C. Rectangles

Codeforces

IDCF10102C

Time2000ms

Memory64MB

Difficulty—

English · Original

Formal · Original

You have n rectangles, each of which is described by three integers i, j and k. This indicates that the lower-left corner of the rectangle will be located at the point (i, 0) and the upper-right corner of the rectangle will be located at the point (j, k). For example, if you have a description of four rectangles like this: 0 2 3 0 3 1 1 2 2 2 4 4 The area occupied by these rectangles will be as follows: The total area in this case will be 14. You are given n and n triples (i, j, k), your task is to compute the total area occupied by the rectangles which are described by the n triples. The first line will contain a single integer T indicates the number of test cases. Each test case will consist of n + 1 lines, the first one will contain the number of rectangles n and the following n lines will be the description of the rectangles. Each description will be a triple (i, j, k) as mentioned above. The Constraints: 10 <= T <= 20 1 <= n <= 100 0 <= i < 100 1 <= j < 100 i != j 1 <= k <= 100 For each test case, print a single integer in a separated line indicates the total area occupied by the rectangles. ## Input The first line will contain a single integer T indicates the number of test cases. Each test case will consist of n + 1 lines, the first one will contain the number of rectangles n and the following n lines will be the description of the rectangles. Each description will be a triple (i, j, k) as mentioned above. The Constraints: 10 <= T <= 20 1 <= n <= 100 0 <= i < 100 1 <= j < 100 i != j 1 <= k <= 100 ## Output For each test case, print a single integer in a separated line indicates the total area occupied by the rectangles. [samples]

**Definitions** Let $ T \in \mathbb{Z} $ be the number of test cases. For each test case: - Let $ n \in \mathbb{Z} $ denote the bit-length of binary strings. - Let $ A, B \in \{0,1\}^n $ be binary strings of length $ n $, representing integers in big-endian format. - Let $ C \in \{0,1\}^n $ be an unknown binary string such that $ A \mid C = B $, where $ \mid $ denotes bitwise OR. **Constraints** 1. $ 1 \le T \le \text{unbounded} $ (implied by "several test cases") 2. $ 1 \le n \le 100 $ 3. $ A, B \in \{0,1\}^n $ — strings of exactly $ n $ bits (no leading zeros assumed; padded to length $ n $) **Objective** For each test case, compute the number of binary strings $ C \in \{0,1\}^n $ satisfying: $$ \forall i \in \{1, \dots, n\}, \quad A_i \mid C_i = B_i $$ If no such $ C $ exists, output `"IMPOSSIBLE"`. Otherwise, output the count modulo $ 1000000007 $. **Bitwise OR Constraint Analysis** For each bit position $ i $: - If $ A_i = 1 $ and $ B_i = 0 $: **IMPOSSIBLE** (since $ 1 \mid C_i \geq 1 $) - If $ A_i = 0 $ and $ B_i = 0 $: $ C_i = 0 $ → 1 choice - If $ A_i = 0 $ and $ B_i = 1 $: $ C_i = 1 $ → 1 choice - If $ A_i = 1 $ and $ B_i = 1 $: $ C_i \in \{0,1\} $ → 2 choices Thus, total number of valid $ C $ is: $$ \prod_{i=1}^{n} \begin{cases} 0 & \text{if } A_i = 1 \text{ and } B_i = 0 \\ 1 & \text{if } A_i = 0 \text{ and } B_i = 0 \\ 1 & \text{if } A_i = 0 \text{ and } B_i = 1 \\ 2 & \text{if } A_i = 1 \text{ and } B_i = 1 \\ \end{cases} $$

API Response (JSON)

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    "name": "C. Rectangles",
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