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problems/MA-06/Atcoder/abc189_e/editorial.md

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+# E - Rotate and Flip (editorial)
+
+Source: https://atcoder.jp/contests/abc189/editorial/590 by en_translator
+
+Each operation is an affine transformation. For example, a symmetric transformation about the line \(x = p\) can be represented as follows:
+
+\( \begin{pmatrix} -1 & 0 & 2p \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \\ 1 \end{pmatrix}= \begin{pmatrix} 2p-x \\ y \\ 1 \end{pmatrix} \)
+
+Therefore, one can calculate the cumulative product of transformation matrices in order to answer each query in an \(O(1)\) time. Since the precalculation of cumulative products takes \(O(M)\) time, the total time complexity is \(O(N+M+Q)\).
+
+More naively, it is sufficient to hold the information of “where (0,0), (1,0) and (1,1) goes after each operation,” in order to find the resulting coordinates of any point after some transformations in a \(O(1)\) time. By actually simulating the operations for those three points beforehand, the total time complexity will be \(O(N+M+Q)\).
+
+![](https://img.atcoder.jp/ghi/68d5f1c6ff95270a3d2f24db7264f4dc.png)

problems/MA-06/Atcoder/abc189_e/judge.json

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+{
+  "time_limit_s": 2.0,
+  "time_limit_source": "task page",
+  "memory_mb": 1024,
+  "memory_source": "task page",
+  "mode": "exact",
+  "validator_flags": "case_sensitive",
+  "comparison": "token-wise, whitespace-insensitive, case-sensitive (AtCoder's checker)",
+  "validator": null,
+  "tests_dir": "data",
+  "answer_suffix": ".out",
+  "n_sample": null,
+  "n_secret": 28,
+  "note": "no sample/secret split: the statement's samples are among the numbered tests"
+}

problems/MA-06/Atcoder/abc189_e/meta.json

@@ -0,0 +1,16 @@
+{
+  "task_id": "abc189_e",
+  "contest": "abc189",
+  "type": "ABC",
+  "letter": "E",
+  "title": "Rotate and Flip",
+  "url": "https://atcoder.jp/contests/abc189/tasks/abc189_e",
+  "difficulty": 1526,
+  "score": 500,
+  "time_limit_sec": 2.0,
+  "memory_limit_mb": 1024,
+  "n_tests": 28,
+  "editorial_url": "https://atcoder.jp/contests/abc189/editorial/590",
+  "code_source": "model",
+  "pid": "A1511"
+}

problems/MA-06/Atcoder/abc189_e/solution.cpp

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+// abc189_e: E - Rotate and Flip
+// Written by a model (gpt-6-astra) from the official editorial, https://atcoder.jp/contests/abc189/editorial/590
+// Judged AC on all 28 tests, max 0.14 s of the 2 s limit (submission 1).
+
+#include <bits/stdc++.h>
+using namespace std;
+
+using ll = long long;
+
+struct Transform {
+    array<ll, 3> x;
+    array<ll, 3> y;
+};
+
+int main() {
+    ios::sync_with_stdio(false);
+    cin.tie(nullptr);
+
+    int N;
+    cin >> N;
+    vector<pair<ll, ll>> points(N + 1);
+    for (int i = 1; i <= N; ++i) {
+        cin >> points[i].first >> points[i].second;
+    }
+
+    int M;
+    cin >> M;
+    vector<Transform> pref(M + 1);
+    pref[0].x = {1, 0, 0};
+    pref[0].y = {0, 1, 0};
+
+    for (int i = 1; i <= M; ++i) {
+        int type;
+        cin >> type;
+        Transform t = pref[i - 1];
+        if (type == 1) {
+            swap(t.x, t.y);
+            for (ll &v : t.y) v = -v;
+        } else if (type == 2) {
+            swap(t.x, t.y);
+            for (ll &v : t.x) v = -v;
+        } else if (type == 3) {
+            ll p;
+            cin >> p;
+            for (ll &v : t.x) v = -v;
+            t.x[2] += 2 * p;
+        } else {
+            ll p;
+            cin >> p;
+            for (ll &v : t.y) v = -v;
+            t.y[2] += 2 * p;
+        }
+        pref[i] = t;
+    }
+
+    int Q;
+    cin >> Q;
+    while (Q--) {
+        int A, B;
+        cin >> A >> B;
+        auto [x, y] = points[B];
+        const Transform &t = pref[A];
+        ll nx = t.x[0] * x + t.x[1] * y + t.x[2];
+        ll ny = t.y[0] * x + t.y[1] * y + t.y[2];
+        cout << nx << ' ' << ny << '\n';
+    }
+    return 0;
+}

