[/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 Kawhi2, Funcshun840, erringbubble, Timmy456, Rounak_iitr, scannose, math90, crocodilepradita, AlienGirl05, cubres
Hello Aops! Since IMO 2024 is less than a week away,
What are your predictions for the category of each problem at IMO 2024?
If you want to write your prediction, please write it in the form ABC DEF
Where A,B,C,D,E,F are problems 1,2,3,4,5,6 respectively. Each letter should be A,C,G or N.
Rules:
1. Problems 1,2,4,5 are distinct categories.
2. Each day consists of 3 distinct categories.
Edit : the answer is ANC GCA
What are your predictions for the category of each problem at IMO 2024?
If you want to write your prediction, please write it in the form ABC DEF
Where A,B,C,D,E,F are problems 1,2,3,4,5,6 respectively. Each letter should be A,C,G or N.
Rules:
1. Problems 1,2,4,5 are distinct categories.
2. Each day consists of 3 distinct categories.
Edit : the answer is ANC GCA
This post has been edited 4 times. Last edited by GreenTea2593, Aug 12, 2024, 2:15 PM
respectively. Will it continue like that ?
which is clearly within realm of guessablity.
