Batch-24 is hosting a "Pre-IUT Meetup" at Smashed Burgers. As part of the registration process, the organizers created registration forms with unique form IDs starting from $$$1$$$ and increasing sequentially.

When registration ended, the organizers found that the number of submissions greatly exceeded the actual number of students in their batch. After the investigation, they discovered that impostors from other batches had also filled out the forms. They examined a few impostor entries and were relieved to find a pattern. An entry is made by an impostor if for the form ID $$$x$$$, $$$x^2 - 1 \text{ is divisible by 24} \quad \text{i.e.,} \quad (x^2 - 1) \equiv 0 \pmod{24}.$$$

Your task is to count the total number of impostors among the first $$$N$$$ form submissions to maintain the Glory of 24!

You are given the first $$$m$$$ elements of an infinite sequence $$$a_1, a_2, \ldots, a_m$$$.

For every index $$$i \gt m$$$, the sequence is extended according to the following rule: $$$$$$ a_i = a_{i-2} \oplus a_{i-1}, $$$$$$ where $$$\oplus$$$ denotes the bitwise XOR operation.

Your task is to determine the value of $$$a_1 \oplus a_2 \oplus \cdots \oplus a_n$$$, that is, the bitwise XOR of the first $$$n$$$ elements of the sequence.

Zunaid is notoriously forgetful. To make his life easier, he always sets his computer password using a combination of his favorite digits: $$$0$$$ and $$$d$$$. He usually sticks to predictable, simple patterns—like a sequence of $$$d$$$s followed by some $$$0$$$s (e.g., $$$dddd0000$$$), or simple repeating arrangements that his forgetful mind can easily retrace (e.g., $$$dd00dd$$$ or $$$d0d0d0$$$).

His roommate, Imtiaz, considers himself a bit of a detective. He has spent months "studying" Zunaid's habits and has successfully cracked the password dozens of times. On a particularly chilly winter night, Zunaid decided he'd had enough. He set a new password, stood by the door, and smirked at Imtiaz.

"I'm heading over to CDS for a bit. I've set a password so strong that you can never deduce it. If you can crack it before I return, I'll treat you to an IUTian's Pizza tomorrow. If not, you owe me."

The inner detective in Imtiaz couldn't refuse. As soon as the door clicked shut, he pulled out a cheap UV flashlight. Shining it over the keypad, he saw deep, glowing marks only on the $$$0$$$ and $$$d$$$ keys, just as he had suspected. He noticed a hint on the lock screen: "Divisible by $$$n$$$." He also knew a password could consist of at most $$$n$$$ digits.

Luck wasn't on Imtiaz's side. Zunaid returned much earlier than expected. However, Imtiaz confidently claimed he had already deduced the password's logic and surely could have cracked it if he had been given five more minutes. Hearing his deductions, Zunaid laughed and replied, "Deducing the properties was the easy part! I'll still buy the pizza if you can actually give me any positive integer that follows the properties:"

• Consists only of digits $$$0$$$ and $$$d$$$.
• Is divisible by $$$n$$$.
• Has a length of at most $$$n$$$ digits.
• Does not contain any leading zero

Time is running out. Can you help Imtiaz get the free pizza?

There are $$$N$$$ players participating in a game involving a glass bridge. The bridge consists of $$$M$$$ steps, and each step has two glass panels: one is safe, and the other is fragile and will break if stepped upon.

The players cross the bridge one at a time, following their position in a queue from $$$1$$$ to $$$N$$$. The rules are as follows:

• Players must cross the bridge in the sequence of their queue positions ($$$1, 2, \dots, N$$$).
• Each step $$$i$$$ has one safe panel and one fragile panel.
• When a player's turn comes, they start from step $$$1$$$ and move forward.
• If a player reaches a step that was previously cleared by a predecessor, they simply step on the known safe panel and continue.
• If a player reaches a step $$$i$$$ that has not been successfully crossed yet:
  • They must choose one of the two panels.
  • If they choose the fragile panel, they are eliminated. Their turn ends, and the next player in the queue begins their attempt from step $$$1$$$, now knowing which panel at step $$$i$$$ is the safe one.
  • If they choose the safe panel, they proceed to step $$$i+1$$$.
• The game ends immediately when a player successfully lands on the safe panel of the $$$M$$$-th step. That player is the sole winner.

Every player is rational and will use all information revealed by previous players. If a player reaches a step where the safe panel is unknown, they will choose one of the two panels with a $$$50\%$$$ probability of success.

You have the chance to bribe the organizers to pick any starting position from $$$1$$$ to $$$N$$$. Your goal is to choose the position $$$k$$$ that maximizes your probability of being the sole winner. If multiple positions yield the same maximum probability, choose any such $$$k$$$.

It was the first AutoCAD lab for the first-year students at Irony University of Technology.

Labib and his friend Jehad were sitting side by side. The lab task was simple: draw a triangle using the given line measurements. Excited to finish the task quickly, they were racing each other instead of listening to the instructor.

Confident in their abilities, both of them ignored the tutorial completely and drew their own triangles much faster than the others.

A few minutes later, panic struck.

Neither of them remembered the actual measurements from the instructions, so they had no idea whether their drawings were correct. But instead of checking the lab sheet like normal students, they became curious about something much more interesting:

"By pure luck, did we draw similar triangles?!"

