Codeforces Round #885 (Div.2) Editorial

#	User	Rating
1	ecnerwala	3649
2	Benq	3581
3	jiangly	3578
4	orzdevinwang	3570
5	Geothermal	3569
5	cnnfls_csy	3569
7	tourist	3565
8	maroonrk	3531
9	Radewoosh	3521
10	Um_nik	3482

#	User	Contrib.
1	maomao90	174
2	awoo	164
3	adamant	161
4	TheScrasse	159
5	nor	158
6	maroonrk	156
7	-is-this-fft-	152
8	SecondThread	147
9	orz	146
10	pajenegod	145

Tutorial

Solution

#include <bits/stdc++.h>
 
using namespace std;
 
#define int long long
 
int32_t main() {
    int t;
    cin >> t;
    while (t--) {
        int n, m, k;
        cin >> n >> m >> k;
        int x, y;
        cin >> x >> y;
        string ans = "YES\n";
        for (int i = 0; i < k; ++i) {
            int xx, yy;
            cin >> xx >> yy;
            if ((x + y) % 2 == (xx + yy) % 2) {
                ans = "NO\n";
            }
        }
        cout << ans;
    }
    return 0;
}

1848B - Vika and the Bridge

Idea: diskoteka

Tutorial

Solution

#include <bits/stdc++.h>
 
using namespace std;
 
int main() {
    int t;
    cin >> t;
    while (t--) {
        int n, k;
        cin >> n >> k;
        vector<int> c(n);
        for (int i = 0; i < n; ++i) {
            cin >> c[i];
        }
        vector<int> last(k, -1);
        vector<int> max_step(k), max2_step(k);
        for (int i = 0; i < n; ++i) {
            int step = i - last[c[i] - 1];
            if (step > max_step[c[i] - 1]) {
                max2_step[c[i] - 1] = max_step[c[i] - 1];
                max_step[c[i] - 1] = step;
            } else if (step > max2_step[c[i] - 1]) {
                max2_step[c[i] - 1] = step;
            }
            last[c[i] - 1] = i;
        }
        for (int i = 0; i < k; ++i) {
            int step = n - last[i];
            if (step > max_step[i]) {
                max2_step[i] = max_step[i];
                max_step[i] = step;
            } else if (step > max2_step[i]) {
                max2_step[i] = step;
            }
        }
        int ans = 1e9;
        for (int i = 0; i < k; ++i) {
            ans = min(ans, max((max_step[i] + 1) / 2, max2_step[i]));
        }
        cout << ans - 1 << "\n";
    }
    return 0;
}

1848C - Vika and Price Tags

Idea: pavlekn

Tutorial

Solution

#include <bits/stdc++.h>
 
using namespace std;
 
#define int long long
 
int gcd(int a, int b) {
    if (a == 0) {
        return 0;
    }
    if (b == 0) {
        return 1;
    }
    if (a >= b) {
        int r = a % b;
        int k = a / b;
        if (k % 2 == 1) {
            return gcd(b, r) + k + k / 2;
        } else {
            return gcd(r, b) + k + k / 2;
        }
    }
    return 1 + gcd(b, abs(a - b));
}
 
int calc(int a, int b) {
    if (a == 0) {
        return 0;
    }
    if (b == 0) {
        return 1;
    }
    return 1 + calc(b, abs(a - b));
}
 
int32_t main() {
    ios_base::sync_with_stdio(false);
    cin.tie(0);
    int t;
    cin >> t;
    for (int _ = 0; _ < t; ++_) {
        int n;
        cin >> n;
        vector<int> a(n);
        vector<int> b(n);
        for (int i = 0; i < n; ++i) {
            cin >> a[i];
        }
        for (int i = 0; i < n; ++i) {
            cin >> b[i];
        }
        set<int> cnt;
        for (int i = 0; i < n; ++i) {
            if (a[i] == 0 && b[i] == 0) {
                continue;
            }
            cnt.insert(gcd(a[i], b[i]) % 3);
        }
        if (cnt.size() <= 1) {
            cout << "YES" << endl;
        } else {
            cout << "NO" << endl;
        }
    }
    return 0;
}

1848D - Vika and Bonuses

Idea: diskoteka

Tutorial

Solution

#include <bits/stdc++.h>
 
using namespace std;
 
#define int long long
 
int f(int s, int k) {
    // (s + 20x) * (k - 4x)
    // (-80)x^2 + (20k - 4s)x + (sk)
    // -b/2a = (5k-s)/40
    int x = (5 * k - s) / 40;
    x = min(x, k / 4);
    int res = s * k;
    if (x > 0) {
        res = max(res, (s + 20 * x) * (k - 4 * x));
    }
    x = min(x + 1, k / 4);
    if (x > 0) {
        res = max(res, (s + 20 * x) * (k - 4 * x));
    }
    return res;
}
 
int32_t main() {
    int t;
    cin >> t;
    while (t--) {
        int s, k;
        cin >> s >> k;
        int ans = s * k;
        if (s % 10 == 5) {
            ans = max(ans, (s + 5) * (k - 1));
        } else if (s % 10) {
            if (s % 2 == 1) {
                s += s % 10;
                --k;
            }
            for (int i = 0; i < 4; ++i) {
                if (k > 0) {
                    ans = max(ans, f(s, k));
                }
                s += s % 10;
                --k;
            }
        }
        cout << ans << "\n";
    }
    return 0;
}

1848E - Vika and Stone Skipping

Idea: diskoteka

Tutorial

Solution

#include <bits/stdc++.h>
 
using namespace std;
 
const int K = 1e6;
 
