and 
. Define 
for 
as 
, 
. Prove that 
exists and find it's value .
be a function that satisfies the conditions:
has Darboux’s property
at 
exist.
![\[ \int_{1}^{2}e^{x( x+\sqrt{x^2-1} )}dx \]](http://latex.artofproblemsolving.com/b/1/d/b1d1f5525bb9cb5fe1f1a7491ff5209cad5438d3.png)
such that
.
is an increasing function on 
,
.
,
for 
using the same 
and 
axes.
.
.
;
be an algebraic extension of 
,
be an algebraic closure of 
is an embedding such that 
for all 
,
extends to an automorphism of 
is an embedding. Then, there exists an embedding 
that extends 
,
is an isomorphism.
and 
that extends 
is also an algebraically closed field.
,
be the field of algebraic numbers, 
.
defined by
Then, according to the exercise above, 
How should we interpret 
?
be a prime number. A flea is staying at point 
of the real line. At each minute,
to the left or to the right.
minutes, it wants to be at 
the number of its strategies to do this
:
be a continuous function such that for all 
and for all 
,
Show that 
Y by
Hello AOPS community,
Should I take the Mathcounts/AMC8 Advanced course over the summer or the AMC 10 Intermediate course?
I'm a 7th grader who is aiming for DHR AMC 8 and AIME qual next year as well as nats mathcounts, but sillies a lot on some stuff.
Scores:
6th grade (2023-2024 year)
Mathcounts Chapter: 33 (21/12)
Mathcounts State: 19 (11/8)
AMC 8: 12 (idk how this happened)
didn't take AMC 10
7th grade (2024-2025 year, so right now)
Mathcounts Chapter: 28 (36 w/o sillies)
Mathcounts State: a score higher than my chapter score (I cant reveal exact until April)
AMC 8: 16 (the problems were randomized cuz online)
AMC 10A: 42 (guessed too much)
AMC 10B: 58.5 (:facepalm:)
In practice AMC 8's I consistently get around 20-22, and on mock AMC 10's I get around 9-12 problems correct (so about low-mid 80s). In mock state Mathcounts I get 26-28s. Several people in my state have said I have a free nats qual next year due to all the competition graduating, but I am aiming for 34+ on state next year for insurance and to see how far I can go. I also hope to get DHR AMC 8 in 2026 and AIME, but not sure whether to take the AMC 8 advanced or AMC 10 course. Should I retake AMC 8 advanced (i took it last summer) to get really good basics or take AMC 10 course, and will the AMC 10 course help me get DHR amc8?
Thank you for your help, its greatly appreciated
please respond I'm begging.
Should I take the Mathcounts/AMC8 Advanced course over the summer or the AMC 10 Intermediate course?
I'm a 7th grader who is aiming for DHR AMC 8 and AIME qual next year as well as nats mathcounts, but sillies a lot on some stuff.
Scores:
6th grade (2023-2024 year)
Mathcounts Chapter: 33 (21/12)
Mathcounts State: 19 (11/8)
AMC 8: 12 (idk how this happened)
didn't take AMC 10
7th grade (2024-2025 year, so right now)
Mathcounts Chapter: 28 (36 w/o sillies)
Mathcounts State: a score higher than my chapter score (I cant reveal exact until April)
AMC 8: 16 (the problems were randomized cuz online)
AMC 10A: 42 (guessed too much)
AMC 10B: 58.5 (:facepalm:)
In practice AMC 8's I consistently get around 20-22, and on mock AMC 10's I get around 9-12 problems correct (so about low-mid 80s). In mock state Mathcounts I get 26-28s. Several people in my state have said I have a free nats qual next year due to all the competition graduating, but I am aiming for 34+ on state next year for insurance and to see how far I can go. I also hope to get DHR AMC 8 in 2026 and AIME, but not sure whether to take the AMC 8 advanced or AMC 10 course. Should I retake AMC 8 advanced (i took it last summer) to get really good basics or take AMC 10 course, and will the AMC 10 course help me get DHR amc8?
Thank you for your help, its greatly appreciated
please respond I'm begging.
