[/quote]
,
such that ![\[f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy\]](http://latex.artofproblemsolving.com/0/6/2/0623b7c328af6d088cfc5dd288afe3e17a94ff29.png)
for all reals 
and 
.
and 
be two sequences of integers such that 
and 
for all 
.
divides 
for an infinity of integers 
such that 
for all 
are coprime. Consider the sequence 
given by 
.
such that 
is even and 
is an integer. Prove that the numerator in its simplest form of 
is an odd integer greater than 
.
be an integer and let 
be 
distinct subsets of 
.
such that the n subsets 
are also disjoint.
s
.
satisfies 
.
,
.
such that for all irrational numbers 
and 
,
where 
satisfy that 
.![$$\sum_{cyc} \sqrt[4]{\frac{a^3}{b+c}}$$](http://latex.artofproblemsolving.com/e/f/0/ef089c4f2915bbf75b0f60b044bbe9273b301a81.png)
compute the smallest possible value of ![\[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\]](http://latex.artofproblemsolving.com/5/6/2/562c27c7779285b6439c4087f52933a41f29ee2e.png)
over all permutations 
of 
satisfying ![\[\min(\sqrt{x+xyz},\sqrt{y+xyz},\sqrt{z+xyz})=\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.\]](http://latex.artofproblemsolving.com/2/6/e/26edbd87f49eb68ec9b662b602db712e5151dc9c.png)
Y by Tem8, smbellanki, Skillfulcotton, aops-g5-gethsemanea2, happypi31415, Rounak_iitr, chewkokwah, anantmudgal09
Hi,
As part of my work with my country’s national olympiad system, I am developing a public resource guide on math contests. This guide covers every stage of contest participation, from complete beginners to those aiming for the highest levels of the USAMO and IMO.
I would be grateful for any feedback; if you have additional resource suggestions, please share them in this thread or by AoPS DM to foster discussion. I also particularly welcome comments as to whether a resource may be under/over-classified in the guide.
The following is the leveling system used, aligned with the corresponding tiers of the US contest framework. Note that the guide indicates the suggested practice contest or textbook for someone aspiring to reach the corresponding level, not someone currently at that level.
J: Strong middle-school student with little math exposure outside the curriculum
1A: Strong high-school student with little math exposure outside the curriculum, or Middle-school student scoring 20-35 at Mathcounts chapter
1B: AMC8* Honor Roll, AMC10 score of ~60-70
2A: AMC8 Distinguished HR, AMC10 score of ~80-90
2B: AIME qualifier (through either AMC10/12), AMC10 Honor Roll
3A: AMC10 Distinguished HR, AMC12 HR
3B: AMC12 Distinguished HR, USAJMO/USAMO Qualifier
4A: USAJMO Honorable Mention, USAMO ~50th percentile
4B: USAJMO Honors, USAMO Bronze, Red/Green MOP Cutoff
5A: USAJMO Top Honors, USAMO Silver, Blue MOP/TST Group Cutoff
5B: USAMO Gold, USAMO RMM/IMO team selection
As part of my work with my country’s national olympiad system, I am developing a public resource guide on math contests. This guide covers every stage of contest participation, from complete beginners to those aiming for the highest levels of the USAMO and IMO.
I would be grateful for any feedback; if you have additional resource suggestions, please share them in this thread or by AoPS DM to foster discussion. I also particularly welcome comments as to whether a resource may be under/over-classified in the guide.
The following is the leveling system used, aligned with the corresponding tiers of the US contest framework. Note that the guide indicates the suggested practice contest or textbook for someone aspiring to reach the corresponding level, not someone currently at that level.
J: Strong middle-school student with little math exposure outside the curriculum
1A: Strong high-school student with little math exposure outside the curriculum, or Middle-school student scoring 20-35 at Mathcounts chapter
1B: AMC8* Honor Roll, AMC10 score of ~60-70
2A: AMC8 Distinguished HR, AMC10 score of ~80-90
2B: AIME qualifier (through either AMC10/12), AMC10 Honor Roll
3A: AMC10 Distinguished HR, AMC12 HR
3B: AMC12 Distinguished HR, USAJMO/USAMO Qualifier
4A: USAJMO Honorable Mention, USAMO ~50th percentile
4B: USAJMO Honors, USAMO Bronze, Red/Green MOP Cutoff
5A: USAJMO Top Honors, USAMO Silver, Blue MOP/TST Group Cutoff
5B: USAMO Gold, USAMO RMM/IMO team selection
This post has been edited 4 times. Last edited by Konigsberg, Jan 28, 2025, 8:07 AM