#define int long long
 
int mod;
 
int dp[K + 1];
int cnt[K + 1];
 
const int MM = 3e6 + 7;
 
int inv[MM];
 
int cc = 0;
 
int mul(int a, int b) {
    int bb = b;
    while (bb % mod == 0) {
        ++cc;
        bb /= mod;
    }
    return (a * (bb % mod)) % mod;
}
 
int dv(int a, int b) {
    int bb = b;
    while (bb % mod == 0) {
        --cc;
        bb /= mod;
    }
    return (a * inv[bb % mod]) % mod;
}
 
 
int32_t main() {
    ios_base::sync_with_stdio(false);
    cin.tie(0);
    int x, q;
    cin >> x >> q >> mod;
    inv[1] = 1;
    for (int i = 2; i < MM && i < mod; i++) {
        inv[i] = mod - inv[mod % i] * (mod / i) % mod;
    }
    int y = x;
    for (int i = 3; i <= K; i += 2) {
        if (dp[i]) {
            continue;
        }
        for (int j = i; j <= K; j += 2 * i) {
            dp[j] = i;
        }
    }
    int ans = 1;
    while (x % 2 == 0) {
        x /= 2;
    }
    for (int i = 3; i <= K; i += 2) {
        while (x % i == 0) {
            ans = dv(ans, cnt[i] + 1);
            ++cnt[i];
            ans = mul(ans, cnt[i] + 1);
            x /= i;
        }
    }
    int f = 1;
    if (x > 1) {
        ans = mul(ans, 2);
    }
    x = y;
    for (int i = 0; i < q; ++i) {
        int d;
        cin >> d;
        while (d % 2 == 0) {
            d /= 2;
        }
        int k = d;
        while (dp[k]) {
            ans = dv(ans, cnt[dp[k]] + 1);
            ++cnt[dp[k]];
            ans = mul(ans, cnt[dp[k]] + 1);
            k /= dp[k];
        }
        if (cc) {
            cout << 0 << '\n';
        } else {
            cout << ans << '\n';
        }
    }
    return 0;
}

1848F - Vika and Wiki

Idea: pavlekn

Tutorial

Solution

#include <bits/stdc++.h>
 
using namespace std;
 
const int MAXN = (1 << 20);
 
int a[MAXN];
 
int solve(int n) {
    if (n == 1) {
        if (a[0] == 0) {
            return 0;
        } else {
            return 1;
        }
    }
    int fl = true;
    for (int i = 0; i < n / 2; ++i) {
        if (a[i] != a[i + n / 2]) {
            fl = false;
        }
    }
    if (fl) {
        return solve(n / 2);
    }
    for (int i = 0; i < n / 2; ++i) {
        a[i] ^= a[i + n / 2];
    }
    return n / 2 + solve(n / 2);
}
 
int32_t main() {
    int n;
    scanf("%d", &n);
    for (int i = 0; i < n; ++i) {
        scanf("%d", &a[i]);
    }
    cout << solve(n) << endl;
    return 0;
}

Comments (208)

Write comment?

bigSchrodinger

9 months ago, # |

+52

A,C,D and E are just math problems. Cf should do better.

→ Reply

yh11TRASH

8 months ago, # ^ |

instead of crying, maybe you should focus on getting good?

leftover_19

+106

Problem statements were too long.

Shonko006

9 months ago, # ^ |

Yeah , I fully agree

FAHEEM_KAHN

even weired

matryoshka03

look at As editorial :0

riverwalk7

+21

I agree with the general mood that A-C of the contest were too hard. D-F were okay, although in E $$$M$$$ should have been set larger than $$$O(Q \log x)$$$ to prevent the edge case where one of the prime powers reaches $$$M$$$.

tfg

+25

Why should setters prevent corner cases instead of contestants preventing them in their code?

← Rev. 5 →

This is a Div 2 codeforces contest. Because this is Div 2 where the contestants aren't necessarily that strong, and because this is codeforces where the pace of the competition is very fast, it should be more about solving the problem than implementation and avoiding corner cases.

Also, it looks like the authors made the same mistake, hence the explanation that Test Case 15 (the first case with that corner case) was incorrect and the solutions had to be rejudged.

+39

This is a Div 1.5 contest, so include purples in your thesis. "not strong div2 participants" wouldn't reach E anyway.

Btw my opinion on this problem is: why the fuck is it variable mod? Why not just use constant mod 1e9 + 7? It seems some coordinators or setters are addicted to using variable mod for no reason other than worsening the runtime of people's solutions.

I didn't know about test case 15, that slipping by testing is a big oof.

← Rev. 2 →

+34

There are problems where variable mod is needed, in particular the problems where there are limited amount of parameters and it's possible to locally compute all the possible answers with a slower code then submit the result.

I agree that for this problem there is no such reason and a constant 1e9+7 is better.

Spectre2017

Div2 contestant also should have rights to solve E with certain wisdom ,instead of getting cornered by this kind of modulo trick. In fact that’s almost irrelevant to the main idea of this problem, which i think is conducting the numbers of factors (forgive my terrible English)

LovelyStar

I think that because the setters want to avoid the solution using inverse modulo to recalculate the product (p/s: sorry for my bad English)

Dominater069

well, stupid edge cases are no fun (the original problem when i tested had a constant mod > 1e9)

also, the setters themsevles got the case wrong....

avnithv

I liked A and B, even though the latter was mostly implementation. C and D were too mathy and guessable though.

chetan__

+13

problem statement of A was pretty confusing for me.

the_knight1

same

pcumamahesh

I like your pfp, I am a veteran of SF3 as well!

ethico

same here :(

Lyrically

+15

There exists an $$$O(n)$$$ solution for problem C, which is better than the editorial's $$$O(n\log a_i)$$$:

Submission

xxballxx

Can you explain your solution, i read but i can't understand

PROELECTRO444

-15

More readable code for same logic.

https://codeforces.com/contest/1848/submission/214137912

I can't prove it, but it's correct:)

The first few steps are the same as editorial. On finding out how to calculate $$$(a,b)$$$'s answer, write a brute force first. Then u can notice the patterns the same in my code:P

If someone can prove it I'll be very grateful.

n.grechiha

Actually it the same as second solution of editorial: "Thus, we can determine the remainder cnt modulo 3 by looking at the pair (a/d, b/d) modulo 2".

Let a = 2**g * ra, b = 2**h * rb

ra, rb — odd numbers.

If g > h, then:

a / gcd(a, b) = 2**(g — h) * (ra / gcd(ra, rb)) — even number

b / gcd(a, b) = (rb / gcd(ra, rb)) — odd number

In case g < h, first will be odd, second even.

In case g == h, both are odd.

-31

This is O(n log(n)) Only to calculate the number of trailing zeroes it takes log(n) times only.