problems/MA-06/Atcoder/abc189_e/statement.md

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+# E - Rotate and Flip
+
+**Score**: 500 points
+
+## Problem Statement
+
+There are \(N\) pieces on a two-dimensional plane. The coordinates of Piece \(i\) are \((X_i,Y_i)\). There may be multiple pieces at the same coordinates.
+
+We will do \(M\) operations \(\mathrm{op}_1, \ldots, \mathrm{op}_M\), one by one. There are four kinds of operations, described below along with their formats in input.
+- `1`: Rotate every piece \(90\) degrees clockwise about the origin;
+- `2`: Rotate every piece \(90\) degrees counterclockwise about the origin;
+- `3 p`: Move each piece to the point symmetric to it about the line \(x=p\);
+- `4 p`: Move each piece to the point symmetric to it about the line \(y=p\).
+
+You are given \(Q\) queries. In the \(i\)-th query, given two integers \(A_i\) and \(B_i\), print the coordinates of Piece \(B_i\) just after the \(A_i\)-th operation. Here, the moment just before the \(1\)-st operation is considered to be the moment just after "the \(0\)-th operation".
+
+## Constraints
+- All values in input are integers.
+- \(1 \leq N \leq 2 \times 10^5\)
+- \(1 \leq M \leq 2 \times 10^5\)
+- \(1 \leq Q \leq 2 \times 10^5\)
+- \(-10^9 \leq X_i,Y_i \leq 10^9\)
+- \(\mathrm{op}_i\) is in the format of one of the four kinds of operations.
+- In an operation with the form `3 p` or `4 p`, \(-10^9 \leq p \leq 10^9\).
+- \(0 \leq A_i \leq M\)
+- \(1 \leq B_i \leq N\)
+
+## Input
+Input is given from Standard Input in the following format:
+```
+N
+X_1 Y_1
+...
+X_N Y_N
+M
+op_1
+...
+op_M
+Q
+A_1 B_1
+...
+A_Q B_Q
+```
+
+## Output
+Print the response to each query in its own line: the \(x\)- and \(y\)-coordinates, in this order, with space in between.
+
+## Sample Input 1
+```
+1
+1 2
+4
+1
+3 3
+2
+4 2
+5
+0 1
+1 1
+2 1
+3 1
+4 1
+```
+
+## Sample Output 1
+```
+1 2
+2 -1
+4 -1
+1 4
+1 0
+```
+Initially, the only piece - Piece \(1\) - is at \((1, 2)\). Each operation moves the piece as follows: \((1,2) \to (2,-1) \to (4,-1) \to (1,4) \to (1,0)\).
+
+## Sample Input 2
+```
+2
+1000000000 0
+0 1000000000
+4
+3 -1000000000
+4 -1000000000
+3 1000000000
+4 1000000000
+2
+4 1
+4 2
+```
+
+## Sample Output 2
+```
+5000000000 4000000000
+4000000000 5000000000
+```

problems/MA-06/Atcoder/abc196_f/editorial.md

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+# F - Substring 2 (editorial)
+
+Source: https://atcoder.jp/contests/abc196/editorial/952 by en_translator
+
+This problem is equivalent to the problem that asks to find the minimum hamming distance between \(T\) and a substring of \(S\) of length \(|T|\).
+ Accordingly, the answer is \(\displaystyle \min_{0 ≤ i ≤ |S| - |T|} \sum_{j=1}^{|T|}(s_{i + j} \oplus t_j)\), where \(s_i\) denotes the \(i\)-th letter of \(S\), \(t_i\) is the \(i\)-th letter of \(T\), and \(\oplus\) is the XOR operator.
+
+Let us consider calculating \(\displaystyle \sum_{j=1}^{|T|}(s_{i + j} \oplus t_j)\) for all \(i\).
+ With a naive way, the time complexity will be \(\Theta(|S||T|)\), which will lead to TLE. Observe that the expression looks like a convolution. By reversing \(T\), we have \(\displaystyle \sum_{j=1}^{|T|}(s_{i + j} \oplus t_{|T|-1-j})\).
+
+Hereinafter we assume that \(T\) is reversed.
+ Let \(C = (c_2, c_3, \dots, c_{|S|+|T|})\) be the convolution of \(S\) and \(T\) by the equation:
+
+\[ c_i = \sum_{i=j+k}s_j \oplus t_k, \]
+
+then \(c_{i+|T|-1} = \displaystyle \sum_{j=1}^{|T|}(s_{i + j} \oplus t_{|T|-1-j})\), so it is sufficient to find this convolution.
+
+If there is multiplication instead of XOR, we can apply Fast Fourier Transform (FFT) to find the convolution, so let us decompose XOR to multiplications.
+
+\[ c_i = \sum_{i=j+k}s_j(1 - t_k) + \sum_{i=j+k}(1 - s_j)t_k \]
+
+FFT enables to find \(C\) in a total of \(O((|S| + |T|) \log(|S| + |T|))\), so the problem has been solved.
+
+For implementation, we can use convolution in [AC-Library](https://github.com/atcoder/ac-library/tree/master/atcoder).

