feecIe6418's blog

By feecIe6418, history, 11 months ago, In English
A Tutorial
A Code
B Hint 1
B Hint 2
B Hint 3
B Tutorial
B Code (Python)
C Hint 1
C Hint 2
C Tutorial
C Code
D Hint
D Tutorial
D Code (Python)
E1 Hint
E1 Tutorial
E1 Code
E2 Hint
E2 Tutorial
E2 Code
 
 
 
 
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11 months ago, # |
  Vote: I like it -23 Vote: I do not like it

Great problems

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11 months ago, # |
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Good stuff, I liked C a lot, it made me think of LCM in a new way

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    11 months ago, # ^ |
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    can you please elaborate your thought about LCM..

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      11 months ago, # ^ |
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      yes plz how u thought of lcm i thought of 3 first then thought for 6 then 12 and come up with factorial approach but it hasn't worked.

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        11 months ago, # ^ |
        Rev. 4   Vote: I like it +8 Vote: I do not like it

        I'll try to explain.

        $$$n \leq 1e16$$$, so just starting a loop and trying to find an answer for each $$$i (1 \leq i \leq n)$$$ is a bad idea. This is why we need to "group" the answers. It is important to understand that the answers cannot be great. Because if $$$f(i) = x$$$, it means that i has divisors $$$2, 3, 4 \dots x - 1$$$, so $$$i \geq lcm (2, 3, 4 \dots x - 1)$$$

        Let's try to figure out what $$$⌊n / lcm (1,2, \dots, i) ⌋$$$ means. This is the number of elements that are divisible by $$$2, 3, 4 \dots i$$$, so we have not counted the answer for them yet. Thus, based on the editiorial "The number of ks such that $$$f (k) = i$$$ is $$$⌊n / lcm (1,2, \dots, i — 1) ⌋ — ⌊n / lcm (1,2, \dots, i)⌋$$$" It means that $$$⌊n / lcm (1,2, ..., i — 1) ⌋$$$ is the number of elements that have "reached" us. And $$$⌊n / lcm (1,2, ..., i) ⌋$$$ is the number of elements for which the answer is greater than $$$i$$$. Thus, subtracting one from the other, we get the number of elements for which the answer is equal to $$$i$$$. And of course, sorry for my bad English, I tried my best.

        My C++ solution:

        121264752

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          11 months ago, # ^ |
            Vote: I like it +1 Vote: I do not like it

          Thanks for the clear explanation. Just one small correction, instead of

          Because if f(i)=x, it means that i has divisors 2,3,4…i−1, so i≥lcm(2,3,4…i−1)

          it probably should be

          Because if f(i)=x, it means that i has divisors 2,3,4… x −1, so i≥lcm(2,3,4… x −1)

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            11 months ago, # ^ |
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            Oh, my fault. I will edit it immediately.

            Thank you!

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          11 months ago, # ^ |
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          thanks man, after reading it looks simple logic but why i can't think of these logics in contest :(

          noob me :{

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            11 months ago, # ^ |
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            I'm really happy that my comment was useful for somebody.

            You are not noob) In my opinion it's absolutely normal, when you can't solve the problem during the contest.

            P.S. I couldn't solve C too))

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          10 months ago, # ^ |
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          Clear explanation.

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          10 months ago, # ^ |
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          Do you think f(k) = i = 6, is possible ??

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            10 months ago, # ^ |
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            f(k) always takes the value which are perfect power of primes. Like 2^1,2^2,2^3, 3^1, 3^2, 3^3. Let's say if f(k)=p*q. Where p and q both are primes. Here p<pq and q<pq so both of them must divide k. And also gcd(p,q)=1 which implies pq must divide k. Which is contradiction. Pardon if anything is not clear since this is my first comment :)

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          10 months ago, # ^ |
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          I was trying to solve this question,everything seems fair to me in my code.But It was giving WA, so I removed only floor and it has passed all TCs. Can you tell me,why?

