did you ask helena for permission to post ^^

for these, i define a_2-a_1=d

1) fairly easy i think? just need to find whatever i,j you can remove to make a 4-term arithmetic sequence

2) i found 3 sequences, maybe try to prove that you can split them up into m arithmetic sequences? basically just need to find a partition of these sequences that makes m sequences in total:

i. a_1, a_3, a_5, ..., a_11
ii. a_4, a_6, a_8, ..., a_(4m+2)
iii. a_(15), a_(17), a_(19), ... a_(4m+1)

the first has 6 terms, the second 2m terms, and the third 2m-6 terms

hope this helps somewhat for this part - i got a bit lazy about finishing the proof but i'll come back and finish later ig

3) no clue i'll come back later and look again

hope this helps!

oops just saw CC_chloe's comment

@helena sry if this goes against your rules oops sry sry

@contactbibliophile for the 2 days i was afk i solved all of them, for parts 1 and 2 i did similar things as u
i assumed d=1 and a_1=1 lol
for part 3, i generalized the solution to part 2

@helena thx for letting me post this and yeah i agree its an interesting problem
i found it on gaokao (chinese college entrance exam)
i'll probably post it on a forum (hsm)

NVM SOMEONE ALREADY POSTED IT
https://artofproblemsolving.com/community/c4h3336148_chinese_gaokao_problem_19
why is it called "lemon sequence" skull

omg help I'm so thankful that I escaped gaokao system
I suck at comp math
very sus
I'll try this today after I finish chem and research hw prob