problems/MA-06/Atcoder/abc196_f/judge.json

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+{
+  "time_limit_s": 3.0,
+  "time_limit_source": "task page",
+  "memory_mb": 1024,
+  "memory_source": "task page",
+  "mode": "exact",
+  "validator_flags": "case_sensitive",
+  "comparison": "token-wise, whitespace-insensitive, case-sensitive (AtCoder's checker)",
+  "validator": null,
+  "tests_dir": "data",
+  "answer_suffix": ".out",
+  "n_sample": null,
+  "n_secret": 38,
+  "note": "no sample/secret split: the statement's samples are among the numbered tests"
+}

problems/MA-06/Atcoder/abc196_f/meta.json

@@ -0,0 +1,16 @@
+{
+  "task_id": "abc196_f",
+  "contest": "abc196",
+  "type": "ABC",
+  "letter": "F",
+  "title": "Substring 2",
+  "url": "https://atcoder.jp/contests/abc196/tasks/abc196_f",
+  "difficulty": 2274,
+  "score": 600,
+  "time_limit_sec": 3.0,
+  "memory_limit_mb": 1024,
+  "n_tests": 38,
+  "editorial_url": "https://atcoder.jp/contests/abc196/editorial/952",
+  "code_source": "editorial",
+  "pid": "A1554"
+}

problems/MA-06/Atcoder/abc196_f/solution.cpp

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+// abc196_f: F - Substring 2
+// Code from the official editorial, https://atcoder.jp/contests/abc196/editorial/952 (by en_translator)
+// Judged AC on all 38 tests, max 0.32 s of the 3 s limit.
+
+#include <iostream>
+#include <vector>
+#include <atcoder/convolution>
+#include <atcoder/modint>
+using namespace std;
+using Modint = atcoder::modint998244353;
+const int INF = numeric_limits<int>::max() / 2;
+void chmin(int& a, int b){ if(a > b) a = b; }
+
+int main(){
+    string S, T;
+    cin >> S >> T;
+    reverse(T.begin(), T.end());
+    const int N = S.size(), M = T.size();
+    vector<Modint> s1(N), s2(N), t1(N), t2(N);
+    for(int i = 0; i < N; i++) (S[i] == '0' ? s1 : s2)[i] = 1;
+    for(int i = 0; i < M; i++) (T[i] == '0' ? t1 : t2)[i] = 1;
+    s1 = atcoder::convolution(s1, t2);
+    s2 = atcoder::convolution(s2, t1);
+    int ans = INF;
+    for(int i = M - 1; i < N; i++) chmin(ans, s1[i].val() + s2[i].val());
+    cout << ans << endl;
+}

problems/MA-06/Atcoder/abc196_f/statement.md

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+# F - Substring 2
+
+**Score :** 600 points
+
+## Problem Statement
+Given are strings \( S \) and \( T \) consisting of `0` and `1`.  
+We will change some of the characters in \( T \) so that \( T \) becomes a substring of \( S \).  
+How many characters do we need to change at least?
+
+<details>
+<summary>What is a substring?</summary>
+
+\( T \) is said to be a substring of \( S \) when some contiguous part of \( S \) matches \( T \).  
+For example, `000` is a substring of `10001`, while `11` is not.
+
+</details>
+
+## Constraints
+- Each of \( S \) and \( T \) consists of `0` and `1`.
+- \( 1 ≤ |T| ≤ |S| ≤ 10^6 \)
+
+---
+
+## Input
+Input is given from Standard Input in the following format:
+```
+S
+T
+```
+
+## Output
+Print the answer.
+
+---
+
+## Sample Input 1
+```
+0001
+101
+```
+
+## Sample Output 1
+```
+1
+```
+Changing \( T \) to `001` makes it match the \( 2 \)-nd through \( 4 \)-th characters of \( S \).
+
+---
+
+## Sample Input 2
+```
+0101010
+1010101
+```
+
+## Sample Output 2
+```
+7
+```
+
+---
+
+## Sample Input 3
+```
+10101000010011011110
+0010011111
+```
+
+## Sample Output 3
+```
+1
+```