          123126566

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        11 months ago, # ^ |
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        it didn't work because

        lcm of 1,2,3,4 is 12

        while the factorial is 24

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      11 months ago, # ^ |
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      to know how many numbers in range 1 to n are divisable by all numbers from 1 to i it equals to n / lcm(1,2,3,..,i) for example :- if n = 15 and you want to know how many number divisable by 1, 2, 3 and 4 floor(15 / lcm(1,2,3,4)) = 1 there is just one number divisable by 1, 2, 3, and 4 and this number is 12.

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        11 months ago, # ^ |
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        tenks man

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        6 weeks ago, # ^ |
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        thanks for your example, but I have a little confusion on the simplified version of the formula in editorial, could you help me explain how does $$$\sum_{i>1} i(\lfloor n/lcm(1,2,...,i-1)\rfloor -\lfloor n/lcm(1,2,...,i)\rfloor )$$$ becomes $$$\sum_{i\ge 1} \lfloor n/lcm(1,2,...,i)\rfloor +n$$$ ?

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11 months ago, # |
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Thanks for the editorial

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11 months ago, # |
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Mathforces!

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11 months ago, # |
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This one requires a lot more math, but it's not friendly to younger participants. And, this is an algorithm competition, not a math competition.

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    11 months ago, # ^ |
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    why is it not friendly? i think everybody should know smth about LCM. Also B was not hard at all. I am a young participant (i'm 13) and ABC weren't hard at all

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      11 months ago, # ^ |
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      smart russians.

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      11 months ago, # ^ |
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      Saying ABC weren't "hard at all" won't cut it mate. They were hard, for the average folks. I've been coding for almost 1.5+ years now, and could only solve A. I found a research paper on OEIS which was directly related to C, still couldn't code it up in time. Thank your stars (or rather your smart ass genes lol) that you're able to do such good level math problems at such a tender age. You probably are meant for this, keep solving!

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        11 months ago, # ^ |
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        Nobody is meant for anything, it's just the timing when they started. And btw you are comparing your present with his, he also learned to check prime from O(n) to O(sqrt(n)) at some point in time.

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          11 months ago, # ^ |
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          i also started about 1.5 years ago)

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            11 months ago, # ^ |
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            I guess he is just frustrated by the fact that the contest was just Math+ DP, out of which people struggle at both a bit... ig..

            PS. like, i like geometry and DS based problems, but it has been a long time i haven't seen such problems in recent contests....

            Anyhow its pretty Subjective, i felt yesterday's contest was pretty doable at least till C.

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      11 months ago, # ^ |
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      Hard is a relative subject, some topics can be hard for someone, and easier for others. Age doesn't matter at all. This was a more math-focused contest (ABC all have math tag), and a mathematical approach is often harder for some.

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      11 months ago, # ^ |
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      Everyone plays games , and only some reach the top but all other plays because they enjoy playing it .

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    10 months ago, # ^ |
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    Beacuse your ability is weak but not the round is not friendly.And I think problem D is easier than B and C.So why this contest unfriendly?

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      10 months ago, # ^ |
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      Hello, Magic_Moon can you explain D. maybe using dry run or any way you are comfortable with. I am having a tough time understanding it.

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11 months ago, # |
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Great Round, Great problems. Statements were short and crisp problem "C" is my personal favourite. Looking forward to see more such rounds. Thank you!!!

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11 months ago, # |
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Although I didn't participate, I also think the problems especially C and D are wonderful!

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11 months ago, # |
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When I was solving B, for some reason I typed "a^x" as "a*x" and spent almost all round to understand why extended euclidian algorithm doesn't work. Moral of the story: Don't tunnel vision, do your math.

Great round!

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11 months ago, # |
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I spent over an hour on b since I put (n % (a ^ k))%b == 0 instead of (n — (a ^ k))%b == 0.

Nonetheless the problems were good, and I enjoyed the round

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    11 months ago, # ^ |
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    I did the same mistake and after seeing other people's code I was like when will these silly mistakes would leave me :(

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11 months ago, # |
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I think B is harder to think than D.

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11 months ago, # |
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Imagine C is harder to think than D...

