CF2264F Deranged Calculator

Hacks are disabled on this problem. We are excited to introduce a new language for this Codeforces round: the Deranged Calculator language (DC). The DC language is a simplified programming language with some weird constraints. It only supports a single expression on one line and only has a single positive integer $ n $ as input. The syntax of the expression is similar to that of most programming languages (in fact, the expression is valid Python), but it is very limited: - Expressions consist of the input parameter $ n $ , parenthesized expressions, and calls to the round function. These are joined together by the operators +, -, \*, and /. - Grouping with parentheses () can be used to change the order of operations, as usual. - The operators \* and / have equal precedence and take precedence over + and -, which also have equal precedence. Operators of equal precedence are evaluated from left to right. - The round function takes a single expression as its argument and rounds its value to the nearest integer, rounding up when the fractional part is exactly $ 0.5 $ . (For example, round(n/(n-n-n-n)) will always evaluate to $ 0 $ , because $ x/(x-x-x-x)=-\frac{1}{2} $ for a positive integer $ x $ . The fractional part is $ 0.5 $ , so it will be rounded up to $ 0 $ .) - Only the input value $ n $ can be used; no numeric constants are allowed in the program. - All operations are performed using exact fractional arithmetic, so division does not round and can produce fractions. A derangement of $ n $ numbers is a permutation of those numbers in which none of the numbers appears in its original position. For example, the derangements of the sequence $ [1, 2, 3] $ are $ [2, 3, 1] $ and $ [3, 1, 2] $ . You are given a single integer $ k $ . Your job is to make a valid DC program that, for every integer $ n $ ( $ 2 \le n \le k $ ), calculates the number of derangements of the sequence $ [1, 2, \ldots, n] $ , when $ n $ is given as input to the DC program. The length of the program should not exceed $ 10\,000 $ characters. A local testing tool is provided to help you develop your solution. It can be found under Contest Materials.

The only line of input contains a single integer $ k $ ( $ k \in \{2,3,50\} $ ). Your DC program should output the correct number of derangements for all integers $ n $ such that $ 2 \le n \le k $ .

Output a valid DC expression on a single line. The expression should only consist of the characters round()+-\*/ and the length of the expression should not exceed $ 10\,000 $ characters. Your expression will be run on all integers $ n $ such that $ 2 \le n \le k $ . For each such run, the DC expression should not divide by $ 0 $ at any time, and its value must be the number of derangements of $ n $ .

For the first example, n-n/n is a valid DC program that calculates $ n-1 $ . The number of derangements for $ n=2 $ is $ 1 $ , so this program correctly computes the answer for $ n=2 $ . In the second example, a slightly overcomplicated expression is used, which shows all the features of the language in action. For $ n=3 $ , the value of the expression is: $ 3+\text{round}(3-3 \cdot(\frac{3}{3+3}))-3 = \text{round}(3 - \frac{3}{2}) = \text{round}(\frac{3}{2}) = \text{round}(1.5)=2 $ . It correctly calculates the number of derangements of the sequence $ [1,2,3] $ . For $ n=2 $ it can be verified that the value of the expression becomes $ 1 $ .