AIMO test hard questions

[wintry nymph]

the questions are really hard :(
(not in order, not 100% accurate, theres 99.99% that some of them are wrong)

🟥 Geometry
🟧 Combinatorics
🟨 Sequences & Series
🟩
🟦
🟪

• 🟥 Find the number of intersections of the diagonals of a regular heptagon

• if a⁴+a³+a²+a+1=0, what is the value of a²⁰²⁵+a²⁰²²+...+a⁶+a³+1 (i think there's a 1)

• 🟧 A,B,C,D and E is passing a ball starting with A, B will always pass to C, D never pass the ball to E, find the number of possible combinations if the ball is passed for 5 times.

• 🟨 3984+4080+4182+4290+...+10032+10320 (probably ends here, i figured that it increases by 96 first time, 102 second time, 108 third time)

• in a (normal) clock, from 10:00 to 12:00, the 🕦 (shorter one) will be perpendicular to the 🕦 (longer one) when it is x:y, find the sum of the possible values

• 🟧 how many ways are possible to use five numbers from 2,3,4,5,6,7 or 8 to form a 5 digit number with no repeating digits which is divisible by 36

• when 423(3×141),564(4×141) and 577(prime i think) is divided by some number, the sum of the remainders is 42(probably), find that number

(there's more)

[leaden raft]

for 🟧 the first one the a b c d one sol: ||this graphe you can graph it where you have options for every one and also make b and c the same node because it is one combination so you have a b d and you want to find the number of combinations with 2 possible operations 2^5 but you will have some under counting you must take that into account you can go strait to c or from b to c so this can be with stuff going to c no b now we have a new problem how many possible 4 pass combination that all the balls go through b we could notice that b may be gone through multaple times so you will have to do for it alot of cases so lets say number of times b was passed by is hx where x is the number b is passed throu we can create this passing operation as a binary string where 0 denotes a pass to a non b number and 1 denotes a pass to b as you can see the only passes that dont go throu a b or h0 the sol is 0000 so there is only one number for that next for h1 it is 4 choose 1 also for the sol hx will be (5-x choose x) we do that for all numbers 1 2 or 3 because you cant pass to yourself for h1 it is 4 for h2 it is 3 for 3 0 so it is 2^5+4+3 or 39|| i didnt prove it and i am not completely sure of it if somone can double check my logic that would be amazing

for the first geometry that is just ||(7 choose 4) so thats your sol||

for the second combanatorics problem ||just take the 7!/2! because you have 7 numbers then you use one you then need 6 then you use 1 then you need 5 so it becomes 7x6x5x4x3 or 7!/2!||

[wintry nymph]

thx

and i somehow got the gold prize
even i had a lot of questions i don't know how to do

[wintry nymph]

but it needs to be divisible by 36

[leaden raft]

Oh forgot that

[leaden raft]

Btw I think for the second problem you just need to take some exponent rules and keep removing zero or remove -1 or smthn like that

[wintry nymph]

i don't really know

[leaden raft]

try to turn the 1+a^2+a^3+a^4 into a form like 1+a^3+a^6 or smthn to simplifu

i cant rly think abt it rn ill do it when i find time

[lethal folio]

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