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Can someone explain the second paragraph of $$$B$$$ tutorial?

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    11 months ago, # ^ |
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    Let us assume n is present in the set. n can be represented as x*a^k + y*b, where x,k,y are whole numbers. Now n%b = (x*a^k + y*b)%b = (x*a^k)%b = x%b * (a^k)%b. x can be any of the previous values in the set, used to generate n. Thus we can assume x = (x')*(a^k') + y*b and x%b = (x')%b * (a^k')%b. Eventually we can see that x%b will be of type a^k where k will be a whole number. Hence we can write n%b = (a^k)%b, given n is present in the set.

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      11 months ago, # ^ |
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      thanks! your explanation is so easy to understand.

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      11 months ago, # ^ |
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      I have an easier explanation. All numbers in the set can be represented as a^x + b*y. Thus, if we subtract a^k from n and it gives a multiple of b, then n belongs to the set. You can iterate over k to check if the number belongs to the set.

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        10 months ago, # ^ |
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        This explanation is so far the easiest one I have found for this problem, I have been trying to understand the other solution for some time now, glad I stumbled upon yours.

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        5 months ago, # ^ |
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        Wow, amazing observation. Thank you very much for sharing this with us!

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11 months ago, # |
Rev. 3   Vote: I like it +28 Vote: I do not like it

I had a slightly different approach for problem C. So I will explain my solution and thinking process.

Initially I was not able to solve the problem. Then I remembered one thing that factorial of even small numbers is very large (like $$$50!$$$). So, this gave me direction to think that $$$f(n)$$$ cannot be very large. So I ran a test to see that when we multiply prime numbers up to $$$50$$$, it will exceed $$$10^{16}$$$.

I still had no clue how to solve this problem. Then I randomly wrote a number in prime factorization form. I saw that $$$f(n)$$$ can only be some power of only a single prime i.e. $$$f(n) = p^k$$$ where $$$p$$$ is some prime and $$$k$$$ is its power. I was able to prove this.

Now, I knew that $$$f(n)$$$ can only be some power of prime up to $$$50$$$. So, now if $$$f(n)=p^k$$$, then $$$n$$$ must have exactly $$$p^{k-1}$$$ in its prime factorization form because otherwise $$$p^{k-1}$$$ will be $$$f(n)$$$. Now, I had to think what else should $$$n$$$ must have in its prime factorization form. I observed that for some other prime $$$q$$$ its power $$$m$$$ must be such that $$$q^{m+1} > p^k$$$ because otherwise $$$q^{m+1}$$$ can be answer. Then I found a number $$$D$$$ satisfying the above requirements. Now, the number of multiples of $$$D$$$ minus the number of multiples of $$$D*p$$$ would give the numbers for which $$$f(n) = p^k$$$. Then I would just add contribution to answer. Note that we have to handle a corner case i.e. when $$$D$$$ exceeds $$$n$$$, then we don't take any contribution from $$$p^k$$$.

My code
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    11 months ago, # ^ |
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    11 months ago, # ^ |
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    I massively overcomplicated my solution (originally in Rust, translated to C++) and reading it again now I realize that it basically amounts to Um_nik's solution except mine is slightly faster because it only evaluates the numbers that change the lcm. Of course, it's nowhere near the TL so that doesn't even matter.

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    11 months ago, # ^ |
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    can someone tell me what's wrong with this approach? if(i==1)sum+=2; else if(i%2)sum+=2; else if(i%2==0 && i%4==0)sum+=3; else if(i%2==0 && i%6==0)sum+=5; feel free to correct me,

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11 months ago, # |
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Any idea why this solution to B gives TLE? Is it some silly mistake? Cause I've pretty much followed the editorial.

void solve(int testcase) {
    int n, a, b; cin >> n >> a >> b;
 
    if (a == 1) {
        cout << ((n - 1) % b == 0 ? "Yes\n": "No\n");
        return;
    }
 
    int m=1;
    bool res=false;
    while (m <= n) {
        if (m % b == n % b) {
            res = true;
            break;
        }
        m *= a;
    }
 
    cout << (res ? "Yes\n": "No\n");
}
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11 months ago, # |
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Can anyone analysis the time complexity of problem C for me pls?