problems/MA-06/Atcoder/abc208_f/editorial.md

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+# F - Cumulative Sum (editorial)
+
+Source: https://atcoder.jp/contests/abc208/editorial/2219 by en_translator
+
+1. Preface
+2. Observation
+3. Formula of sum of \(K\)-th powers
+4. Properties of \(f(n, m)\)
+5. Lagrange interpolation
+6. Evaluation of \(f(N)\) using Lagrange interpolation
+7. Summary
+8. Bonus
+
+### 1. Preface
+
+This problem treats techniques that are frequently seen as a subproblems of high-level combinatory problems. We hope the knowledge obtained throughout this problem will be a footstep for getting AC for more advanced combinatory problems in the future.
+
+This problem asks to find the \(n\)-th term of a sequence. When it comes to problems that requires faster computation than linear time, it is crucial to **understand the properties of the sequence**. Shown in the table below is a summary of properties and computational complexity of the sequences that have commonly been seen in the past ABCs.
+
+   | type of sequence | example | precalculation | query |
+ |  |  |  |  |
+ | powers | \(n ^ k\) | None | \(\mathrm{O}(\log k)\) |
+ |  |  |  |  |
+ | factorials / binomial coefficients | \(n!,\binom{n}{k}\) | \(\mathrm{O}(n)\) | \(\mathrm{O}(1)\) |
+ |  |  |  |  |
+ | sequences whose transitions are expressed as a \(d \times d\) matrix | Fibonacci sequence | None | \(\mathrm{O}(d^3 \log k)\) |
+
+Now, what kind of property does the sequence in the problem satisfies?
+
+### 2. Observation
+
+When you are stuck in such kind of problem, one of the strategy is to **observe the behaviors for the simple cases and find good properties**.
+
+Let us consider the cases with small \(m\). First of all, \(m = 0\) is trivial; it can be computed easily by
+
+\[ f(n, 1) = n^k.\]
+
+Next, consider the case where \(m = 1\); we have
+
+\[ \begin{aligned} f(1, 1) &=& 1^K \\ f(2, 1) &=& 1^K + 2^K \\ f(3, 1) &=& 1^K + 2^K + 3^K \\ \vdots \end{aligned} \]
+
+so we can figure out that \(f(n, 1)\) (which we will denote \(S_K(N)\) for convenience) can be expressed as a sum of \(K\)-th powers, namely
+
+\[ S_K(n) = \sum_{i = 1} ^ n i^K. \]
+
+Especially for \(K = 1, 2\) and \(3\), the value \(S_K(n)\) can be expressed as a well-known elementary formula, that is,
+
+- If \(K = 1\), \(S_1(n) = \frac{n(n + 1)}{2} \);
+- If \(K = 2\), \(S_2(n) = \frac{n(n+1)(2n+1)}{6} \);
+- If \(K = 3\), \(S_3(n) = \frac{n^2 (n+1)^2}{4} \).
+
+Observing the formula above, we can propose an important hypothesis that **\(S_K(n)\) is a polynomial of degree \(K + 1\)**. We will prove this hypothesis in the next chapter.
+
+### 3. Formula of sum of \(K\)-th powers
+
+We will prove the following proposition.
+
+\(\displaystyle S_K(N) = \sum_{n=1}^N n^k\) can be expressed as a polynomial of \(K + 1\) degrees.
+
+There are several ways to prove this. Here we will show it recursively.
+
+Define a function
+
+\[T_K(n)=n^K - (n-1)^K.\]
+
+Here \(T_{K+1}\) can be expressed in the two ways:
+
+\[\displaystyle \sum_{n=0}^N T_{K+1}(n) = N^{K+1},\]
+
+and at the same time
+
+\[\displaystyle \sum_{n=0}^N T_{K+1}(n) = \sum_{n=0}^N Kn^K + (\text{polynomial in }n\text{ of degree }(K-1)).\]
+
+If the proposition holds for every integer less than or equal to \(K\), we have
+
+\[\sum_{n=0}^N T_{K+1}(n) = (\text{polynomial in }n\text{ of degree }K) + K \sum_{n=0}^N n^K,\]
+
+so
+
+\[S_K(N) = \sum_{n=0}^N n^K = \frac{N^{K+1} + (\text{polynomial in }n\text{ of degree }K)}{K}\]
+
+follows. Accompanied by the case \(K=1\), the proposition has been proved recursively.
+
+### 4. Properties of \(f(n, m)\)
+
+Now let’s get back to the original problem. By the fact shown in Chapter 3, the following property of \(f(N, M)\) can be proved:
+