+10

Hi, __builtin_ctz() is an inbuilt function of G++ that only have $$$O(1)$$$ complexity as far as I know. Correct me if not:)

-23

https://codeforces.com/blog/entry/59268

Though it does the bitwise calculation, it is too fast to get noticed, but the complexity is still O(log(n)). You can verify this by manually writing the code for it, and you will see that it takes the same time.

Click on compare the previous solution in the below submission.

I can't really agree. It seems on my local machine that __builtin_ctz(x) is at least 30 times faster than the handwritten one. as the handwritten one has the complexity of $$$O(\log n)$$$, I would say that __builtin_ctz(x) has to be something like $$$O(1)$$$.

Here's my code and results:

Test 1:

__builtin_ctz() running 1e7 times, 0.02 s on average

#include<bits/stdc++.h>
using namespace std;
#define int long long
#define pii pair<int,int>
#define pb push_back
#define rep(i,n) for(int i=0;i<n;i++)
#define rep1(i,n) for(int i=1;i<=n;i++)
int read(){int x;scanf("%lld",&x);return x;}
void print(int x){printf("%lld\n",x);}
void file(string s)
{
	freopen((s+".in").c_str(),"r",stdin);
	freopen((s+".out").c_str(),"w",stdout);
}
const int mod=998244353;
int get(int x)
{
	return __builtin_ctz(x);
}
signed main()
{
	int lim=10000000;
	double st=clock();
	for(int i=1;i<=lim;i++)
	{
		get((1<<30));
	}
	double ed=clock();
	cout<<(ed-st)/CLOCKS_PER_SEC;
	return 0;
}

handwritten __builtin_ctz in your previous submission running 1e7 times, 0.6s on average

#include<bits/stdc++.h>
using namespace std;
#define int long long
#define pii pair<int,int>
#define pb push_back
#define rep(i,n) for(int i=0;i<n;i++)
#define rep1(i,n) for(int i=1;i<=n;i++)
int read(){int x;scanf("%lld",&x);return x;}
void print(int x){printf("%lld\n",x);}
void file(string s)
{
	freopen((s+".in").c_str(),"r",stdin);
	freopen((s+".out").c_str(),"w",stdout);
}
const int mod=998244353;
int get(int x)
{
	int cnt=0;
	while((x&1)==0)
	{
		cnt++;
		x>>=1;
	}
	return cnt;
}
signed main()
{
	int lim=10000000;
	double st=clock();
	for(int i=1;i<=lim;i++)
	{
		get((1<<30));
	}
	double ed=clock();
	cout<<(ed-st)/CLOCKS_PER_SEC;
	return 0;
}

Test2:

change the lim in the first code to $$$10^8$$$, and it runs 0.2 s on average; change the lim in the second one to $$$10^8$$$, and it runs 6.1 s on average.

What do u think about that?

upd: i did another test, i changed the get(x) to:

int get(int x)
{
    return 2*x;
}

and $$$lim$$$ is still $$$10^8$$$. the average time is 0.24s! That means the inbuilt function of g++ seems to be even faster than adding two numbers together! What's ur opinion?

A5HU705H

theoretically, it is still n log n but practically it is O(n)

idk how __builtin_ctz() is implemented,but i still think its real complexity should be real close to $$$O(n)$$$.

+16

Godbolt says there's special "bsf" assembler instruction for it.

ShaoNianTongXue5307

-22

Stop posting misleading content you are discussing popcount but ctz only needs x&-x

and popcount isn't O(log n) too it's O(log w) like O(log log n).

kgb_70

how did you arrive at the intuition of using 2 highest power in solving it.

+18

I'm stupid, I don't have intuition, I write brute force, I observe patterns, I draw conclusions, I implement, I pass

hai hai sensei!

AdityaTakkar

What was the logic in your brute force code that showed some pattern?

FOX-index

+27

This contest problem statements are too long like a reading comprehension.

I think C is more difficult than D

May be I'm too foolish , but what is the tutorial of C's meaning???

Plums

+11

Personally I don't think problem A is too hard for its position. The idea is fairly common in game theory puzzles like https://www.youtube.com/watch?v=dh4nEuhZBgg and just requires a bit of good intuition.

Ritwin

+69

diskoteka [user:pavlovn]

My thoughts on this round:

The problems themselves were fine.

I do think A was not suitable to be a div2A, but that’s because I think div2A’s should be “submitbait”. The problem itself was fine, and may have done better as div2B.

Perhaps D was too math-y than ideal, but that’s fine.

Perhaps E could’ve had $$$M \geq 10^8$$$ or have been fixed, but it wasn’t much extra (and it is not too hard to generalize to all $$$M$$$). It is bad that the sequence is on OEIS though, so ideally this problem shouldn’t have been used.

Also, don’t listen to the haters, and please keep setting contests. Over time you’ll get better, and this contest probably provided good feedback!

diskoteka

+28

ty very much

alemontc7

Make more contests!!

shreyas_v

True! A problem based on chess with mathematical proof indeed shows that Diskoteka is a smart person. Also with experience, they will get great at problem setting too!

pranjaljha

could you link some resource or some other questions like A which use similar logic.

jayantjha1109

-20

C was a beautiful problem!

-9

It's a good thing beauty is subjective.

swayingdream

+32

Your tutorial for problem D should write (b+20x)(a-4x), not (b+20x)(a-x) in the 4th paragraph, as reflected in your code.

+35

Easier for F is just doing binary lifting as soon as you know the power of 2 thing. It's basically the same solution but O(NlogN) and you don't have to think about reducing it to an array of size N/2.

russell_anu24

For problem A, vika's friends can see her(vika) right? So they see her and try to modify their movement right? Can vika see her friends as well?

d1amondzz

If we color the rooms like a chessboard, and at least one of her friends starts on the same color as her, that friend has the strategy, and can react to her movement(because she moves first) in such a way that no matter what she does, she eventually will be caught, this is explained in the editorial. So, whether Vika tries to avoid her friends or not, it doesn't matter.

Isa_m

please explain me this(Problem A):

1 2 1

1 1

1 2

I think when vita goes to right , friend will catch him . Perhaps, i misunderstood something.

0YassineMouadden0

both vika and her friend should perform a move .So they're just going to swap rooms forever and never meet each other.

a friend catches vika only if they find themselves in the same room after performing the move

During contest, I understood from statement so:

First vika goes, then her friend.

