[/quote]
,
,![$P_{i}(x)\in \mathbb{R}[x], \forall i=\overline{1,n}.$](http://latex.artofproblemsolving.com/0/5/1/0514f85850830af9a9e1a4dd5e7a4e8e68b2dff9.png)
.
polynomial with only one root.
be real numbers. For 
,![\[\{a_n^{(i)}\}_{n \geq 0}\]](http://latex.artofproblemsolving.com/4/c/7/4c74470e7f3812966ba17e6fca535f085dced237.png)
be an infinite real sequence satisfying![\[a_0^{(i)} = 0.\]](http://latex.artofproblemsolving.com/0/f/6/0f6a0e940e48c2bf62fc12579f2396fb86d7b896.png)
It is known that:![\[a_1^{(1)}, a_1^{(2)}, \cdots, a_1^{(2025)}\]](http://latex.artofproblemsolving.com/7/9/5/795b375a183f2c553294be354f67826951092b84.png)
are not all zero.
and any ![\[p_i \cdot a_n^{(i+1)} = a_{n-1}^{(i)} + a_n^{(i)} + a_{n+1}^{(i)},\]](http://latex.artofproblemsolving.com/1/a/c/1ac323cfc66546d8172f307b020d955b086d99f8.png)
![\[\{a_n^{(2026)}\}\]](http://latex.artofproblemsolving.com/b/6/e/b6e825cc8d5a1a8b2a56cf20ecde79c4c51ba9a6.png)
satisfies![\[a_n^{(2026)} = a_n^{(1)}, \, n = 0, 1, 2, \cdots.\]](http://latex.artofproblemsolving.com/c/d/f/cdf1e2f03ae1c343f34a017157886e4b6fca2d21.png)
such that for infinitely many positive integers 
,![\[\max \left\{ |a_n^{(1)}|, |a_n^{(2)}|, \cdots, |a_n^{(2025)}|\right\} \geq r.\]](http://latex.artofproblemsolving.com/2/5/5/255ba07c42ee84b59da2d3f1959a95d81c1c6b99.png)
for which there exists an infinite subset 
of the positive integers such that: for any pairwise distinct positive integers 
,
and the product 
are both square-free.
be a finite set of real numbers, 
be a real number, and 
be 2025 non-zero real numbers. Define![\[A =
\left\{
(x_1, x_2, \cdots, x_{2025}) : x_1, x_2, \cdots, x_{2025} \in X \text{ and } \sum_{i=1}^{2025} \lambda_i x_i = d
\right\},\]](http://latex.artofproblemsolving.com/2/e/1/2e18b6d25bfaa3e3d52f7efe5e4cba7a9cb87db9.png)
![\[B =
\left\{
(x_1, x_2, \cdots, x_{2024}) : x_1, x_2, \cdots, x_{2024} \in X \text{ and } \sum_{i=1}^{2024} (-1)^i x_i = 0
\right\},\]](http://latex.artofproblemsolving.com/4/1/f/41f6b126cc3cf24757b37147906c767d49a0fdd9.png)
![\[C =
\left\{
(x_1, x_2, \cdots, x_{2026}) : x_1, x_2, \cdots, x_{2026} \in X \text{ and } \sum_{i=1}^{2026} (-1)^i x_i = 0
\right\}.\]](http://latex.artofproblemsolving.com/d/e/6/de6586b88004e3f5aa7774525b00ee975c072ff9.png)
Show that 
.
and 
satisfies the following recursion formulas: 
,
.
, 
.
and prove 
be a finite, nonempty set of points in the plane such that, for every point 
in 
in 
.
,![\[ a_n > 0,\qquad\ b_n>0,\qquad\ a_{n+1}=a_n+\dfrac{1}{b_n},\qquad\ b_{n+1} = b_n+\dfrac{1}{a_n}. \]](http://latex.artofproblemsolving.com/4/f/c/4fc9fd1cb31a25378268f20e019e639667e52910.png)
Prove that 
.
and centroid 
.
is the reflection of 
and 
.
over 
be an isosceles triangle with 
,
be a point inside side 
such that 
.
and 
be two points inside sides 
and 
,
.
meet line segment 
at 
,
meet again at point 
,
.
Y by aidan0626, Hachi123, ys-lg, scannose, Alex-131, Sedro, KAME06, Mquej555, Awesomeness_in_a_bun, IMUKAT, MatSeFner, Siddharth03, CatsMeow12, User21837561, EpicBird08, GeoKing, meduh6849, qwerty123456asdfgzxcvb, lksb, Maximilian113, valenbb
I have noticed thousands upon thousands of inequalities that you have posted to HSO and was wondering where you get the inspiration, imagination, and even the validation that such inequalities are true? Also, what do you find particularly appealing and important about specifically inequalities rather than other branches of mathematics? Thank you 
