[/quote]
,
Prove that
be a 
-
.
,
in 
such that the sets ![\[ A_j=\{x+t_j\mid x\in A\},\qquad j=1,2,\ldots,100 \]](http://latex.artofproblemsolving.com/d/5/c/d5c2a7b943e8c7200361698f3eb6f83700ad68d6.png)
are pairwise disjoint.
be a triangle. Let 
be the feet of the altitudes from 
respectively. Let 
be the projections of 
.
.
and 
Prove that
and 
be two sequences defined by:![\[
a_{n+1} = \frac{1 + a_n + a_n b_n}{b_n} \quad \text{and} \quad b_{n+1} = \frac{1 + b_n + a_n b_n}{a_n}
\]](http://latex.artofproblemsolving.com/a/a/0/aa0a5c5acedc990eb345fb50a8395db05307a4e7.png)
for all 
,
and 
.![\[
a_{2024} < 5.
\]](http://latex.artofproblemsolving.com/f/c/a/fca6e6630f33429b5a7ea867d639a2e29ff881f2.png)
such that the following equation holds:![\[
7^a + 1 = 2b^2
\]](http://latex.artofproblemsolving.com/e/0/8/e08d3fc7feefd7b4acc4a0f9c7e8bcfa328b5791.png)
such that
such that
Prove that
such that
Y by
Info:
In 8th grade so I'm really hoping I can make nats now
I currently mock around 38 - 40 on nationals questions from 2015+
I mock anywhere from 37 - 42 on state questons from 2020+
For the sprint round I also have noticed that the difficulty jump from questions around 19 and 20 to questions around 22 and 23 has been really large (starting from 2023). I've also noticed that the last three questions (also from 2023 ->) are IMO impossible to do in the 40 minutes.
On target I can get 7/8 or even 8/8 if I'm lucky but it's possible for me to get 6/8
I'm in MA :sob: really hard state so do I have a chance
Edit: Just mocked the 2022 state round and got a 41 (29 sprint, 12 target :sob:)
Currently putting around 3 hrs or so a day and I have been for the past 2 months
States is 3/1 for me :sob:
so am i cooked
In 8th grade so I'm really hoping I can make nats now
I currently mock around 38 - 40 on nationals questions from 2015+
I mock anywhere from 37 - 42 on state questons from 2020+
For the sprint round I also have noticed that the difficulty jump from questions around 19 and 20 to questions around 22 and 23 has been really large (starting from 2023). I've also noticed that the last three questions (also from 2023 ->) are IMO impossible to do in the 40 minutes.
On target I can get 7/8 or even 8/8 if I'm lucky but it's possible for me to get 6/8
I'm in MA :sob: really hard state so do I have a chance
Edit: Just mocked the 2022 state round and got a 41 (29 sprint, 12 target :sob:)
Currently putting around 3 hrs or so a day and I have been for the past 2 months
States is 3/1 for me :sob:
so am i cooked
This post has been edited 2 times. Last edited by stjwyl, Feb 22, 2025, 8:51 PM