Thanks

← Rev. 3 →

It is my dreadful carelessness,thanks.

visheshgautam.official

yeah they will be in the same room no doubt but her friend has to make move thats why being on the same room doesnt matter as long as that friend has perform her move

i_love_fumeShroom

-103

This is the WORSR contest i've ever seen. Can you just stop writing such long description of problems?? If you want to show your ability of writing, you don't need codeforces. And problems DEF are so easy to think, the difficulty is too low to be the div2 DEF, it can be used in div3. The quality of the problems are also too low I think. And the awful test cases of problem E, can you just check your test cases before the contest? To my surprise, this contest has not been unrated, but I don't want to see such contest like this anymore. YOU DO NOT NEED TO CREATE THE CONTEST IF YOU DO NOT HAVE THE ABILITY OR YOU DO NOT WANT TO DO IT WELL!!!

when problem A has the longest tutorial, you know you did something wrong.

arda

wthforces

SuperJ6

I don't get why so much hate for this round. A is better than normal, I liked problem C quite a bit, and rest of problems were fine. (yet another round with no datastructure tho ;-;)

You could've used a segment tree in order to avoid the corner case in E lol

I wonder why they even used prime mod. Either used fixed or force this....

The5threich

Problem — A was quite lengthy from a beginner perspective and difficult to prove.
Otherwise the rest of the contest (I mean A-D, I haven't read E and F) was fine.
(I struggled a little bit to decipher the language in A (and B for that matter) The story of friends catching "Vika" but having a greater affinity to make a move seems inconsistent and looks like a bad attempt to make the problem look real worldish when it isn't).

I agree A could've been worded better, but I don't think it's bad enough to greatly worsen round.

CallmeSourav

Sorry ,but I feel A lengthy and language is very tricky

akash456

In solution1 of problem C , it will be ai-4*bi in place of ai-4ai

_simpleThing_

can someone explain me the problem statement of B in easy way.. i have read it multiple times but i am not able to understand

thanks..

destroyer_sam

vika can paint any particular plank of her choice. you have to figure out which one so that she has to step over minimum number of plank

condition is that she will be jumping on only same color planks.

got it... thanks :)

i_ujjwal

Problem A was way too unbalanced for its position and difficulty range. Its not a great idea for begineers

in task B, isnt it necessary to mention in the problem statement that k distinct colourst will definitely be present.

for problem A, i figured that if there are odd number of cells between Vika and friend, they will meet.

And, to calculate the number of cells between them, i thought of using this formula:

d = |vx-fx|+|vy-fy|.

for ex, if this value d is 1, means they are in adjacent cells(or there are no cells between them), so they wont ever meet

prabhanshubhowal

No, for d=0 it means they are in the same cell (You have taken absolute value) . For d=1, it means adjacent cells

yep! Thanks for pointing

bhatte2.0

My solution for A :
So we need to check for every position (i,j) in n*m grid, if there is a position where both main person and anyone among the k people have the same distance to (i,j). So we iterate over every position and check whether the distance of main person is same as distance of any of k friends to that position. TC : O(n*m*k).
AC code : 214108505

I used the same implementation.

papa-ka-para

After reading editorial for problem A, do you still feel like this is DIV-2 A problem ?

Only in Vika-land, problem would need this much logical thinking and proof for problem A solution.

I think, by considering it as a problem that could be solved by brute force thus being easy to implement, it's a good Div 2A.

teraqqq

Because of problems like A, people start to hate math in cf-contests. The problem would be great if it were at the math contest, but not at cf.

FlyingElephant

Exactly, and also if you make A like this people leave the contest, so the round receive less partecipation as well.

Noluck_167

If someone is making a video tutorial on A and B please share it

Osama_Fahim

have you found a good video on it ?

not until now

lookcook

+57

Personally, I loved this round! I only solved ABCD, but this round had better problems than most recent Div2s in my opinion. I don't understand why the announcement blog post was downvoted. Keep making rounds!

kunalc1

ive done other coding contests but this was my first cf contest and i solved 0 questions, so skill issue for me ig

notkai

as you can tell from the malding in the comments this contest was harder than usual, esp A

n1k1450

auto itr = Map.rbegin();
++itr;

to get access of previous element of itr, do ++itr instead of --itr! due to this, I was not able to submit problem B :(

_MASTER__

can any one explain problem D editorial please? I noticed the cycles before reading the editorial, but the formula??

tankman890

← Rev. 4 →

In problem C, I have a different way of finding the $$$cnt_i\% 3$$$ mentioned in the editorial.

Similar to solution 2, we will build the sequence from the end. It will be of the following form:

$$$... x, x/2, x/2, 0 ...$$$

Now try to build a tree of all possible values before $$$x$$$. You will notice values in a particular level are either of the type $$$odd * x / 2$$$ or $$$k * x$$$ where $$$k$$$ is a positive integer. You will also notice that $$$k * x$$$ has a period of $$$3$$$, same as the period of $$$0$$$.

So the problem reduces to finding which of the $$$a_i, b_i, c_i$$$ is of the type $$$k * x$$$.

Here's my submission.

i am still not able to understand your approach, can you explain in more simple way. thanks.

Think of the first time $$$a_i$$$ becomes zero. It will be of the following form:

$$$...k, k, 0...$$$

What would come before $$$k$$$? Let it be $$$y$$$, solving $$$|y - k| = k$$$ we would get $$$y = 2*k$$$. So the sequence now becomes

$$$... 2*k, k, k, 0 ...$$$

Similarly, try to find what would come before $$$2*k$$$. From here, multiple values might be possible for each position. For example, before $$$2*k$$$, both $$$3*k$$$ and $$$k$$$ are possible values.

Try to draw a tree of these values. $$$2*k$$$ will be the root and $$$3*k$$$ and $$$k$$$ are its children. It will be something like this: https://binary-tree-builder.netlify.app/?indextree=2*k,3*k,k,5*k,k,3*k,null,8*k,2*k,4*k,null,4*k,2*k.

In this tree, at each level, the values are either of the type $$$even*k$$$ or $$$odd*k$$$. You will also notice $$$even*k$$$ has a period of $$$3$$$, just like the period of $$$0$$$.

Now consider a situation where $$$a_1 = even*k$$$ and $$$a_2 = odd*k$$$. Then we know that both $$$a_1$$$ and $$$a_2$$$ won't become $$$0$$$ at the same time.

