[/quote]
,
be nonnegative real numbers such that 
.
,
is the incenter, and the incircle touches sides 
,
,
at 
,
,
,
parallel to 
and 
at 
and 
,
.
and 
Prove that
,![$P_{i}(x)\in \mathbb{R}[x], \forall i=\overline{1,n}.$](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 
.
Y by
Find the smallest positive integer n such that n is a multiple of 11, n +1 is a multiple of 10, n + 2 is a
multiple of 9, n + 3 is a multiple of 8, n +4 is a multiple of 7, n +5 is a multiple of 6, n +6 is a multiple of
5, n + 7 is a multiple of 4, n + 8 is a multiple of 3, and n + 9 is a multiple of 2.
I tried doing the mods and simplifying it but I'm kinda confused.
multiple of 9, n + 3 is a multiple of 8, n +4 is a multiple of 7, n +5 is a multiple of 6, n +6 is a multiple of
5, n + 7 is a multiple of 4, n + 8 is a multiple of 3, and n + 9 is a multiple of 2.
I tried doing the mods and simplifying it but I'm kinda confused.
This post has been edited 1 time. Last edited by jag11, Mar 22, 2025, 10:41 PM
Reason: edit
Reason: edit
satisfies the condition except that 
is negative. What is the smallest positive number you can add to 
is the multiple of 11, 10, 9, 8, 7, 6, 5, 4, 3, 2
is lcm(11, 10, 9, 8, 7, 6, 5, 4, 3, 2)-11