+When \(M\) is fixed, \(f(N,M)\) can be expressed as a polynomial in \(N\) of degree \(M + K\).
+
+This fact can be shown recursively. Assuming that \(f(N,m)\) can be expressed as a polynomial in \(N\) of degree \(m+K\), due to the fact obtained in Chapter 3 it follows that
+
+\[ \begin{aligned} f(N, m+1) &=& \sum_{n = 1} ^ N f(n, m) \\ &=& \sum_{n=1}^N (\text{polynomial in }n\text{ of degree }(m+K)) \\ &=& (\text{polynomial in }N\text{ of degree }(m+K+1)), \end{aligned} \]
+
+so, accompanied by \(f(n,0)=n^K\) for \(m=0\), it has been proved.
+
+Therefore, we can now see that the desired sequence can be expressed as a polynomial of \(N\) for fixed \(M\). However here comes another issue: how can \(f(N)\) be computed fast when \(f(n)\) is known to be a polynomial?
+
+If we know the coefficients of the polynomial, it can be computed in an \(O(N)\) time with a naive algorithms like Horner’s method. However, in this problem it seems difficult to reconstruct the coefficients of \(f(N, M)\).
+
+In the following chapters, we will explain some properties of polynomials and explain the algorithm to calculate \(f(N)\) fast.
+
+### 5. Lagrange interpolation
+
+In fact, a polynomial has the following property.
+
+When \(x_0, x_1, \dots,x_d\) and \(y_0, \dots,y_d\) is given (where \(x_i\) are pairwise distinct), a polynomial \(f(x)\) of degree no more than \(d\) such that
+
+\[f(x_i) = y_i (i=0,\ldots,d)\]
+
+is uniquely determined.
+
+Let take linear functions as an example: “A polynomial with degree no more than \(1\) such that \(f(0) = b\) and \(f(1) = a + b\)” is uniquely determined as \(f(x) = ax + b\). The proposition above is its generalization.
+
+The proposition implies the following important fact:
+
+- Given a polynomial \(f(N)\) of degree \(d\) with unknown coefficients, we want to compute \(f(N)\). Here, even if we do not know the coefficients, **one can reconstruct \(f(N)\) as long as we know \(f(0), f(1), \dots, f(d)\)**.
+
+Look back on the original problem; \(f(0, M), f(1, M), \dots, f(d, M)\) can be computed easily with cumulative sums. In such problems, it is essential to hold a polynomial as a form of \(f(0), f(1),\dots, f(d)\) in order to reduce the computational complexity.
+
+Here, the proposition can be solved with the help of a matrix. Let
+
+\[f(x) = f_0 + f_1 x + f_2 x^2 + \dots + f_d x^d,\]
+
+then \((f_0, f_1,\dots,f_d)\) and \((y_0,\dots,y_d)\) can be expressed with the aid of Vandermonde matrix:
+
+\[ \left( \begin{array}{ccccc} 1 & x_0 & x_0^2 & \cdots & x_0^d \\ 1 & x_1 & x_1^2 & \cdots & x_1^d \\ 1 & x_0 & x_0^2 & \cdots & x_0^d \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & x_d & x_d^2 & \cdots & x_d^d \\ \end{array} \right) \left( \begin{array}{c} f_0 \\ f_1 \\ f_2 \\ \vdots \\ f_d \\ \end{array} \right) = \left( \begin{array}{c} y_0 \\ y_1 \\ y_2 \\ \vdots \\ y_d \\ \end{array} \right). \]
+
+Since the determinant of Vandermonde matrix is
+
+\[ \left \vert \begin{array}{ccccc} 1 & x_0 & x_0^2 & \cdots & x_0^d \\ 1 & x_1 & x_1^2 & \cdots & x_1^d \\ 1 & x_0 & x_0^2 & \cdots & x_0^d \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & x_d & x_d^2 & \cdots & x_d^d \\ \end{array} \right \vert = \prod_{j \lt i} (x_i - x_j) \neq 0, \]
+
+we have shown that one can uniquely determine \(f_0,\ldots,f_d\) that satisfies the conditions.
+
+Thus, we have seen that a polynomial \(f(x)\) of degree \(d\) is uniquely determined when \(f(0),f(1),\ldots,f(d)\) are given. However, multiplying the Vandermonde matrix to the vector of \(y\) to obtain \(f(N)\) costs too much, so we introduce polynomials called **Lagrange basis polynomials**.
+
+Let us define Lagrange basis polynomial \(L_i (x)\) for given \(x_0,\dots,x_d\) as follows:
+