So we need all $$$a_i$$$ to be $$$even*k$$$ or all $$$b_i$$$ to be $$$even*k$$$ or all $$$c_i$$$ to be $$$even*k$$$. Only then we will get all zeroes at the same time.

The best explanation I found so far. But why (all ai or all bi or all ci) need to even*k. why not just check if all ci are even*k only? why we have to check ai and bi too?

We might have something like this:

$$$a = [odd*k_1, even*k_2, odd*k_3, ...]$$$ $$$b = [even*k_1, odd*k_2, even*k_3, ...]$$$ $$$c = [odd*k_1, odd*k_2, odd*k_3, ...]$$$

As you can see, this will not result in all zeroes occurring at the same time.

Now consider the following:

$$$a = [odd*k_1, odd*k_2, odd*k_3, ...]$$$ $$$b = [even*k_1, even*k_2, even*k_3, ...]$$$ $$$c = [odd*k_1, odd*k_2, odd*k_3, ...]$$$

This will result in all zeroes occurring at the same time. So its important that you check all $a_i$, $$$b_i$$$, $$$c_i$$$.

Koyote

You're my savior

Thanks a lot !!

Avik_AB17

-8

Real Div.2 contest<:

The_Evening_Nightingale

What a round!

arnavwinner

diskoteka, in the editorial of C, while writing Solution 1: I believe where you wrote

(ai, bi, ai — bi, ai — 2 * bi, ai — 3 * bi, ai - 4 * ai)

(and should have written: ai - 4 * bi)

Six_Seconds

Have anyone solved Problem-A using BFS without getting TLE ? looking at the constraints, I thought it was a graph related problem but getting TLE on pretest 3.

Siddharth_78

Same I too tried to implement a bfs/dfs/backtracking soln, but the actual soln is game/mathematical based!

Hamloo

Thank you diskoteka for great math problems, better than previous div(1,2) speedforces

DottedCalculator

+36

E and F are well known math problems.

E: 2016 AMC 12B P16

F: Trivial special case of 2023 USA TSTST P9

IAmTheHazard

It has been pointed out to me that the setup for F is actually identical to USA TSTST 2023/9 with $$$p^e=2$$$, although I’m not sure how much knowing the solution to the TSTST problem helps with the codeforces and vice versa. At the very least, one of the solutions to the TSTST problem proves that you will never print $$$-1$$$ (see the first claim in post #5 by a familiar-looking user).

Edit: sniped by the person who pointed this out lol

The first part of the TSTST problem tells you what the operation becomes after applying it $$$2^i$$$ times.

Oh true—the explicit computation of the $$$k$$$-th finite difference implies that for $$$k=2^i$$$, we have $$$\Delta^{2^i} f(n) \equiv f(n+2^i)+f(n) \pmod{2}$$$, since by Kummer's theorem/Legendre $$$\binom{2^i}{a}$$$ is even unless $$$a \in {0,2^i}$$$, which more or less reduces this problem to a straightforward binary search

Tom66

Great round! Like these problems very much :3

We've found some alt ways to solve F, but they have higher time complexity.

luanmenglei

A is too hard...

lory1608

I think problem A is harder than problem B.

-18

Different Approach For Problem C. Code

Adunama

Can you explain how it works, like why does the highest power of 2 matters?

Arjun_Nair_7

In the first question(Vika and her friends)

How does the area decrease when vika moves in the opposite direction and the friend moves towards her, shouldn't it remain the same??

On a long enough time-frame, vika will reach the border & cannot move in the opposite direction & will be forced to move in a direction that reduces the distance between her & her friends, the catch to the problem is that this cat-mouse game happens infinitely!

virapaksa

Problem provider 114514

CaveNightingale

Then, if the cycle is "scroll" x times, the discount will be (b+20x)(a-x)

WHY NOT (b+20x)(a-4x)?

ashutosh_saini

In C, I tried to find the number of operations to make ai zero manually in the hope of getting some pattern or some formula but all in vain. May be they should try not to give such mathematical things till C.

C is one of those problems which gives neither the pleasure when I solve them nor guilt when I could not.

Feng_D

The background of the problem is just too long, and it's really frustrating! I completely understand your pain. Why do they have to pile up so much unnecessary information? It could have been expressed concisely and clearly, but instead, they make us read a whole paragraph just to understand the problem. It's such a waste of our precious time and energy! And sometimes, the background information has absolutely nothing to do with solving the problem. It's like a test of our patience and endurance. The problem should be about solving the actual problem, not reading a background story! I really wish the problems could be more direct and concise, so that we can focus more on solving the challenging tasks at hand!

Chymera

Problem A statement was so confusing to me, also IMHO this can't be a DIV2 A. All candidate masters and above in the comments trying to justify why it's not bad, of course you don't mind it because if you do it is a problem. Go justify Div1 A and leave Div2

syxwz

problem A is the most difficult ever. I can solve B,C, but A is a impossible task for me.

Thus we call a hall with coordinates (c,d) a neighbouring for it if |a−c|+|b−d|=1. Why this statement is even in the question it confused me a lot , I thought that in few test cases Vika canthave an adjacent neighbouring room so she cannot move anywhere

sanidhyas3s

Can someone break down C for me, I observed a few things during the contest.

Let a>b for all below expressions, if not we can just move to the next stage where b and |a-b| will fit the scenario(moving one parity ahead).

Every pair will eventually reach a loop of GCD(a,b), GCD(a,b), 0. So essentially we needed to check if every pair have the same parity (%3 values of the number of steps they reach 0). I also noted that we can go from a,b to a%b,b keeping the same parity, except for when (a/b ==1). And that is where it stopped for me.

I saw some people use a%2b to move a parity ahead, but where does that 2 come from?

Fanarill

I'll use pair notion meaning $$$(a, b)$$$ pair becomes $$$(b, |a - b|)$$$.

Consider some large $$$y$$$ and small $$$x$$$: $$$(x, y) \Rightarrow (y, y - x) \Rightarrow (y - x, x) \Rightarrow (x, y - 2x)$$$.

We made 3 operations meaning it wont change amount operations modulo 3, therefore you need to find largest $$$k$$$ i.e. amount of operations that: $$$0 <= y - k(2x) < 2x$$$ which is literally a definition of remainder.

