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Artificial Intelligence Methods in the Environmental Sciences 1

如果n是一个整数且n的平方是偶数，则n也是偶数。这个定理可以通过反证法证明。假设n是奇数，则n可以表示为2k+1的形式，其中k是整数。那么n的平方可以表示为(2k+1)²=4k²+4k+1的形式。由于4k²和4k都是偶数，它们的和也是偶数，因此n²是奇数。这与n²是偶数的假设矛盾，因此假设不成立，n必须是偶数。
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这个结论可以证明如下：假设存在一个线性码 C，它是一个 [n,k,d]q 码，即它有 q^k 个码字，每个码字长度为 n，最小距离为 d。我们可以构造一个新的码 D，它的码字集合为 C 中所有长度为 d 的码字的补集。也就是说，D...

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Using the generating function for the Fibonacci numbers, prove the identity f0 +f2 +...+f2n =f2n+1 for any n ≥ 0.

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On the Literature lesson Sergei noticed an awful injustice, it seems that some students are asked more often than others. Seating in the class looks like a rectangle, where $ n $ rows with $ m $ pupils in each. The teacher asks pupils in the following order: at first, she asks all pupils from the first row in the order of their seating, then she continues to ask pupils from the next row. If the teacher asked the last row, then the direction of the poll changes, it means that she asks the previous row. The order of asking the rows looks as follows: the $ 1 $ -st row, the $ 2 $ -nd row, $ ... $ , the $ n-1 $ -st row, the $ n $ -th row, the $ n-1 $ -st row, $ ... $ , the $ 2 $ -nd row, the $ 1 $ -st row, the $ 2 $ -nd row, $ ... $ The order of asking of pupils on the same row is always the same: the $ 1 $ -st pupil, the $ 2 $ -nd pupil, $ ... $ , the $ m $ -th pupil. During the lesson the teacher managed to ask exactly $ k $ questions from pupils in order described above. Sergei seats on the $ x $ -th row, on the $ y $ -th place in the row. Sergei decided to prove to the teacher that pupils are asked irregularly, help him count three values: 1. the maximum number of questions a particular pupil is asked, 2. the minimum number of questions a particular pupil is asked, 3. how many times the teacher asked Sergei. If there is only one row in the class, then the teacher always asks children from this row.Write C++code

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