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    11 months ago, # ^ |
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    Someone correct me if I'm wrong, I think it's O(nlogn) because for every i computing lcm involves computing gcd which is logn with Euclidean algorithm

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      11 months ago, # ^ |
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      So you mean there are 10^16 log 10^16 operations in worst case @@?

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        11 months ago, # ^ |
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        no, max(N) is about 40-50

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        11 months ago, # ^ |
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        Sorry that was a horrible mistake. The loop runs while lcm <= n, and I think by lcm(1,...,40) you passed 10^16, so it should be O(40 log(something))

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11 months ago, # |
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The introduction of hints was really very good. Thanks for such a great editorial.

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Can someone please guide me through the equivalence of both formulas in C?

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    10 months ago, # ^ |
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    I also want to know about this... I've got the first equation but can anyone explain how the

    SUM [ n/lcm(1,2,...,i) ] + n

    comes from SUM [ i { (n/lcm(1,2,...,i−1)−(n/lcm(1,2,...,i) } ) ?

    I figured out the front part but how the (+ n) coming out from it? feecIe6418

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11 months ago, # |
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math math math math math maTh MaTH MATH MATH MATH MATH MATH MATH MATH AAAAAAAAAAAAAAAAAAAA!!!!1!!!11!!!

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11 months ago, # |
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These format of spoilers hurt my eye really bad, can you make it like this ? It will look much better.

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11 months ago, # |
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Really a great contest! Kudos!

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11 months ago, # |
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11 months ago, # |
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Nice contest.
- Team Atcoder Fan, 2021

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11 months ago, # |
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in the C Tutorial : "Since f(n)= i means lcm(1,2,...,i−1) ≤n " why? I don't understand this, can you explain it to me?

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    11 months ago, # ^ |
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    because 'i' is the minimum number not divisor of n. and n is divisible by all number (1 to i-1). Like what is the minimum number that divisible by all number (1 to i-1) ??? it's g = Lcm(1,,,,i-1). Now, if f(n) = i, that means (n % g == 0) and g <= n.

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    11 months ago, # ^ |
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    I think what they are actually trying to say is that it doesn't make sense to go to such an i where lcm(1,2,...,i-1)>n since the contribution for such an i would be zero.

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11 months ago, # |
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why chinese people think,that math problems are more interesting than algorithmic or data structure?

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11 months ago, # |
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Can someone help me understand how they thought of this DP (and it's states) in problem D?

The current editorial focuses more on implementation, less on the idea.

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    11 months ago, # ^ |
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    watch galen colin video.

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    11 months ago, # ^ |
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    When you have these expected value problems and you have to compute the total value (or expected value) of all sums, you can generally compute the total contribution of each term and add it separately.

    I think this can somewhat be seen by putting all the subsets of sums with the # of times they get counted vertically. When you add all of them up, you just get one equation in the $$$n$$$ variables (for whatever $$$n$$$ is, here it's the number of terms that start with $$$+$$$). The coefficient of each variable here is the number of times it's counted across all subsets -- which is the same as its independent contribution. Then, you can proceed as above.

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Look at my code for B, it's very funny compared to the original solution, but still my code passed all the tests. Here is the link: https://codeforces.com/contest/1542/submission/121210576

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What are $$$p1$$$ and $$$q1$$$ in E1 editorial? And what do you mean by enumerating $$$p1$$$ and $$$q1$$$? feecIe6418

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    11 months ago, # ^ |
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    $$$p_1$$$ is the first element of $$$p$$$

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Hi, I don't understand the editorial of problem B. Can someone please explain it to me?