Thanks! I understood the pattern. During the contest I just mistook (I was trying to guess it rather than calculate it down ;_;) the pattern to be repeating with (a,b) -> (a%b,b) which is wrong I guess, which is why it will fail for floor(a/b) = 1.

FO7

can u prove doing this way + perform the said operation will not be more than logarithmic

I don't understand why do you need to prove this fact since it's literally gcd-like + you literally can implement it and test on big random numbers but here you are:

Since operation in pairs are not simmetrical you can't assume WLOG $$$x \leq y$$$ so I'll let you do $$$x > y$$$ by yourself ;)

So I'll be doing $$$x \leq y$$$ case:

Now consider two cases:

$$$4x \geq y$$$, then pair $$$(x, y)$$$ becomes $$$(y, y - x)$$$ where $$$y - x \leq \frac{3y}{4}$$$, $$$(y, y - x) \Rightarrow (y - x, x) \Rightarrow (x, y - 2x)$$$. Congrats! we have reduced second term by at least 3 quarters, since $$$y - x \leq \frac{3y}{4} \Rightarrow |y - 2x| \leq \frac{3y}{4}$$$
$$$4x \leq y$$$, then $$$(x, y)$$$ becomes $$$(x, y \ \% (2x))$$$. Congrats! we reduced second term by at least half

This yields next conclusion: rough estimate for upper bound of steps is $$$3*log_{4/3}(y)$$$ which is good enough. The rest of the proof is up to you

Kohinoor

Typo in d editorial:

$$$(b+20x)*(a-4x)$$$

magnesium12

I found a very interesting pattern for problem C. For every i, we count number of trailing zeroes in binary representation of A[i] and B[i], now if A[i] has more, then this pair can be made 'dull' in 3k steps, if B[i] has more, then 3k+1 steps, and if they both have equal trailing zeroes, then 3k+2 steps. And we just have to ignore A[i]=B[i]=0 case.

something like this — code

practis44

I think my solution is slightly similar in intuition to problem A, but is much simpler.

For any friend to be able to reach Vika, the Manhattan distance between them must be even. So, what I mean is, if the distance between Vika and all friends are odd, then they'll never reach her, but if any one distance is even, that implies that the friend will be able to reach Vika at least at some point of time in future.

I'm not exactly sure yet how to explain or prove this, but I feel it can be better understood if one can just draw out some testcases and check.

In case the claim I made is incorrect, do provide your suggestions, I'll be grateful :)

Here is my submission

etahull

Can someone provide me with clear insights on Problem C? Or provide resources to learn the topic. It is clear to me that the only thing that matters is the modulo 3 of the first occurrence of zero for each position of the array. But I find the tutorial very confusing on how to proceed further (maybe I am missing some number theory / modular arithmetic prerequisites)?

Consider several cases of a[i] and b[i]. Such as:

Both are odd. [O,O]
- Then the first difference ($$$O-O = E$$$) will be an even number. [O,E]
- second difference ($$$O-E = O$$$) will be an odd number. [E,O]
- third difference will be an odd number. [O,O]
- Now, we know that the sum of the numbers keeps reducing and one of them reaches $$$0$$$ eventually.
- So at what parity can a[i] = 0 occur from above observations?
- Ans: At all "second differences" , because zero is even
Similarly analyse cases of [O,E] and [E,O]
The case of [E,E] is interesting. You can observe that they will mimic the behave of the numbers we obtain by doing successive division by 2.
- Eg: [4,6] will mimic the behaviour of [2,3].
- So case 3 can be collapsed into cases 2 and 1.

You know the story after this.

sleeeep

Hi every one in problem B i didn't use k in solving problem

rigvedinho

Can someone pls tell me what is wrong in this solution for C? 214173357

Nvm silly mistake figured it out, but now weirdly enough this submission is giving a RTE for some reason, can someone tell me why? 214174585

At this point, I don't even want to read the problem. It looks like a marathon.

animish01

Did everyone lose rating? Half of the people must have gained acc to algo

paddakartheek

I didn't understand the solution explained for problem C. Can someone explain it clearly.

Zuneve

me too

kronikle

LC Better

Who can explain the tasks C, D? I didn't understand the author's decision at all

1xx55

problem C:

Hint1

Hint2

how to prove complexity of urs code is log

It's similiar to the complexity of GCD and that's log.

But that we can prove .Here (a,b) where a>=2*b or b>=2*a remains in the same state for more times.Its not exactly same .from this state to say them same is not reasonable

Assume that a>=2*b.(if 2*b>a>=b , My code do the same thing as GCD)

GCD : (a,b) → (b,a%b) , 1 step makes the biggest num decrease from a to b.

Mycode :(a,b) → (a%(2*b),b).

I. 0< a%(2*b) < b. 1 step too , same as GCD.

II. b<= a%(2*b) <2*b . Then we go the second step: (a%(2*b),b) → (b,a%(2*b)-b). 2 steps makes the biggest num decrease from a to b.

so Mycode's complexity is no more than twice of GCD's . That's log.

← Rev. 8 →

(a,b)

three cases :

1) a>=2*b

2) 2*b>a>=b

3) a<b

fourth thing each state will be one of three

first case u proved in 1 or 2 steps will be same as (b,a%b)

second case u also proved in 1 step can get to (b,a%b)

third case can be reduced to (b,b-a) one of case 1 or 2(can take 3 steps)

a %b==a-b only if 2*b>a>=b

VERY WELL DONE

here is my implementation of the same approch (not even a single change ) https://codeforces.com/contest/1848/submission/214252557

-16

if wanted its complexity can also be O(2*logn) or O(3*logn) -complexity at the heart

thank you very much!

Flvx

Hey man, how were you specifically able to come up with the a >= 2 * b condition. Can you please explain your thought process behind it?

hzy99999

The tutorial is clear.

A.Hazem_2003

any one solve b by using Binary search ??

LextyleO_o

-10

WTF you haven't said that sum of Ks over all test cases is at most 200000 in B!

mym0404

It is unnessesary. k <= n is stated

GusFring

Is the kind of reasoning used in problem E familiar to high-rated coders? Looking at the tutorial, I don't see any way I could have solved the problem on my own, so I want to know if people who solved it are well-versed with such kinds of reasoning or if it came to them naturally.

