Output looks like Codeforces titles.

Codeforces rating -> title

The title and the fact that the input is represented by $$$c$$$ is a hint.

What well-known function has a fixed point at $$$-40$$$?

Celsius -> Fahrenheit

Sample 2 has output $$$28$$$, which is $$$\binom{8}{2}$$$.

$$$28$$$ is the number of nonempty substrings of sample $$$2$$$.

Number of distinct nonempty substrings.

We are looking for a word from the grids.

Exactly one word appears in all the grids with answer YES and does not appear in all the grids with answer NO.

Output YES iff "vapid" can be found in the grid (vertically, horizontally or diagonally).

The title is asking you to factor the sample outputs and spot a pattern.

Look for factors coming from well-known sequences.

The answer is Fib(n)^2*Catalan(n). Originally I planned to use Fib(n)*Catalan(n) but unfortunately it appears in OEIS for some reason.

The queries look like subrectangle queries. Play around with the samples. # contributes to the answer somehow. Answer is always even.

Compute the "perimeter" of the # region in the query subrectangle.

The input makes no sense and seems unrelated to the title. The title seems to be our only clue?

Notorious coincidence.

Google “Coloring Trees codeforces” (or something similar) will lead you to https://codeforces.net/problemset/problem/711/C

Input $$$1$$$ is $$$\frac{1}{6} \pmod{998244353}$$$.

We can guess that the output is a rational number mod $$$998244353$$$.

By brute forcing all fractions with numerator and denominator $$$\le 3000$$$ (sth that can run in $$$\tilde{O}(n^2)$$$) tells us that the answer to sample $$$2$$$ is $$$\frac{691}{2730}$$$.

Google $$$\frac{691}{2730}$$$.

Output the absolute value of the $$$(2n-2)$$$-th Bernoulli number (using the indexing from Wikipedia). For $$$n \le 15$$$ you can just hard code it. For $$$n \le 2 \cdot 10^5$$$, you can compute it using its EGF (Exponential Generating Function) in $$$O(n\log n)$$$.

While Eulerian paths and the input format might suggest this is a graph problem, the ? in the title suggests that we should think twice (deja vu from season 1?).

// is very weird. Why is it in the title?

It's a geometry problem.

Output seems to contain two pairs of triangles.

You can reinterpret the title as "Parallel Euler Lines".

Find two pairs of triangles from the set of points with parallel Euler lines.

You are playing a two-player game on a $$$n$$$ by $$$n$$$ grid where on each turn, you can color a $$$1 \times k$$$ or $$$k \times 1$$$ subrectangle. The game ends when all cells are colored, and you lose if you can’t make a move. From here, it is a normal CP problem.

Symmetric Strategy

You can always win as P1 when $$$n$$$ or $$$m$$$ is odd by starting with the center row/column and using a symmetric strategy. Similarly, you win as P2 if $$$n, m$$$ are even.

What does $$$a,b,c$$$ stand for in math?

Side lengths of triangle

Given $$$X$$$, find a triangle with integer side lengths with area $$$\sqrt{X}$$$.

The score here only depends on $$$|s|$$$. $$$|s| = 81$$$ for full points. You get higher partials if the length is a square, otherwise the partial credit roughly indicates how close you are to the nearest square number below. Trying strings of different lengths will eventually lead you to this.

You need to submit a string $$$s$$$ of perfect square length to get nonzero points in B or C. This suggests that we need to use the perfect square condition somehow.

Since the length is a square number, we can arrange the digits in a square grid (for length=81, 9x9 grid).

This criterion counts the number of rows and columns which contain all distinct digits and gives a higher score if you have more such rows/columns. It also gives a higher score for larger grids. To get full score, you need a $$$9 \times 9$$$ Latin square.

This criteria is a bit more complicated. Having a lot of composite numbers will tend to give you a higher score.

First, read each row and column as a (9-digit) number. Your score depends on the average number of (not necessarily distinct) prime factors is (higher if more prime factors). To get a full score, you need an average of $$$\ge \frac{100}{9}$$$.