+\[ \begin{aligned} L_i(x) &=& \prod_{j \neq i} \frac{x - x_j}{x_i - x_j} \\ &=& \frac{(x - x_0) (x - x_1)\dots(x-x_{i-1})(x-x_{i+1})\dots (x-x_d)}{(x_i - x_0) (x_i - x_1)\dots(x_i-x_{i-1})(x_i-x_{i+1})\dots (x_i-x_d)} \end{aligned} \]
+
+Then, using \(L_i(x)\), \(f(x)\) is expressed as follows:
+
+Given \(x_0, x_1, \dots,x_d\) and \(y_0, \dots,y_d\), the polynomial \(f(x)\) of degree no more than \(d\) satisfying
+
+\[f(x_i) = y_i (i=0,\ldots,d)\]
+
+can be expressed as follows, using Lagrange basis polynomials \(L_i(x)\):
+
+\[f(x) = y_0 L_0(x) + y_1 L_1(x) + \dots + y_d L_d(x)\]
+
+We will justify the claim above. A Lagrange basis polynomial
+
+\[ L_i(x) = \frac{(x - x_0) (x - x_1)\dots(x-x_{i-1})(x-x_{i+1})\dots (x-x_d)}{(x_i - x_0) (x_i - x_1)\dots(x_i-x_{i-1})(x_i-x_{i+1})\dots (x_i-x_d)} \]
+
+satisfies the following property:
+
+\[L_i(x_j) = \begin{cases} 1 & i = j \\ 0 & i \neq j \end{cases}. \]
+
+By the equation above, it follows that
+
+\[ y_0 L_0(x_i) + y_1 L_1(x_i) + \dots + y_d L_d(x_i) = y_i,\]
+
+so the claim is justified, with the fact that the degree of \(L_i(x)\) is at most \(d\).
+
+Therefore, we have shown that given multiple points \((x, y)\), the polynomial of minimum degree that passes through all the given points is uniquely determined. A discovery of a polynomial that passes through all the given points is called **interpolation**, and particularly an interpolation with the help of Lagrange basis polynomials is called **Lagrange interpolation**.
+
+### 6. Evaluation of \(f(N)\) using Lagrange interpolation
+
+Now that all the preparation is set, we will explain a fast algorithm computing \(f(N)\) from \(f(0),f(1),\dots,f(d)\).
+
+For a polynomial \(f(x)\) of degree at most \(d\), where \(y_0 = f(0),y_1 = f(1) \ldots,\) and \(y_d = f(d)\) are known, \(f(N)\) can be computed in a total of \(\mathrm{O}(d)\) time.
+
+Using the Lagrange basis polynomials for \(x_i = i\), we obtain
+
+\[f(N) = y_0 L_0(N) + y_1 L_1(N) + \dots + y_d L_d(N),\]
+
+so all we need is to compute \(L_i(N)\) fast enough. \(L_i(N)\) can be transformed as
+
+\[ \begin{aligned} L_i(N) &=& \frac {(N - 0) (N - 1)\dots(N-(i-1))(N-(i+1))\dots (N-d)} {(i - 0) (i - 1)\dots(i-(i-1))(i-(i+1))\dots (i-d)} \\ &=& \frac {\displaystyle \prod_{0 \leq j \leq d, j \neq i} (N - j)} {i! \cdot (-1)^{d - i} (d - i)!}. \end{aligned} \]
+
+If we precalculate the cumulative products of \(N-i\) from the left and from the right and \((1!)^{-1},(2!)^{-1},\dots,(d!)^{-1}\) in an \(\mathrm{O}(d)\) time, \(L_i(N)\) can be computed in an \(\mathrm{O}(1)\) time. Therefore \(L_i(N)\) has been computed in a total of \(\mathrm{O}(d)\), and \(f(N)\) has been found in a total of \(\mathrm{O}(d)\) time too.
+
+### 7. Summary
+
+By the observations above, we can see that the problem can be solved in a total of \(\mathrm{O}(K \log K + (K + M)M)\) with the following simple algorithm:
+
+- find the first \(K+M+1\) terms of \(f(n, 0) = n^K\) naively;
+- calculate the cumulative sums \(M\) times to find the first \(K+M+1\) terms of \(f(n, M)\);
+- find \(f(N, M)\) with Lagrange’s interpolation.
+
+The implementation is as follows. Since the time limit is a bit strict, acquiring AC with PyPy may require careful implementation such as optimization of division by JIT compile and avoiding bottlenecks of big integers, or it may lead to TLE.
+
+### 8. Bonus
+
+1. Although we did not calculate the coefficients sequence of the polynomial \((f_0, f_1,\dots,f_d)\) because we did not need them, if we do, it is known that \((f_0, f_1,\dots,f_d)\) can be obtained from \((y_0,\dots,y_d)\) in a total of \(\mathrm{O}(d \log^2 d)\) time with Vandermonde matrix. The Polynomial Interpolation problem in Library Checker has a sample code, so take a look at it if you are interested in.
+2. For \(N\leq 10^{18}, M \leq 10^7\), and \(K \leq 10^6\), find \(f(N, M) \bmod 998244353\).