C probelm solution

Can someone please explain what does area to Vika mean? Is it something other that the number of blocks between Vika and one of her friends?

__Timu__

Although i was unable to solve any problem during the contest, But i enjoyed a lot. Specially, problem A. Moreover, i felt a bit uneasy to read the statements.

SAT2020

Simpler Solution for A:

Every move, the parity of R+C changes by 1 for both Vika and her friends. Therefore, a friend can catch up to Vika if and only if the sums of their coordinates have the same parity.
Because the friend moves after Vika, she can always decrease the distance between the two of them. By contrast, Vika can only increase distance by moving in the same direction as the gap between her and her friend (e.g. if Vika is three units to the right of her friend, she can only decrease horizontal distance by moving to the right). Because the map is finite, Vika will be forced to decrease the distance between her and her friend after a maximum of R+C moves, and so she will eventually be captured.

Nice .In both dimensions the distance cannot remain same .just follow vika steps .he will be caugth

__hermit__

Problem E Observation: # of odd divisors = # of distinct initial f can also be proved in the following manner:

Consider all the pairs of [l,r] such that $$${\sum_{i = l}^r i = x}$$$
We are sure the the lengths len of all these segments have to distinct
So we need to find what all valid lengths are there. We have the following formula

$$$x = len \cdot ((len+1) / 2) + len \cdot k$$$

Consider one of the case where len is even. The case when len is odd is similar to prove.
let len = 2p and p>=1

$$$x \equiv len \cdot ((len+1) / 2) \pmod{ len }$$$ $$$x \equiv p \cdot (2 \cdot p + 1) \pmod{ 2 \cdot p }$$$ $$$x \equiv ((2 \cdot p) \cdot p) + p \pmod{ 2 \cdot p }$$$ $$$x \equiv p \pmod{ 2 \cdot p }$$$ $$$x = (2 \cdot t + 1) \cdot p $$$

Codertang

why is question b tagged binary search

where did the formula (b+20x)(a-x) in D come from?

In any case, except for 5 and 0, the last digits will start following the pattern of 2,4,6,8 after a certain point. so we can make the formula as for not taking a discount 4 times we will have an increase in b by 20(2+4+6+8) and a decrease in the number of purchases left by 4.

if we do this process x times then the discount i can get is (b+20*x)*(a-4x).

and the formula in the editorial is partially wrong(missed 4*x).

bhaarat123

I have created video editorial for problem A,B and C in english link. If you understand it, upvote.

BlueGlaucus1

Nice! TY. I actually understood A from this video.

Thanks :-)

Mayank_Codes

Why is this giving TLE in problem C? Submission ID:https://codeforces.com/contest/1848/submission/214346070 But this is not. Submission ID: https://codeforces.com/contest/1848/submission/214346500 I am not able to figure out the cause.

C_LiHua

Why do we need to compare x and x+1? x = (5*k-s)/40; x = min(x,k/4); x = min(x+1,k/4); Isn't the first x already the largest？

ImplementingForces

Can anyone please give me a more intuitive idea of the induction of problem F? I mean how can we see it happen beside trying to prove the equation?

sevlll777

Try to solve problem Xor pyramid. This induction thing in F is just a special case $$$(n = 2^k)$$$ of Xor pyramid.

Spoiler to solution

Error_Yuan

Maybe there is something wrong with problem A statement. What will happen when $$$n=1,m=1$$$?

In this case none of the girls can move to the neighbouring rooms, which violates your statement.

diskoteka Could you please, explain this case. Seems that the answer of MCS is No.

Irmuun.Ch

Finding $$$cnt_i\%3$$$ in problem C is not that hard I think.

$$$[a,b]$$$ will change like this $$$[odd,odd] \to [odd,even] \to [even,odd] \to [odd,odd] \to ... $$$

And $$$a$$$ can be $$$0$$$ in only $$$[even,odd]$$$.

(of course if both number is $$$0$$$ a is always $$$0$$$ and otherwise we can divide numbers by $$$gcd(a,b)$$$ so we can remove $$$[even,even]$$$ case)

Akshay184

Can anyone share there solution for D using ternary search. Mine is failing. My solution for D

memoryallocator

My ternary search solution for D (in Rust).

Also, the link you provided leads to a solution for C, not D.

denilb

Solution written in python for problem D, using ternary search.

In Problem D why are we only subtracting 1 from k after every iteration, shouldn't we take into account the fact that we are performing 4 * x number of operations and adjust k accordingly?

yes typo in editorial

z-force

cf should do better

BF_OF_Priety

A can be done through brute force too , just go to each cell , if distance from vika's starting cell == distance from any of her friend's cell , then vika can't escape

Fundrush

I didn't quite understand the line Can someone explain it :

If Vika and the friend are in the same column, then the area to Vika is equal to the sum of the areas of all rows in the direction of Vika and the row where the friend is located.

Am I the only one not getting it ?

tanishq2507

in problem C ,We will build the sequence from the end. Let's find the first moment when we obtain 0 . Before this zero, there is some number d . It can be easily proven that d is exactly equal to gcd(ai,bi).

How to prove it?

HyPraT_

great contest, do consider making more contests please. [user:diskoteka][user:pavlekn]

nonrice

I spent quite a long time upsolving C in coming up with the formulas to accelerate the procedure when $$$a>b$$$... I have no knowledge of gcd algorithms... how can it make the problem easier?

LazyBong104

How to approach solve B by binary search??

Bouttouwa

the friends of vika try to checkmate her xD.

Sarthak_78

Has anyone solved question B using binary search? If so, could you please share the solution? My solution getting wrong on test 2. My solution for B

aleke

I have a solution using binary search by answer https://codeforces.com/contest/1848/submission/214106278

ccpp

The solution of C is poorly written. "Then, the sequence of numbers will have the form: (ai,bi,ai−bi,ai−2⋅bi,bi,ai−3⋅bi,ai−4⋅bi,bi,…) ." is wrong take ai = 5, bi = 3: sequence will be :5,3,2,1,1,0 clearly bi(3) is not appearing again and again as written int tutorial.

shiv_prime94

Anyone who implemented the binary search solution of B. My Binary Search solution is failing on test case 2. Can someone help me out with this?

for each color, we save their coordinates including the last n in the array and then check the difference between them if it is greater than the middle, we paint this board once, if we have already painted the board, then we continue to look for this one in another color or in the answer

This is not my solution I saw from another user.

ahsoltan

Alternative solution to F using SOS DP:

Notice that the value of $$$a_0$$$ after $$$k$$$ operations is the xor of all $$$a_i$$$ such that $$$\binom{k}{i}$$$ is odd.