You are given the three side lengths of both triangles. Determine whether the two triangles are similar*.

*Two triangles with side lengths $$$(a, b, c)$$$ and $$$(d, e, f)$$$ are considered similar if there exists a permutation $$$(x, y, z)$$$ of $$$(d, e, f)$$$ such that

$$$$$$\displaystyle \dfrac{a}{x} = \dfrac{b}{y} = \dfrac{c}{z}$$$$$$

While doing group projects together in their university life, Meena and Kabir used to dream of opening a vintage-style book-cafe. Years later, their dream is finally coming true. The heart of their cafe is a Shelf, holding $$$n$$$ of their favorite books.

Kabir has just finished unboxing the books into an initial order $$$a$$$. Each book is identified by an ID number; since they have several copies of certain favorites, multiple books may share the same ID.

As always, Meena suggests a game: they win if they can reach their "dream" arrangement $$$b$$$—the one from their university notebooks—by only swapping books that are exactly $$$m$$$ slots apart. Formally, they can choose an index $$$i \le n-m$$$, and swap the books at index $$$i$$$ and $$$i+m$$$. As they sip their coffee and enjoy the quiet atmosphere of their new cafe, they wonder if the IDs on the shelf will allow them to win their game.

Moushi is a very serious competitive programmer. As ICPC is approaching, her coach assigns her a strict training rule: she must solve an even number of problems every day.

To stay on track, she writes down her target solve count for the next $$$T$$$ days $$$X_1, X_2, \dots, X_T$$$ and goes to sleep.

However, she wakes up to find that the digit 9 has completely vanished from the world, making the number system consisting only of digits $$$0$$$ to $$$8$$$. Because of this change, she can no longer decide whether a number is even or odd by simply checking its last digit.

To ensure that her target of the day complies with the coach's requirement, Moushi needs to re-evaluate all the numbers under this new base-$$$9$$$ numeral system.

Tariq needs to solve the following math problem for his homework:

He decided to consult with ChatGPT for help.

Figure: Conversation with ChatGPT. May 20, 2026, 10:07 AM.

However, Tariq does not fully trust ChatGPT's answer, so he came to you. Can you provide the correct answer?

Denji and Reze went to explore a school building at night. The building has $$$4$$$ rooms arranged in a $$$2 \times 2$$$ grid. Each room contains a light bulb, but the electrical system is malfunctioning.

Each room has a switch, but the switches are broken. Pressing the switch in a room does not affect the light in that room. Instead, it toggles the lights in the two adjacent rooms, namely the rooms directly above, below, to the left, or to the right. Toggling a light means changing its state from ON to OFF, or OFF to ON.

Reze asked Denji to turn all the lights OFF. Denji can press any switch any number of times and in any order. Given the initial state of the lights, determine whether it is possible for him to turn all the lights OFF at the same time.

The New World is a chaotic sea, but even the most notorious pirate crews have a rhythm to their madness. "Big News" Morgans, president of the World Economy News Paper, has discovered that every pirate crew follows a bizarre, endlessly repeating cycle of daily activities. He is utilizing this predictability to track their movements, anticipate clashes, and write his next front-page headline.

There are $$$K$$$ pirate crews currently being tracked. Each crew $$$i$$$ enters the New World on a specific starting date and follows an infinitely repeating activity cycle represented by a string $$$P_i$$$. The string consists of the characters S (Sailing), F (Fighting), P (Partying), and D (Docked).

For example, if a crew's pattern is SFFD, they will spend their first active day sailing, the next two days fighting, the fourth day docked, and on the fifth day, the cycle restarts with sailing. A crew is considered strictly inactive on any date prior to their starting date.

Morgans has a series of $$$Q$$$ queries to track the fleets. You must process three types of queries:

• Type 1 (1 DD/MM/YYYY C): Find all active crews engaged in a specific activity on a given date. Here, DD/MM/YYYY is the date to check, and C is a single character (S, F, P, or D) representing the target activity.
• Type 2 (2 $$$N_i$$$ MM YYYY): Calculate the total number of days a specific crew spent Sailing (S), Fighting (F), and Partying (P) during a given calendar month. Here, $$$N_i$$$ is the name of the crew, MM is the two-digit month (e.g., 01 to 12), and YYYY is the four-digit year.
• Type 3 (3 $$$N_1$$$ $$$N_2$$$ DD/MM/YYYY): Morgans anticipates a clash. Here, $$$N_1$$$ and $$$N_2$$$ are the names of two distinct crews, and DD/MM/YYYY is the starting date. Starting from this date, check a window of exactly $$$69$$$ days (the given date and the $$$68$$$ days following it). Find the first date within this window where both specified crews are active and Fighting (F) on the exact same day.

In an advanced material science lab, researchers are studying the crystal structure of alkali metals, such as sodium and potassium. These metals form a body-centered cubic (BCC) lattice, which is a repeating three-dimensional arrangement of atoms.

In a BCC lattice, each unit cell consists of one atom at the center of the cube and eight atoms at the corners of the cube. Each corner atom is shared by eight adjacent unit cells.

The domain of an atom is the region where its force strength is stronger than that of any other atom. In other words, the domain of an atom is the set of points that are closer to that atom than any other atom.

Your task is to calculate the volume of the domain of a center atom of a BCC lattice given its lattice spacing (side length of the unit cells) as a fraction $$$\dfrac{p}{q}$$$.