It's time to submit (your ac solutions)!

Teamscode has had a history of setting problems where outputting the sample output gives AC on the first problem. Examples of past problems include "Meeting Minutes," "Best Waifu," "Are You Busy," etc.

Now, it is time to submit to this new first problem of the Spring $$$2024$$$ contest. Is outputting the sample output enough to get AC?

Arararagi has an array $$$a$$$ of length $$$n$$$ and two monkeys. The first monkey can take an index $$$i$$$ and set $$$a_i = a_i | x$$$ where $$$x$$$ is any nonnegative integer in one step. The second monkey can take an index $$$i$$$ and set $$$a_i = a_i \& x$$$ where $$$x$$$ is any nonnegative integer in one step. Remember $$$|$$$ is bitwise OR and $$$\&$$$ is bitwise AND.

Parallegi likes arrays when the bitwise AND of all its elements is equal to the bitwise OR of all its elements. Claragi will make two copies of his array and hand one to each monkey. He will ask each monkey to turn his array into one he likes by having them repeatedly apply their operation. Rararagi likes competition and will reward the monkey that uses fewer operations with a piece of candy (but in the case of a tie he will eat the candy and both monkeys will be sad). Help Doalagi predict which monkey will win.

Disclaimer: the only part about this statement that is real is the part where I lose against all real participants. — Alternet

lethan3 is organizing a lockout tournament!

lethan3 has a tournament of $$$N$$$ participants ($$$N$$$ is a power of 2). The tournament follows a specific match format: participant 1 plays against participant 2, participant 3 plays against participant 4, and so on. After the round, the losers step out of the tournament and the winners progress to the next round in the same order. In the second round, the winner of match 1 and the winner of match 2 play against each other, and similarly, the winners of matches 3 and 4 play against each other, etc. Rounds are played until there is only one winner left.

Alternet is participating in lethan3's lockout tournament. Unfortunately for him, he has no skill and is dominated by all of the participants. Fortunately, he has many friends (also in the tournament) who will help him win. Because Alternet knows the order in which participants will solve problems, he can instruct his friends to snipe them.

Formally:

There are three types of participants: real participants, Alternet's friends, and Alternet.

When two real participants compete, the one with a lower seed wins (The $$$i$$$th participant has seed $$$i$$$).

Alternet can teach each one of his friends individually to beat exactly one real participant (they will lose to all other real participants). Alternet's friends will let Alternet win against them, and when Alternet's friends compete against each other, Alternet can decide who wins.

Alternet loses against real participants and only wins against his friends.

Alternet really wants to win the tournament. So, he asks you to determine if it's possible to win the tournament.

In the Araragi household, there is a fridge that contains an infinite amount of Haagendaz numbered $$$1, 2, 3, \dots $$$. Koyomi and Tsukihi both love Haagendaz but have some level of guilt ($$$K$$$ for Koyomi and $$$T$$$ for Tsukihi) that dictates the maximum amount of Haagendaz they can eat in one sitting. Both Koyomi's and Tsukihi's guilt levels start at $$$1$$$ and will increase when they see the other person eating Haagendaz. In particular, whenever someone sees the other person eats $$$x$$$ Haagendaz in one sitting, they will increase their own guilt level by $$$x$$$. Tsukihi will eat the maximum number of Haagendaz her guilt level allows, the following day, Koyomi will eat the maximum number of Haagendaz his guilt level allows, then the next day Tsukihi will eat the maximum number of Haagendaz, etc.

Handle $$$n$$$ questions of the form $$$x$$$, where you must determine who will eat the $$$x$$$'th Haagendaz.

Jinshi loves his toys, and his two favorite toys are an array $$$a$$$ of length $$$n$$$ and a sequence of $$$m$$$ pairs of integers, the $$$i$$$'th being $$$x_i$$$ and $$$y_i$$$ $$$(1 \leq x_i, y_i \leq n)$$$. Whenever Jinshi is bored, he likes to try and sort $$$a$$$ with his $$$m$$$ pairs of integers. Jinshi will loop over the pairs in order from $$$1$$$ to $$$m$$$, and swap $$$a_{x_i}$$$ and $$$a_{y_i}$$$. He will be happy if after all $$$m$$$ swaps, $$$a$$$ becomes sorted in increasing order. Afterward, Jinshi will restore $$$a$$$ to what it was before all the swaps happened.

Maomao will mess with Jinshi over the following $$$q$$$ days, where on the $$$i$$$'th day, she will swap $$$a_{l_i}$$$ and $$$a_{r_i}$$$ $$$(1 \leq l_i, r_i \leq n)$$$. Without knowing Maomao has messed with his array, Jinshi will try to sort it every day after Maomao does the swap. Maomao will not restore $$$a$$$ after doing the swap. On each day, find out if Jinshi will successfully sort his array.

You are given a string $$$s$$$ of length $$$n$$$. Determine if all subsequences of $$$s$$$ that have length $$$k$$$ are unique.

A subsequence of $$$s$$$ is defined as a sequence that can be obtained from $$$s$$$ by deleting some elements (possibly none), without changing the order of the remaining elements.

For instance, the subsequences of $$$trust$$$ of length $$$4$$$ are $$$trus$$$, $$$trut$$$, $$$trst$$$, and $$$rust$$$, all of which are unique.

But when considering subsequences of $$$threes$$$ of length $$$4$$$. The subsequence $$$tres$$$ occurs twice ($$$\underline{t}h\underline{re}e\underline{s}$$$ and $$$\underline{t}h\underline{r}e\underline{es}$$$).