Why?

Lucas's theorem states that $$$\binom{p}{q}$$$ is odd if and only if $$$q$$$ is a submask of $$$p$$$. Thus, the value of $$$a_0$$$ after $$$k$$$ operations is the xor of all $$$a_i$$$ such that $$$i$$$ is a submask of $$$k$$$. This can be computed in $$$O(n \log n)$$$ using SOS DP. We know that after $$$n$$$ operations all $$$a_i$$$ become zero, so the answer is the first moment that $$$a_0$$$ becomes zero and stays zero.

umbral

Yeah this was the solution I found (admittedly I didn't realize its truth was due to Lucas's Theorem). Let to a very beautiful implementation :)

zhyh

F: Could the answer be -1? Why?

Rushdi

6 months ago, # ^ |

3 2 2 0

at this testcase isn't it -1?

Krzychuo

I'm wondering how is one suppossed to come up with a proof such as one presented in tutorial E (or even think of proving the idea that number of odd divisors affects the answer)? Is that even Div2E level? The ideas used in the proof such as considering the parity of $$$cnt$$$ seem so random to me. I'd be grateful if any math gods would explain to me how to think about such problems or if you would show me more intuitive proof than the one in tutorial.

diskoteka could you suggest any learning resource or problems that could help me solve problems similar to problem A in this contest.

karandh

8 months ago, # |

Could anyone please explain why in Problem C it is necessary to do x=(x%2*y), what is it essentially doing to find the first occurence of 0;

ism_hatti

for the problem C what sum decreases there i don t understand it really.

ComboGirl

6 months ago, # |

Here is another way to think about the operation in Problem F which I like.

Since $$$\oplus$$$ is the same as addition $$$\pmod{2},$$$ let us consider the operation: $$$a_1, a_2, \ldots, a_n \to a_1 + a_2, a_2 + a_3, \ldots, a_{n} + a_1$$$ where addition is taken modulo $$$2.$$$ Indices mod $$$n.$$$

Let us iterate it and see what happens: After two iterations we get $$$a_1 + 2 a_2 + a_3, \ldots, a_{n} + 2 a_1 + a_2.$$$ After three iterations, we have $$$a_1 + 3 a_2 + 3 a_3 + a_4, \ldots, a_{n} + 3 a_1 + 3 a_2 + a_3.$$$

It looks like after $$$j$$$ iterations, the sequence looks like

$$$\binom{j}{0} a_i + \binom{j}{1} a_{i+1} + \cdots + \binom{j}{j} a_{i+j}$$$

Indeed, you can prove this using induction with pascal's identity. So what we really are looking at is Pascal's triangle mod 2, ie. Sierpinski triangle.

Now, we have nice properties of $$$\binom{n}{m} \pmod{2},$$$ such as $$$\binom{2^k}{m}$$$ is even for all $$$1 \leq m \leq 2^k-1,$$$ and $$$\binom{2^k - 1}{m}$$$ is odd for all $$$m.$$$ You can prove them using Lucas' theorem.

It follows that after $$$2^k$$$ iterations, the terms are $$$a_i + a_{i + 2^k}$$$ mod 2. So since $$$n = 2^k$$$ is a power of two we know that after $$$n$$$ iterations they all become zero, $$$a_i + a_{i + n} = 2 a_i = 0.$$$ I initially just showed it for $$$2^{k+1} - 1$$$ iterations, which is also not hard to see since all the binomial coefficients are odd.

It's also interesting to note that this is a linear map $$$f : (\mathbb{F}_2)^n \to (\mathbb{F}_2).$$$ Additionally, if you write out the matrix of the linear map, $$$M = \begin{pmatrix} 1 & 1 & 0 \ldots & 0 & 0 \newline 0 & 1 & 1 \ldots & 0 & 0 \newline & \vdots & \ddots \newline 0 & 0 & 0 \ldots & 1 & 1 \newline 1 & 0 & 0 \ldots & 0 & 1 \end{pmatrix}$$$

it is a Circulant Matrix so one may write out the eigenvalues explicitly and consider the restriction to the field $$$\mathbb{F}_2.$$$ This also makes it easy to compute the powers of the matrix. $$$\newline$$$ $$$\newline$$$ Finally, here's an nice mathematics problem which is (somehow) closely related to this problem; a friend shared this actually right around mid July, same week as this CF round.

Let $$$\varphi : \mathbb{Z}^n \to \mathbb{Z}^n$$$ be a function mapping on tuples of integers such that:

$$$\varphi(x_1, \ldots, x_n) = (|x_1 - x_2|, |x_2 - x_3|, \ldots, |x_n - x_1|)$$$

For some inputs, repeatedly iterating eventually yields the zero vector $$$(0,0,\ldots,0).$$$ ie. there exists some $$$k$$$ for which $$$\phi^k(x_1,\ldots,x_n) = \phi(\phi(\cdots\phi(x_1,\ldots,x_n) \cdots )) = (0,\ldots,0).$$$

For which positive integers $$$n$$$ is it true that repeatedly iterating $$$\phi$$$ eventually yields the zero vector, no matter what the input is?

B_Consistent

can anyone tell what bug is in this code I tried using shadow clone jutsu on umnik submission but CF is finding the original one here

variety-jones

Take a look at Ticket 17141 from CF Stress for a counter example.

AC Submission with a one line change.

can anyone tell what bug is in this code I tried finding bug using sharigan jutsu on my submission but I am unable to find one here

Take a look at Ticket 17143 from CF Stress for a counter example.

can anyone tell what bug is in this code I tried summoning jutsu on variety-jones to help find bug on my submission but I am unable to summon him here

Take a look at Ticket 17145 from CF Stress for a counter example.

I was using mind transformation jutsu into problem and placed a wrong link.can you raise a Ticket for this?

Ianlsam

nvm

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