[/quote]
,
be a set of integers with the following properties:
.
and 
,
.
,
is composite, then all positive divisors of 
and 
be positive integers with 
.
be a polynomial of degree 
such that the polynomial 
divides 
is zero. Prove that 
,
.
lies on 
such that 
.
passing trough 
at 
.
.
.
,
and 
such that 
.
Compute -![\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}x} \bigg[ \int_{0}^{e^x} \log ( t ) \cos^4 ( t ) \mathrm{d}t \bigg]
\end{align*}](http://latex.artofproblemsolving.com/8/6/f/86fb48db976012106bd30f74cb6ada8c6e6d7800.png)
For 
define 
Determine the open interval(s) (if any) where 
is decreasing and all the open interval(s) (if any) where 
Determine all the local minima of 
of positive integers such that
,
.
and 
Prove that
with circumcenter 
,
be an arbitrary point on the circumcircle of triangle 
and that line 
is not parallel to line 
.
be the point on the circumcircle of triangle 
is perpendicular to 
be the point on line 
.
be the points on the circumcircle of triangle 
is a diameter, and 
and 
are parallel. Let 
be the midpoint of 
.
and 
are perpendicular.
and 
are perpendicular.
be nonnegative real numbers such that 
.
Y by mathmonkey14
1: Little bit harder than most Nr 1's, but you can just bash this out either way.
2. Kinda annoying, but once u break it down it isnt that bad. Average nr2.
3. Cool problem, just break it down into the 3 cases and it isnt too bad. Tiny bit easy for a n3.
4. Dividing by xy, x^2, or y^2 makes this problem a lot easier. You can also factor. avg no 4.
5. Easily sillyable again, kinda annoying for a nr 5. Honestly I would switch no 5 and 6.
6. Pretty simple if you use pithot. If not it can be difficult. Avg no 6.
7. Very hard for a nr7. The whole problem itself is not bad, just it is extremely sillyable
8. Very bashy, and not super easy to solve, but putting this one on a cartesian plane makes it easier. Harder than last years nr8 by a mile.
9. Easy if you see the trick, impossible if not. Without the trick, this problem becomes super bashy, but probably avg nr9. I spent like 1hr on this to absolutely no avail. I got into an equation with degree 6 bc I didnt see the trick.
10. Not very easy, as you have to break it down well for the solution to flow nicely. Quite hard for a nr10 imo
11. Difficult to understand, but once you have the hang of it down it is not horrible. Despite this, many struggled to understand it in the first place. Fairly hard for a nr 11.
12. Lotta people struggle to graph inequalities in 3d planes (So do I). little hard for a nr 12.
13. Very confusing for a bunch of people (including me). Avg for a nr 13 tho.
14. Super hard problem, but extremely elegant. Fermats point is a cool concept here. Hard for a nr 14 tho
15. LTE helps a lot. Honestly I would switch 14 and 15.
Overall, I think problems 1-6 were avg for an aime, but after that the problem got significantly harder. Much harder than last yrs imo. Tough to say what cutoffs will be given all that has gone last year. Tbf I predict sub 220 for both 10a and 10b, but I could be wrong. We kinda just have to wait and see.
2. Kinda annoying, but once u break it down it isnt that bad. Average nr2.
3. Cool problem, just break it down into the 3 cases and it isnt too bad. Tiny bit easy for a n3.
4. Dividing by xy, x^2, or y^2 makes this problem a lot easier. You can also factor. avg no 4.
5. Easily sillyable again, kinda annoying for a nr 5. Honestly I would switch no 5 and 6.
6. Pretty simple if you use pithot. If not it can be difficult. Avg no 6.
7. Very hard for a nr7. The whole problem itself is not bad, just it is extremely sillyable
8. Very bashy, and not super easy to solve, but putting this one on a cartesian plane makes it easier. Harder than last years nr8 by a mile.
9. Easy if you see the trick, impossible if not. Without the trick, this problem becomes super bashy, but probably avg nr9. I spent like 1hr on this to absolutely no avail. I got into an equation with degree 6 bc I didnt see the trick.
10. Not very easy, as you have to break it down well for the solution to flow nicely. Quite hard for a nr10 imo
11. Difficult to understand, but once you have the hang of it down it is not horrible. Despite this, many struggled to understand it in the first place. Fairly hard for a nr 11.
12. Lotta people struggle to graph inequalities in 3d planes (So do I). little hard for a nr 12.
13. Very confusing for a bunch of people (including me). Avg for a nr 13 tho.
14. Super hard problem, but extremely elegant. Fermats point is a cool concept here. Hard for a nr 14 tho
15. LTE helps a lot. Honestly I would switch 14 and 15.
Overall, I think problems 1-6 were avg for an aime, but after that the problem got significantly harder. Much harder than last yrs imo. Tough to say what cutoffs will be given all that has gone last year. Tbf I predict sub 220 for both 10a and 10b, but I could be wrong. We kinda just have to wait and see.
This post has been edited 1 time. Last edited by isache, Feb 11, 2025, 3:36 AM
Reason: because
Reason: because
) 3 cases and boom
,