Thanks

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    11 months ago, # ^ |
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    Here is my approach — Every element of the set can be expressed in the form $$${a^p + qb}$$$ where p and q are whole numbers. You can try writing the elements of the set in a tabular form to see it for yourself, intuitively. $$$1,1+b,1+2b,1+3b...$$$
    $$$a,a+b,a+2b,a+3b...$$$ $$$a^2,a^2+b,a^2+2b,a^2+3b...$$$

    We want to check if n is expressible in this form. To do this, iterate through p from 0 to any large enough value. For each value of p, check if $$$n = a^p + qb$$$, that is, ($$$n - a^p$$$)%b == 0. If this is true, the answer is YES. The loop terminates when $$$a^p$$$ becomes greater than n itself.

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Does anyone have a different solution for E1?

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11 months ago, # |
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froggyzhang had given a solution for e2 by using generating functions which i think is much easier than the official tutorial(which is also great!). Unluckily, it's in Chinese. And if anyone want to read it you can find it here.

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    11 months ago, # ^ |
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    My English is very bad, can anyone help me translate it into English? awa

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.

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Is there any recursive approach to solve D

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In B, is it true that if $$$n$$$ is equal to some power of $$$a$$$ then $$$n$$$ belongs to the set?

I had the following condition in my code:

if( n % a == 0) {
    cout << "Yes\n";
    continue;
}

and for some reason this gives a wrong answer. My reasoning was the following. If $$$1$$$ and $$$1 \cdot a$$$ are in the set then $$$1 \cdot a \cdot a$$$ must also be in the set and other powers e.g. $$$1 \cdot a \cdot a \cdot a$$$ as well. So if $$$a$$$ divides $$$n$$$ i.e. $$$n \pmod a == 0$$$ then $$$n$$$ should be in the set. Am I missing something trivial ????

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Can someone explain in better/different words what the states in the DP for D actually represent? The editorial is not at all clear to me.

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    11 months ago, # ^ |
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    For each number x, to determine whether it's in the final set or not, we only need to consider those numbers that are smaller than x. Because those larger ones will always be popped after x is popped. If there are more than one x, we assume the one that appears first is smaller.

    So, for each number x, define f[i][j] represent: after considering the ith operation, how many subsequences are there satisfies that there are j numbers in the set are smaller than x. (Remember to ignore the operation for the number x).

    Then do the dynamic programming as the tutorial shows.

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      11 months ago, # ^ |
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      "If there are more than one x, we assume the one that appears first is smaller." Can you explain why so?

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        11 months ago, # ^ |
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        Actually, this is just an assumption to prevent some number being counted twice.

        For example, the whole sequence is +1 +1 -.

        After processing the whole sequence +1 +1 -, the final set is 1.

        When we count the first +1, we will not count the sequence +1 +1 -, cause the first +1 is smaller, according to the assumption. The 1 in the final set is the second +1, we'll count this one when processing the second +1.

        If you don't set the rule for equal elements, +1 +1 — will be counted for both +1, which is not what we want.

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          11 months ago, # ^ |
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          Now I got the intution behind it! Thnx!

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      10 months ago, # ^ |
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      Thanks got it!

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    11 months ago, # ^ |
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    You can have a look here and the parent comment, I tried to explain it with a picture:

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What a nice math contest!(C is really nice)

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11 months ago, # |
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Brilliant Problems!! Thanks for such a nice contest.

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11 months ago, # |
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Could anyone explain why I got a time limit exceed in problem B test 2?

Thank you so much.

#include <iostream>
using namespace std;

int main() {
    int tcs, tc, n, a, b, i;
    bool f;
    cin >> tcs;
    for (tc = 0; tc < tcs; tc++) {
        cin >> n >> a >> b; 
        if (a == 1) cout << (n % b == 1 ? "Yes" : "No") << endl;
        else {
            i = 1; f = false;
            while (i <= n) {
                if ((n - i) % b == 0) {f = true; break;}
                i *= a;
            }
            cout << (f ? "Yes" : "No") << endl;
        }
    }
    return 0;
}
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    11 months ago, # ^ |
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    Because you used int instead of long long.

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      11 months ago, # ^ |
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      thank you, but I got a wrong answer in test 2 now.

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        11 months ago, # ^ |
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        Try this data, the answer is Yes.

