If there are two consecutive zeroes, can you ever change either of them?

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Which animatronic should you choose with your flashlight?

How many animatronics are worth incrementing when there are $$$k$$$ flashes remaining?

Write an expression for the ending danger level in terms of the night length and the danger levels of the animatronics that get flashed.

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Solve the case of one drain first! You should aim for $$$O(n)$$$.

If you pick two drains, what's the shape of their overlap?

Brute force.

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There are tons of ways to do this problem, and chances are you found a different one than most other contestants! This problem is solvable in subquadratic time however, can you do it?

Also, the diagrams in the problem, as well as the ones in all editorials and other problems, are completely drawn natively in tikzpictures. Did you know you can cram 3-D plots onto 2-D to make textured filling?

Let's first think about path graphs. For some $$$k$$$, what are the $$$n$$$ for which an $$$n$$$-path wins for Cyndaquil?

Between every pair of Cyndaquil-winning nodes a distance of $$$k$$$ apart, all the nodes on the path between them also win. How can we find these efficiently?

Why did we ask for only vertex $$$v$$$, and not for all nodes?

Tree DP.

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Write some bounds for the $$$a_i$$$. What's the biggest/smallest each one can be?

Try to pick only numbers that aren't in the $$$a_i$$$, and $$$+\infty$$$.

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Read E1's hints first.

There are actually a very well-constrained set of working solutions. How many times do we pick each number not present in the $$$a_i$$$?

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Think about how you'd play in the case of only one color. Once you have that, how would you do two colors?

Each color has some sequence of clues that will get them to be played correctly. How do we coalesce them into one string of clues?

Some rank clues must be given for all cards to be played. If two consecutive forced rank clues are for ranks $$$a, b$$$, is it ever optimal to only rank clue some of the ones between $$$a, b$$$?

There is an $$$O(n^2)$$$ dp. How can we speed it up?

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As suggested by the editorial, this problem was originally about Squid Game! Until we made the round about the 2010s, that is.

There's actually a way to do this in $$$O(nm \sqrt{nm})$$$ or faster with flows! Can you represent the problem as a flow or matching instance?

Use Pigeonhole to first solve for the case of no needy cells.

How can you adjust your solution when there are needy cells?

The graph is planar. Recall the Euler characteristic formula.

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It is open to us whether the following cheese is actually provably correct:

• First attempt to fill in the cells (blind to whether they are special or not) in reading order, adding if it doesn't form a cycle.

• Second attempt to fill in the cells, but with special cells first, in reading order, adding if it doesn't form a cycle.

We conjecture that one of the two always passes.

Use 0-1 sorting to make the structure a little simpler. What's a concise way to visualize a min max function?

The min max function is a tree that alternates min/max across depths. Can you determine the type of the top operation?

You will need to try a good number of schemes, but try to find a point where you can split $$$f = \min(g, h)$$$ or similar. Then, divide and conquer will win.

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Read the H1 hints.

Go for $$$O(n \log n)$$$; how can we optimize the previous approach?

Determine the top node in $$$O(\log n)$$$ time.

Scan for your split point from both ends simultaneously.

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The music choice across H1, H2, H3 actually follows the level of the final castle in World 8 of New Super Mario Bros Wii.

Read H1 and H2 first.

What is $$$f(1, \dots, n)$$$?

If $$$t = f(1, \dots, n)$$$, then $$$x_t$$$ is the leaf at an alternating path from root (max goes right, min goes left.)

What is $$$f(x_1, \dots, x_t, +\infty, \dots, +\infty)$$$? How does it relate to the tree structure of $$$f(x_1, \dots, x_n)$$$?

You know the number of variables in each child of $$$f(x_1, \dots, x_t, +\infty, \dots, +\infty)$$$ and $$$f(-\infty, \dots, -\infty, x_t, \dots, x_n)$$$. Can you use these children to assemble $$$f(x_1, \dots, x_n)$$$?

Work from the inside (two nearest pieces to $$$x_t$$$) to the outside with two pointers. How many queries do we use? The recurrence may look quite terrifying, but...

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The solution due to awesomeguy856 and incidentally the one in contest found by ecnerwala actually uses only $$$2n$$$ queries!

A naive entropy bound gives a lower bound of $$$n / \log n$$$ queries; can we push this to linear so our upper bound of $$$O(n)$$$ queries becomes asymptotically tight?