problems/MA-06/Atcoder/abc208_f/judge.json

@@ -0,0 +1,15 @@
+{
+  "time_limit_s": 2.0,
+  "time_limit_source": "task page",
+  "memory_mb": 1024,
+  "memory_source": "task page",
+  "mode": "exact",
+  "validator_flags": "case_sensitive",
+  "comparison": "token-wise, whitespace-insensitive, case-sensitive (AtCoder's checker)",
+  "validator": null,
+  "tests_dir": "data",
+  "answer_suffix": ".out",
+  "n_sample": null,
+  "n_secret": 32,
+  "note": "no sample/secret split: the statement's samples are among the numbered tests"
+}

problems/MA-06/Atcoder/abc208_f/meta.json

@@ -0,0 +1,16 @@
+{
+  "task_id": "abc208_f",
+  "contest": "abc208",
+  "type": "ABC",
+  "letter": "F",
+  "title": "Cumulative Sum",
+  "url": "https://atcoder.jp/contests/abc208/tasks/abc208_f",
+  "difficulty": 2772,
+  "score": 600,
+  "time_limit_sec": 2.0,
+  "memory_limit_mb": 1024,
+  "n_tests": 32,
+  "editorial_url": "https://atcoder.jp/contests/abc208/editorial/2219",
+  "code_source": "model",
+  "pid": "A1626"
+}

problems/MA-06/Atcoder/abc208_f/solution.cpp

@@ -0,0 +1,71 @@
+// abc208_f: F - Cumulative Sum
+// Written by a model (gpt-6-astra) from the official editorial, https://atcoder.jp/contests/abc208/editorial/2219
+// Judged AC on all 32 tests, max 0.36 s of the 2 s limit (submission 1).
+
+#include <bits/stdc++.h>
+#include <atcoder/modint>
+using namespace std;
+using mint = atcoder::modint1000000007;
+
+int main() {
+    ios::sync_with_stdio(false);
+    cin.tie(nullptr);
+
+    long long N;
+    int M, K;
+    cin >> N >> M >> K;
+
+    const int x = static_cast<int>(N % mint::mod());
+    if (M == 0) {
+        cout << mint(x).pow(K).val() << '\n';
+        return 0;
+    }
+    if (x == 0) {
+        cout << 0 << '\n';
+        return 0;
+    }
+
+    // f(n, M) is a polynomial of degree K + M, which is below the modulus.
+    const int degree = K + M;
+    const int d = min(degree, x);
+    vector<mint> y(d + 1);
+    for (int i = 1; i <= d; ++i) {
+        y[i] = mint(i).pow(K);
+    }
+    for (int m = 0; m < M; ++m) {
+        for (int i = 1; i <= d; ++i) {
+            y[i] += y[i - 1];
+        }
+    }
+
+    if (x <= degree) {
+        cout << y[x].val() << '\n';
+        return 0;
+    }
+
+    vector<mint> invfact(d + 1);
+    mint fact = 1;
+    for (int i = 1; i <= d; ++i) fact *= i;
+    invfact[d] = fact.inv();
+    for (int i = d; i >= 1; --i) {
+        invfact[i - 1] = invfact[i] * i;
+    }
+
+    vector<mint> prefix(d + 2);
+    prefix[0] = 1;
+    for (int i = 0; i <= d; ++i) {
+        prefix[i + 1] = prefix[i] * (x - i);
+    }
+
+    mint answer = 0;
+    mint suffix = 1;
+    for (int i = d; i >= 0; --i) {
+        mint term = y[i] * prefix[i] * suffix * invfact[i] * invfact[d - i];
+        if ((d - i) & 1) answer -= term;
+        else answer += term;
+        suffix *= (x - i);
+    }
+
+    cout << answer.val() << '\n';
+    return 0;
+}

