[/quote]
,
with 
odd, which satisfy the equation:
denote the set of all real numbers. Find all functions 
such that ![\[ f\left( x^{2}+f(y)\right) =y+\left( f(x)\right) ^{2}\hspace{0.2in}\text{for all}\,x,y\in \mathbb{R}. \]](http://latex.artofproblemsolving.com/8/5/c/85c37ee472f918eace12c8cd7bef6766e617d2aa.png)
(where 
are the digits and 
)
For example the permutation sum of 20253 is 
Let 
and 
be two fivedigit integers with the same permutation sum.
.
rest on a plane. A sphere with radius 
is tangent to all of them, but does not intersect nor lie on the plane. A sphere with radius 
lies on the plane, and is tangent to all three spheres with radius 
If the shortest distance between the sphere with radius 
,
pairwise distinct positive integer numbers 
,![\[
\frac{a_i^2 + a_j^2}{a_i + a_j}
\]](http://latex.artofproblemsolving.com/3/5/3/3537ddec6d793d3831ba82a1e13b98d43b1c29bc.png)
be the incircle of triangle 
.
is parallel to side 
and tangent to 
is parallel to side 
and tangent to 
,
satisfy all conditions below:
for all 
for all 
for all 
that are solutions of the equation 
.
![\[
2^a + 3^b + 1 = 6^c
\]](http://latex.artofproblemsolving.com/1/9/e/19eb5efa2ccdf54cd277339f8d3b99f75168779d.png)
Y by littlebigbull
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.
