H. Pencil Game

Codeforces

IDCF10054H

Time3000ms

Memory256MB

Difficulty—

English · Original

Formal · Original

Minh has a box of pencils. The box is a rectangle of size M * N, where position (i, j) has a pencil with a length of exactly i * N + j (0 ≤ i ≤ M - 1, 0 ≤ j ≤ N - 1). Note that position (0, 0) does not have any pencil hence having a length of 0. He wonders if he could select a sub-rectangle of the box and join all the pencils within that sub-rectangle together, to get a new long pencil that has a specific length L that he wants. Your task is to find a sub-rectangle of the box in which the total length of the contained pencils is L and return the area of that the sub-rectangle. If there are multiple solutions, return the smallest possible area. If there’s no such sub-rectangle, return - 1. The input file consists of several datasets. The first line of the input file contains the number of datasets which is a positive integer and is not greater than 150. The following lines describe the datasets. Each dataset contains three space-separated numbers M, N and L (1 ≤ M, N ≤ 106, 1 ≤ L ≤ 1012) written in one line. For each dataset, write in one line the smallest possible area of the sub-rectangle in which the total sum of pencil lengths is L. Write in one line - 1 if there is no such sub-rectangle. ## Input The input file consists of several datasets. The first line of the input file contains the number of datasets which is a positive integer and is not greater than 150. The following lines describe the datasets. Each dataset contains three space-separated numbers M, N and L (1 ≤ M, N ≤ 106, 1 ≤ L ≤ 1012) written in one line. ## Output For each dataset, write in one line the smallest possible area of the sub-rectangle in which the total sum of pencil lengths is L. Write in one line - 1 if there is no such sub-rectangle. [samples]

**Definitions** Let $ M, N \in \mathbb{Z}^+ $ denote the dimensions of the grid, with positions $(i, j)$ for $ 0 \leq i \leq M-1 $, $ 0 \leq j \leq N-1 $. The pencil at position $(i, j)$ has length $ i \cdot N + j $. Let $ L \in \mathbb{Z}^+ $ be the target total length. **Constraints** 1. $ 1 \leq M, N \leq 10^6 $ 2. $ 1 \leq L \leq 10^{12} $ **Objective** Find the smallest area $ A = (i_2 - i_1 + 1) \cdot (j_2 - j_1 + 1) $ of a sub-rectangle defined by $ i_1 \leq i \leq i_2 $, $ j_1 \leq j \leq j_2 $, such that: $$ \sum_{i=i_1}^{i_2} \sum_{j=j_1}^{j_2} (i \cdot N + j) = L $$ If no such sub-rectangle exists, return $-1$.

API Response (JSON)

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    "name": "H. Pencil Game",
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      "content": "Minh has a box of pencils. The box is a rectangle of size M * N, where position (i, j) has a pencil with a length of exactly i * N + j (0 ≤ i ≤ M - 1,  0 ≤ j ≤ N - 1). Note that position (0, 0) does n",
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    "platform": "Codeforces",
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      "content": "Minh has a box of pencils. The box is a rectangle of size M * N, where position (i, j) has a pencil with a length of exactly i * N + j (0 ≤ i ≤ M - 1,  0 ≤ j ≤ N - 1). Note that position (0, 0) does n...",
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      "content": "**Definitions**  \nLet $ M, N \\in \\mathbb{Z}^+ $ denote the dimensions of the grid, with positions $(i, j)$ for $ 0 \\leq i \\leq M-1 $, $ 0 \\leq j \\leq N-1 $.  \nThe pencil at position $(i, j)$ has lengt...",
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