        1
        2 1 1
        
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          11 months ago, # ^ |
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          thx, I didn't think about this. You help me a lot

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11 months ago, # |
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Someone please explain solution of D. I am unable to comprehend the tutorial solution.

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11 months ago, # |
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Can anyone explain me div 2 D solution

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11 months ago, # |
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when coming up with dp solutions of taskE1 would you write the initial statements first(for(int i=1;i<=n*(n-1)/2;i++)s[0][i]=1;) or figure it out after the recursive statements? Because I had a hard time writing the initial statements,not knowing how many elements in the array should I initialize(Like why is s[0][0]=1 not needed). Please share some tips! and sorry for my bad English

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11 months ago, # |
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I'M THE ONLY ONE FST ON D!!!!!!!!!!!!! :(

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11 months ago, # |
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Let me introduce froggyzhang's idea of problem E in English. You can find the original edition here(in Chinese).

I recommend you read the first and second paragraph of the official tutorial of E2 and the official tutorial of E1 before reading the following part of this article.

Assume $$$p_{i+1}=a$$$ and $$$q_{i+1}=b$$$, $$$a,b$$$ are their rank in their permutations.

So $$$inv(p)> inv(q)\Leftrightarrow inv(p[i+2,\dots n])-inv(q[i+2\dots n]\ge b-a+1$$$. So we just want to find the number of the pairs of permutations $$$p,q$$$ of length $$$n-i-1$$$ and $$$inv(p)-inv(q)\ge b-a$$$.

Precalculate $$$f(i,j)$$$ the number of the pairs of permutations $$$p,q$$$ and $$$inv(p)-inv(q)=j$$$. You can insert the numbers to the permutation from small to big and you can do it in $$$O(n^4)$$$ or $$$O(n^5)$$$ by using brute force(which is also mentioned in the official tutorial of E2).

You will find inserting number $$$i$$$'s generating function is as follow:

$$$ \sum_{j=0}^{i-1}x^j\sum_{j=0}^{i-1}x^{-j}=\frac{x^i-1}{x-1}\cdot\frac{x^{-i}-1}{x^{-1}-1}=\frac{x^{i+1}+x^{-i+1}-2x}{(x-1)^2} $$$

About the above part we can write it in to DP functions($$$a\to b$$$ means we add the value of $$$a$$$ to $$$b$$$).

$$$ f[i-1][j]\to f[i][j+i+1]\\ f[i-1][j]\to f[i][j-i+1]\\ -2f[i-1][j]\to f[i][j+1]\\ $$$

And $$$\frac{1}{(x-1)^2}$$$ means we want to roll back twice. Just think what do you do when you $$$\times (x-1)^2$$$. So we can calculate $$$f$$$ in $$$O(n^3)$$$(both time and memory). We can only keep two layers of transition to optimize the memory to $$$O(n^2)$$$. And we can calculate suffix sum and the answer of the same prefix in $$$O(n^2)$$$.

You can read froggyzhang's code in his blog. You can find other details of coding in his code if you need.

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11 months ago, # |
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In the implementation of problem D, why is the additional if (i!=t) : f[i][j]=(f[i][j]+f[i-1][j])%mod there at the end of innermost for loop? Isn't the case where we skip the ith element already considered in each case?

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    11 months ago, # ^ |
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    Please read the definition of the dp statement again. It means the number of sequences for $$$x$$$ exit at last if we add option $$$t$$$ to the sequences. So that we needn't consider option $$$t$$$.

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11 months ago, # |
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Can someone explain to me how the formula $$$f(n,k)=\sum_{|i|<n} f(n-1,k-i)\times (n-|i|)$$$ (from the editorial for E2) works? I don't get which "first and second permutation" is the author talking about. Thanks.

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11 months ago, # |
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Can anyone explain problem D. I didn't understand the solution in editorial.

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11 months ago, # |
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This is my code for Div2 D

Spoiler

Someone please provide me any small testcase for debugging.

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11 months ago, # |
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Thank you! This is very helpful!