problems/MA-06/Atcoder/abc208_f/statement.md

@@ -0,0 +1,82 @@
+# F - Cumulative Sum
+
+**Score:** 600 points
+
+## Problem Statement
+
+For non-negative integers \(n\) and \(m\), let us define the function \(f(n, m)\) as follows, using a positive integer \(K\).
+
+\[
+\displaystyle f(n, m) = 
+\begin{cases} 
+0 & (n = 0) \\ 
+n^K & (n > 0, m = 0) \\ 
+f(n-1, m) + f(n, m-1) & (n > 0, m > 0) 
+\end{cases}
+\]
+
+Given \(N\), \(M\), and \(K\), find \(f(N, M)\) modulo \((10^9 + 7)\).
+
+## Constraints
+
+- \(0 \leq N \leq 10^{18}\)
+- \(0 \leq M \leq 30\)
+- \(1 \leq K \leq 2.5 \times 10^6\)
+- All values in input are integers.
+
+## Input
+
+Input is given from Standard Input in the following format:
+
+```
+N M K
+```
+
+## Output
+
+Print \(f(N, M)\) modulo \((10^9 + 7)\).
+
+## Sample Input 1
+
+```
+3 4 2
+```
+
+## Sample Output 1
+
+```
+35
+```
+
+When \(K = 2\), the values \(f(n, m)\) for \(n \leq 3, m \leq 4\) are as follows.
+
+|       | \(m = 0\) | \(m = 1\) | \(m = 2\) | \(m = 3\) | \(m = 4\) |
+|-------|-----------|-----------|-----------|-----------|-----------|
+| \(n = 0\) | 0         | 0         | 0         | 0         | 0         |
+| \(n = 1\) | 1         | 1         | 1         | 1         | 1         |
+| \(n = 2\) | 4         | 5         | 6         | 7         | 8         |
+| \(n = 3\) | 9         | 14        | 20        | 27        | 35        |
+
+## Sample Input 2
+
+```
+0 1 2
+```
+
+## Sample Output 2
+
+```
+0
+```
+
+## Sample Input 3
+
+```
+1000000000000000000 30 123456
+```
+
+## Sample Output 3
+
+```
+297085514
+```

problems/MA-06/Atcoder/abc212_h/data/1.in

@@ -0,0 +1,2 @@
+2 2
+1 2

problems/MA-06/Atcoder/abc212_h/data/1.out

@@ -0,0 +1 @@
+4

problems/MA-06/Atcoder/abc212_h/data/10.in

@@ -0,0 +1,2 @@
+199991 16
+1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768

problems/MA-06/Atcoder/abc212_h/data/10.out

@@ -0,0 +1 @@
+684344764

problems/MA-06/Atcoder/abc212_h/data/11.in

@@ -0,0 +1,2 @@
+199959 16
+65534 65533 65531 65527 65519 65503 65471 65407 65279 65023 64511 63487 61439 57343 49151 32767

problems/MA-06/Atcoder/abc212_h/data/11.out

@@ -0,0 +1 @@
+558071059

problems/MA-06/Atcoder/abc212_h/data/12.out

@@ -0,0 +1 @@
+892944731

problems/MA-06/Atcoder/abc212_h/data/13.out

@@ -0,0 +1 @@
+987709985

problems/MA-06/Atcoder/abc212_h/data/14.out

@@ -0,0 +1 @@
+701323473

problems/MA-06/Atcoder/abc212_h/data/15.in

@@ -0,0 +1,2 @@
+199823 14313
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