/9/9/5/995e8eec93828b82901b0639463f907f5ff95d08.png)
![\[\hspace*{3.3em} \vdots \]](http://latex.artofproblemsolving.com/e/e/1/ee15e1fefe2ad550d3c9b1c51207eaf66fbfa41f.png)
![\[a_{n}^2 + a_n - 1 = a_1\]](http://latex.artofproblemsolving.com/7/c/7/7c727d1f2044c1bf7c80981e1074bb3cde807b8f.png)
of positive integers is called central if for every positive integer , the arithmetic mean of the first 
terms of the sequence is equal to .
of positive integers such that for every central sequence 
there are infinitely many positive integers with 
.
be the circumcircle of triangle 
, 
is the midpoint of 
and 
be the second intersection of 
and . Tangent line of at intersects at 
, let 
be a transversal of and 
be intersections of 
with . Let 
be a tangent line of . Prove that 
is cyclic
be positive integers that satisfy the inequality![\[
\frac{a}{b} < \frac{x}{y} < \frac{c}{d}
\]](http://latex.artofproblemsolving.com/8/d/9/8d982d52e655546ac29039fa6e440e68821bfe7f.png)
with 
. If 
, determine the minimum value of 
.
. Let 
and be the feet of the altitudes from 
to and from 
to 
respectively.
and 
as the reflections of across lines 
and , respectively. Let 
be the circumcircle of 
. Denote by the second intersection of line 
with , and by 
the intersection of ray 
with .
is the circumcenter of , prove that , , and are collinear if and only if quadrilateral 
can be inscribed within a circle.
and 
Prove that:
where 
where 
.
and Prove that:where where .
https://artofproblemsolving.com/community/q4h1187317p5776869
and 
Prove that
Prove that
Prove that
![$$1 \le \sqrt[3]{\frac{a}{a+3b}} + \sqrt[3]{\frac{b}{b+3a}} \le \frac{2}{\sqrt[3]{4}} $$](http://latex.artofproblemsolving.com/8/c/d/8cd7bf385a28b4dd8c1589454cdf5792c39425a8.png)
When does the equality hold ?
![$$T= \sqrt[3]{\frac{a}{a+kb}} + \sqrt[3]{\frac{b}{b+ka}} $$](http://latex.artofproblemsolving.com/1/f/8/1f8f4e2d285eda70d965dfd3ffc5c626d5a82523.png)
then![$$ 1 \le T \le \frac{2}{ \sqrt[3]{k+1}} $$](http://latex.artofproblemsolving.com/0/5/c/05c005d8a3df625ad706e113a0545750008f0912.png)
where 
![$$ min( 1, \frac{2}{ \sqrt[3]{k+1}} ) \le T \le \sqrt[3]{\frac{k+1}{k-1}}$$](http://latex.artofproblemsolving.com/1/0/1/101057031d70ba7c1d3eaeec4e1d8ace191d89b5.png)
where 
![$$ \frac{2}{12^{1/5}} \le \sqrt[5]{\frac{a}{a+11b}}+ \sqrt[5]{\frac{b}{b+11a}} \le \frac{5}{10^{3/5}} $$](http://latex.artofproblemsolving.com/8/7/d/87db2b15a3775a710de8a5e71436c0a3d347edd0.png)
When does the equality hold ?
and ![$$ T= \sqrt[5]{\frac{a}{a+kb}}+ \sqrt[5]{\frac{b}{b+ka}} $$](http://latex.artofproblemsolving.com/c/6/9/c69e8c2302e57b3f6ec7e46df75b8a706bc938c2.png)
then:![$$1 \le T \le \frac{2}{ \sqrt[5]{k+1} } $$](http://latex.artofproblemsolving.com/c/1/4/c14f64b66ec49625a19f15f4bede531e3ce9f78a.png)
where 
![$$ min (1, \frac{2}{ \sqrt[5]{(n+2)(n+3)}} ) \le T \le \sqrt[5]{ \frac{(n+4)^2}{(n+1)(n+3)} } $$](http://latex.artofproblemsolving.com/6/9/1/691f7a090cf902719e57387649b397218b8fbad9.png)
where 
& 
.![$$1 \le \sqrt[n]{\frac{a}{a+nb}}+ \sqrt[n]{\frac{b}{b+na}} \le \frac{2}{\sqrt[n]{n+1}} $$](http://latex.artofproblemsolving.com/1/d/d/1dd7ef9aa3f7cf58d9533a775b4ee553e09e1c84.png)
![$$\sqrt[n]{\frac{a}{a+\left(2^n-1\right)b}}+\sqrt[n]{\frac{b}{b+\left(2^n-1\right)a}}\geq1.$$](http://latex.artofproblemsolving.com/8/7/9/87985b2d3e42985ef426840be20b929f83b64a3a.png)
Prove that![$$1 \le \sqrt[3]{\frac{a}{a+3b}} + \sqrt[3]{\frac{16b}{b+3a}} \le \frac{1+3\sqrt[3]{3}}{2} $$](http://latex.artofproblemsolving.com/c/e/6/ce675d1190a4e01be6bc0f1c76030cb31c0184c7.png)