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10 months ago, # |
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In Question B it is clear that every number x which is in the set is of the form 'x mod b = a^k mod b' where k is a non negative integer but how have we proven that every number of such form will be in the set?

That is how we know that the condition 'x mod b = a^k mod' is necessary and sufficient and not just necessary?

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10 months ago, # |
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Hey I don't know why I am wrong answer on test 2 of Problem B. This is my submission: https://codeforces.com/contest/1542/submission/121246560 Can you tell me what I am wrong. Thanks you

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10 months ago, # |
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the D is pretty good

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10 months ago, # |
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the solution of D:

#include<iostream>
#include<string>
#include<algorithm>
#include<cstring>
#include<cstdio>
using namespace std;
typedef long long ll;
const int mod = 998244353;
const int maxn = 505;
int dp[maxn][maxn];
int num[maxn];

void ini()
{
	memset(dp, 0, sizeof(dp));
	dp[0][0] = 1;
}
int main()
{
	int n;
	char ope;
	ll ans = 0;
	scanf("%d", &n);
	for(int i = 1; i <= n; i++)
	{
		scanf(" %c", &ope);
		if(ope == '-') num[i] = -1;
		else  scanf("%d", &num[i]);
	}
	for(int i = 1; i <= n; i++)
	{
		if(num[i] == -1) continue;
		ini();
		for(int z = 1; z <= n; z++)
		{
			if(i == z)
			{
				for(int g = 0; g <= z; g++)
					dp[z][g] = dp[z-1][g];
			}
			else if(num[z] == -1) 
			{
				if(z < i) 
					dp[z][0] = (0ll + dp[z-1][0]*2 + dp[z-1][1])%mod;
				else 
					dp[z][0] = (0ll + dp[z-1][0]+dp[z-1][1])%mod;
				for(int g = 1; g <= z; g++)
				{
					dp[z][g] = (0ll + dp[z-1][g+1] + dp[z-1][g])%mod;
				}
					
			}
			else if(num[z] < num[i] || (num[z] == num[i] && z < i))
			{
				dp[z][0] = dp[z-1][0];
				for(int g = 1; g <= z; g++)
					dp[z][g] = (0ll + dp[z-1][g] + dp[z-1][g-1])%mod;
			}
			else
			{
				for(int g = 0; g <= z;g++)
					dp[z][g] = (2ll*dp[z-1][g])%mod;
			}
		}
		for(int g = 1; g <= n; g++)
			dp[n][g] = (dp[n][g] + dp[n][g-1])%mod;
		ans = (1ll*dp[n][n]*num[i] + ans)%mod;
	}
	printf("%lld", ans);
	return 0;
}
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10 months ago, # |
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这代码写的好称头哦

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10 months ago, # |
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In C it was possible calculate everything before sending solution as in my solution ,then only modulo by every long long int in vector :-)

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10 months ago, # |
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i submitted 123918769 for problem[problem:1542A]. This submission is giving run time error. Can anyone please tell my mistake. Thanks in advance.

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9 months ago, # |
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so I spent 4 hours understanding problem C :)

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9 months ago, # |
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Edit: Understood my mistake.

Can someone tell me what's wrong in my logic for B: I observed a pattern for f(n) over numbers. Till every 6th number f(n) is 2 3 2 3 2, and for every 6th number, if the divisor upon dividing the number by 6 if odd then f(n) is 4 otherwise 5. So if n = 12, f(n) = 2 + 3 + 2 + 3 + 2 + 4 + 2 + 3 + 2 + 3 + 2 + 5 = 33. This logic is not working for the pretest 1 (10^16). Can anyone tell what's wrong in this?

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    8 months ago, # ^ |
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    When n is large enough,f(x) can be 7、9 or larger for some x,and it’s hard to find the law.

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9 months ago, # |
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in tutorial for C, why ∑i>1i(⌊n/lcm(1,2,...,i−1)⌋−⌊n/lcm(1,2,...,i)⌋) is equal to ∑i≥1⌊n/lcm(1,2,...,i)⌋+n?