You received a shipment of $$$N$$$ pandas labelled with numbers $$$x_1, x_2, \ldots, x_N$$$ from Mars ($$$N \leq 10^5$$$, $$$1 \leq x_i \leq 10^9$$$). Additionally, you have a magical zookeeper who has a favorite number $$$K$$$ ($$$K \leq 10^9$$$). The zookeeper will do the following exactly once:

1. Pick two distinct pandas, say panda $$$i$$$ and panda $$$j$$$, and puts them to sleep.
2. Concatenate the numbers on the labels of panda $$$i$$$ and panda $$$j$$$ to form a new number $$$y$$$.
3. If $$$y$$$ is not divisible by $$$K$$$, then the zookeeper ships pandas $$$i$$$ and $$$j$$$ back to Mars.

Your task is to find the number of ordered pairs $$$(i, j)$$$ such that after performing the operation above, the zookeeper does not ship pandas $$$i$$$ and $$$j$$$ back to Mars.

Note: $$$i$$$ is not necessarily less than $$$j$$$. Also, the concatenation of $$$i$$$ and $$$j$$$ is different from the concatenation of $$$j$$$ and $$$i$$$.

A concatenation of numbers $$$x$$$ and $$$y$$$ is the number that is obtained by writing down numbers $$$x$$$ and $$$y$$$ one right after another without changing the order. For example, a concatenation of numbers $$$12$$$ and $$$3456$$$ is a number $$$123456$$$.

Korosensei has decided to hold a dance for Class E. Unfortunately, no one showed up, so he is forced to dance with himself. He has created two lines of $$$n$$$ afterimages each (where $$$n$$$ is even), the $$$i$$$'th in the first line having height $$$a_i$$$ and the $$$i$$$'th in the second line having height $$$b_i$$$. Korosensei will then have all his afterimages sing $$$k$$$ songs and dance to them.

Every time a song starts, for all $$$i$$$ from $$$1 \dots n$$$, the $$$i$$$'th afterimage in the first line will dance with the $$$i$$$'th afterimage in the second line. Every time a song ends, all the afterimages in the first line will cyclically shift to the right by $$$1$$$, and the $$$n$$$'th afterimage will go to the front of the line ($$$a_2 = a_1, a_3 = a_2 \dots a_{n} = a_{n-1}, a_1 = a_n)$$$. At the same time, all the afterimages in the second line will cyclically shift to the left by $$$1$$$, and the first afterimage will go to the back of the line ($$$b_1 = b_2, b_2 = b_3 \dots b_{n-1} = b_{n}, b_n = b_1)$$$.

Even though he can travel at Mach 20, even Korosensei has trouble dancing with someone taller (or shorter) than him. The awkwardness of a dance between two afterimages with heights $$$x$$$ and $$$y$$$ is $$$abs(x - y)$$$ where $$$abs$$$ denotes the absolute value function. For each afterimage (in both lines), determine the maximum awkwardness of any dance they are a part of. As you, the participant, cannot write at Mach 20 (but can probably add at Mach 20?), you are only required to find the total sum of awkwardness for each afterimage.

You are given $$$n$$$ numbers $$$a_1, a_2, \dots, a_n$$$. How many numbers between $$$l$$$ and $$$r$$$ can be expressed in the form $$$\displaystyle\sum_{i=1}^{n} (a_i \oplus x)$$$, where $$$x$$$ is a nonnegative integer, and $$$\oplus$$$ denotes the bitwise XOR operation?

The difference between the Easy and Hard versions is the constraints on $$$T$$$ and $$$R$$$.

Utena really likes the number $$$3$$$. Let $$$f(x)$$$ count the number of $$$3$$$'s in the base-$$$10$$$ representation of $$$x$$$.

Given two integers $$$l$$$ and $$$r$$$ where $$$l \leq r$$$, compute $$$g(l, r)$$$, the sum of $$$f(x)$$$ for all $$$x$$$ such that $$$l \leq x \leq r$$$ and $$$0 \equiv x \pmod{3}$$$ (note that $$$0 \equiv x \pmod{3}$$$ means that $$$x$$$ is divisible by $$$3$$$). That would have been the problem but Utena forgot what $$$l$$$ and $$$r$$$ were.

So given $$$L$$$ and $$$R$$$, compute the sum of $$$g(l, r)$$$ for $$$L \leq l \leq r \leq R$$$. Since this number may be very large, please find it modulo $$$998244353$$$.

Kyousuke is ordering sushi for Kirino. Sushi is free, but toppings cost money. The menu has $$$n$$$ toppings, each represented with a number $$$i$$$, where $$$i$$$ is a number from $$$1$$$ to $$$n$$$. Each topping has some price $$$a_i$$$. However, Kirino has a rule for each topping, where if Kyousuke orders some topping $$$i$$$ on the sushi he can't put some other topping $$$b_i$$$ on the sushi, because Kirino will slap him. What is the most expensive sushi Kyousuke can order that can satisfy Kirino?

Waymo and Brian are discussing the winter 2023 seasonsals a string $$$s$$$ and will have $$$q$$$ different conversations. In some conversation, Brian will start talking about $$$s_{l \dots r}$$$, but Waymo will attempt to gaslight him into thinking he was talking about $$$s_{l' \dots r'}$$$ where $$$s_{l \dots r}$$$ and $$$s_{l' \dots r'}$$$ differ by exactly 1 element. Waymo loves to gaslight but he doesn't know the string well enough, so he asks you to construct an $$$l'$$$ and $$$r'$$$ for each $$$l \ r$$$ Brian talks about. Note that $$$r' - l' + 1 = r - l + 1$$$.

Note that two strings $$$a$$$ and $$$b$$$ of the same length differ by exactly $$$1$$$ element if and only if there exists some index $$$i$$$ such that $$$a_i \neq b_i$$$ and for all $$$j \neq i$$$, $$$a_j = b_j$$$.