Hello, Codeforces!

On wednesday, 10 june 2020 y. on 17:35 (UTC + 3) will take place End of the learning — Beginning of the tour contest (Div 3).

This is my first round (mashup, because for creating trainings I haven't got rating, because I need to stop writing contests on the phone) with my tasks. Competition will be held according to the rules of ICPC. Penalty for every wrong submission is 20 minutes. 20 minutes before contest table will be freezed.

You will be given 6 tasks and 2 hours. I wish you like it. I have high hopes for E task.

Ahmad Ahmadsm2005 Said, Ahmed fouad. Fouad, Matvey Asgio Kulinich and Maxim Xennon Karpuk helped me with testing tasks. Also thanks to Mike MikeMirzayanov Mirzayanov for systems Polygon and Codeforces!

I wish you luck!

I almost forgot about link:

https://codeforces.com/contestInvitation/4036ec99932a47484351d57a812de34c7a4fbb2c

Is the time UTC+9?

UTC+3

Isn't the penalty set to 20 minutes (in the contest)?

why?

I'm not saying that it should be, I'm saying that it

is, even though you mention the penalty is 10 minutes.For example, see my score: (if it's blurry, https://imgur.com/a/JpaNEYU)

My solve times are 1 + 3 + 5 + 7 + 13 + 18 = 47, meaning that my 3 penalties are each causing 20, not 10 as they should.

It may be. I will correct it in blog

How to solve F?

I will public solves later

egor.anisevich, is it okay if I post a brief editorial here for everyone (since submissions are public now)?

Okay, I will do it too) there would be 2 editorials)

Slight editorial (with author permission):

AThis is mostly about knowing how your language works. Division, however, gets picky. It's a good idea to

not use doublesbecause they have precision issues (read). Instead, you can use integer division, which automatically truncates (in C++, at least), possibly having to be careful with negatives in other languages.For the actual implementation, it seems like the test cases are designed so overflow won't happen and you won't have to mod with negative numbers, so the rest is simple.

My submission: 83308763

BConsider all numbers that have to be odd. Because numbers are nonnegative, each odd number must be worth at least $$$1$$$. Similarly, each even number must be worth at least $$$0$$$. So the minimum possible number you can create is $$$c_o$$$ — the number of required odd numbers. You can create any $$$s \geq c_o$$$ where $$$s$$$ and $$$c_o$$$ have the same parity (just add $$$2$$$ until you get there), and you can't create any $$$s$$$ of different parity. So the final conditions to check are that $$$s \geq c_o$$$ and $$$s \mod 2 = c_o \mod 2$$$.

My submission: 83309018

CStore any form of set (

`std::map`

in C++,`java.util.TreeSet`

in Java, a dictionary in Python, I don't know about other languages) that supports fast insertion and checking if an element is in the set. Then the first type of query is insertion, and the second type of query is checking if an element is in the set, which is perfect.My submission: 83309210

DThis is a direct application of segment trees. I... don't have more to say about this one.

My submission: 83309468

EThis is a problem with a simple solution, but one that may be hard to come up with.

First, we must note that this problem is exactly equivalent to finding the number of paths from $$$(1, 1)$$$ to $$$(i, j)$$$ in a rectangular grid, where you can only move down or right (read). Why is this so? It's given in the statement that each value at $$$(i, 1)$$$ and $$$(1, j)$$$ is $$$1$$$. Conveniently, there's always exactly $$$1$$$ path to each of these cells. For all other cells, to get to $$$(i, j)$$$ (in the path problem) there are two places you can come from — either $$$(i - 1, j)$$$ or $$$(i, j - 1)$$$. Thus the number of ways to get to $$$(i, j)$$$ is the sum of the number of ways to get to the two places you can come from.

This gives us a dynamic programming recurrence: $$$dp[i][j] = dp[i - 1][j] + dp[i][j - 1]$$$ where $$$dp[i][j]$$$ is the number of ways to get to $$$(i, j)$$$. Note that this is exactly the same as the one given in the problem.