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8 months ago, # |
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Sorry,I'm not sure abount Problem D.If there are multiple smallest elements and then we execute an erasing operation,what will happen?Will we erase all the smallest elements or just erase one of them?

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    7 months ago, # ^ |
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    just erase one of them.

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      7 months ago, # ^ |
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      Thank you and I have accepted now.That's a great problem. :)

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4 months ago, # |
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In C code, why do we break if (G > n) ?

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3 months ago, # |
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Problem $$$E1$$$ is very nice. I have an alternative solution, which to me at least makes more sense.

In a lot of these problems where we have $$$p < q$$$ for some permutation/array $$$p, q$$$, we can break up $$$p, q$$$ into some interval $$$[1, \dots i]$$$ in which $$$p_j = q_j \forall j \in [1, i]$$$, $$$p_{i + 1} < q_{i + 1}$$$ and then an arbitrary permutation from $$$i + 2$$$ onwards. Perhaps my notation is a bit pedantic; maybe an example will help:

$$$p = 1 3 4 5 2$$$

$$$q = 1 3 2 4 5$$$

In this case, $$$p$$$ and $$$q$$$ share a common prefix of length $$$2$$$. Then, after that, we have that $$$4 > 2$$$, so we know that $$$p > q$$$. $$$[5, 2]$$$ and $$$[2, 5]$$$ can be respectively remapped to a new permutation ($$$[2, 5] \to [1, 2]$$$).

Okay, so let's assume that $$$p, q$$$ share some prefix of length $$$len$$$. Then, we know that $$$p_{len + 1} > q_{len + 1}$$$, so we can iterate over pairs $$$p_{len + 1} < q_{len + 1}$$$, which serve as a sort of distinguisher. If we let $$$dp[n][k]$$$ be the number of permutations of length $$$n$$$ with $$$k$$$ inversions, then we see that $$$ans[inv]$$$ — the number of pairs $$$p, q, p < q$$$ such that $$$\text{inv} (p) > \text{inv} (q)$$$ — can be written as:

$$$ans[inv] = ans[inv] + dp[N - len - 1][inv + q_{ind + 1} - p_{ind + 1}] \cdot \frac{N!}{(N - len)!}$$$

It remains to find $$$dp[n][k]$$$. The editorial has a good explanation on that, so I omit that detail.

Also needs some optimization since this is $$$\mathcal{O}(N^5)$$$. Can be optimizes if we iterate over $$$q_{ind + 1} - p_{ind + 1}$$$ instead.

Cheers!

EDIT: SAME IDEA CAN SOLVE E2 LET'S GOOOO: 145998673 First 2700 problem solved w/o editorial!!

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3 months ago, # |
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please explain problem B in simple way i'm not able to understand to editorial...

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2 months ago, # |
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In problem C,

I think there is an interesting pattern to observe...

Look...

For all Odd numbers,

x will always be 2.

For even numbers,

Please correct me If I am wrong...

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    2 months ago, # ^ |
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    This is wrong. For example, $$$f(60)=7$$$.

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      2 months ago, # ^ |
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      I didn't notice that:"

      Thank you so much. From now onwards, I will think about such cases...

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    12 days ago, # ^ |
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    I solved this by observing such patterns (probably bad idea, but this all I can come up)

    We can see 4 is alternating with 5, but the after the forth 4 is not 5 but 7 (at 60)

    Still, 4 occurs every 6 path minus every 12 path (which can be 5, or 7, or anyelse)

    Let's write g(x): times occurrences of a number x in length n.

    g(4) = n/6 - n/12

    Let's check g(x) for some other x.

    g(2) : something like n/2
    g(3) = n/2 - n/6
    g(4) = n/6 - n/12
    g(5) = n/12 - n/60
    g(6) = 0
    g(7) = n/60 - n/420
    g(8) = n/420 - n/840
    

    We can see a pattern here, but why g(6) = 0? also what's up with g(8)?

    Turns out g(x) = 0 for all x which has more than one prime factor. (in this case 6 = 2*3)

    And g(8)'s pattern is different because the pattern actually concerns with its prime factor (in this case 8's sole prime factor is 2).