Now that we've reduced the problem, let's attack the path problem a bit harder. Currently, our solution is $$$O(i \cdot j)$$$, which will certainly not pass. Let's think of the path from $$$(1, 1)$$$ to $$$(i, j)$$$ as a string of the moves we make:

`D`

for down and`R`

for right. This string must be of length $$$(i - 1) + (j - 1) = i + j - 2$$$, have exactly $$$i - 1$$$ of`D`

, and exactly $$$j - 1$$$ of`R`

. And our goal is now to count the number of different strings. A bit of combinatorics tells us that there are exactly $$$\dbinom{i + j - 2}{i - 1}$$$ or $$$\dbinom{i + j - 2}{j - 1}$$$ possible strings.Now we just have to compute the answers under the tight memory conditions. $$$\dbinom{i + j - 2}{i - 1}$$$ is equal to $$$\dfrac{(i + j - 2)!}{(i - 1)!(i + j - 2 - (i - 1))!} = \dfrac{(i + j - 2)!}{(i - 1)!(j - 1)!}$$$. Computing factorials and inverse factorials can be done iteratively with $$$O(1)$$$ space and $$$O(i + j)$$$ time (although $$$O((i + j)\log{(10^9 + 7)})$$$ time is simpler).

My submission: 83310114

FWe can maintain the probability of ending in each tunnel iteratively. Let $$$a_i$$$ be the given probabilities for each turn $$$i$$$ from the input, and $$$p_i$$$ be the probability of ending at tunnel $$$i$$$ (for $$$i$$$ from $$$1$$$ to $$$n + 1$$$). Then $$$p_1$$$ is just $$$a_1$$$ because we have exactly that chance of ending in that tunnel. Otherwise, we have a $$$100 - a_1$$$ chance of advancing to tunnel $$$2$$$. Then, at tunnel $$$2$$$ we have a $$$(100 - a_1)a_2$$$ chance of stopping in tunnel $$$2$$$, and a $$$(100 - a_1)(100 - a_2)$$$ chance of advancing to tunnel $$$3$$$. More generally, the probability of stopping at tunnel $$$i$$$ is $$$p_i = (100 - a_1)(100 - a_2) \dots (100 - a_{i - 1})a_i = a_i \cdot \prod\limits_{k = 1}^{i - 1}(100 - a_k)$$$, where $$$a_{n + 1}$$$ is just $$$1$$$ because we have to stop there.

The final answer is then just the expected value, which is $$$\sum\limits_{k = 1}^{n + 1} k \cdot p_k$$$. Be careful with output (use

`setprecision`

with a large number of digits in C++) to make sure the jury knows your answer is correct because small-number accuracy is important here.My submission: 83310636

Hello! How you did these opening triangles?

These are spoilers:

oh look another oneOk! Thank u very much!

You can public it in your blog, I will add link

I will do editorial on russian lenguage then

For F when you turn right you go to another tunnel.So for p1 shoudn't it be 100-a1 for ending in that tunnel and then advancing for 2nd tunnel you should have a1 chance for that ? Or I have misunderstood the question

The statement's not the clearest, but I interpreted "turn right" from

`On every turn person can turn right with probability a [i]`

as "end at that tunnel with probability $$$a_i$$$" and "go straight" as "continue to the next tunnel"Yes, you're right!

"the tunnel with n turns, which lead right, in other tunnels" This statement made me think you have to turn right to go to another tunnel

I won't do such mistakes later

Why did you took the path from (1,1) to (i,j) ?I mean what's the reason or intuition behind this.Thanks in advance

Unfortunately, there isn't much intuition. The best I can say is "I've seen this problem before, so I recognized it quickly."

Can you make it a blog?

I think you meant editorial here too instead of parse.

Yes

Done!

Some suggestions:

Try to use MathJax format. For example, something like

`1<=n<=1e5`

should be`$$$1 \le n \le 10^5$$$`

(use single dollar sign, but not 3 dollar signs) which will be displayed as $$$1 \le n \le 10^5$$$Try to spell key words correctly. (

guaranteedbut notgaranted)Hope you can make better contests next time :D

Thank u very much!