結果

問題 No.1501 酔歩
コンテスト
ユーザー qwewe
提出日時 2025-05-14 13:19:19
言語 PyPy3
(7.3.15)
結果
AC  
実行時間 205 ms / 2,000 ms
コード長 5,509 bytes
コンパイル時間 182 ms
コンパイル使用メモリ 82,836 KB
実行使用メモリ 101,936 KB
最終ジャッジ日時 2025-05-14 13:20:23
合計ジャッジ時間 12,999 ms
ジャッジサーバーID
（参考情報） 		judge5 / judge1
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ファイルパターン 結果
sample AC * 3
other AC * 53
権限があれば一括ダウンロードができます

ソースコード

diff # 

import sys
# Import the Fraction class for precise rational number arithmetic
from fractions import Fraction

def solve():
    # Read N (number of squares) and K (starting square) from input
    N, K = map(int, sys.stdin.readline().split())
    
    # Handle the edge case where there is only one square (N=1).
    if N == 1:
        # If N=1, K must also be 1. The piece starts at square 1, which is also square N.
        # It has already reached N. The probability is 1.
        print("1/1")
        return

    # Read the array A containing integers for each square.
    # This line is executed only if N > 1.
    # The problem guarantees N integers are provided.
    A = list(map(int, sys.stdin.readline().split()))

    # Handle the trivial cases where the piece starts at an absorbing square (1 or N).
    if K == 1:
        # If the piece starts at square 1, it is absorbed immediately.
        # It cannot reach square N (unless N=1, which was handled above).
        # The probability of reaching N is 0.
        print("0")
        return
    
    if K == N:
        # If the piece starts at square N, it is absorbed immediately.
        # The probability of reaching N is 1.
        print("1/1")
        return

    # The general case: the piece starts at square K where 1 < K < N, and N >= 2.
    
    # Use 0-based indexing for the list A for convenience in Python.
    # A_list[i] corresponds to the value A_{i+1} in the problem statement.
    A_list = A 

    # Initialize the running sum for calculating the denominator of the final probability.
    # Use Fraction for precise arithmetic with rational numbers.
    current_sum = Fraction(0)
    # Initialize sum_K, which will store the partial sum needed for the numerator.
    sum_K = Fraction(0) 

    # Check if A_list has at least 2 elements. Since N >= 2 in this code path,
    # len(A_list) is N >= 2, so this check is implicitly satisfied.
    # However, for robust code, consider explicitly checking N >= 2 if N=1 case wasn't handled.
    # Ensure A_list has elements at index 0 and 1.
    if len(A_list) < 2:
         # This block should technically not be reached due to N>=2 condition.
         # If it were possible, an error or default handling would be needed.
         # For this problem's constraints, it's safe to assume len(A_list) >= 2.
         pass 

    # Precompute the numerator term A_1 * A_2. 
    # A_1 is A_list[0], A_2 is A_list[1].
    # Constraints state A_i >= 1, so A_list[0] and A_list[1] are valid indices and values are >= 1.
    A1A2_num = A_list[0] * A_list[1]
    
    # Calculate the first term in the sum, C'_2.
    # The formula for C'_i is (A_1 * A_2) / (A_{i-1} * A_i).
    # For i=2, C'_2 = (A_1 * A_2) / (A_1 * A_2).
    # Since A_i >= 1, A_1 * A_2 >= 1, so C'_2 is always 1.
    C_prime_2 = Fraction(1) 
    # Add C'_2 to the running sum.
    current_sum += C_prime_2
    
    # If the starting position K is 2, the sum required for the numerator (sum_K) is just C'_2.
    if K == 2:
        sum_K = current_sum

    # Iteratively calculate the remaining terms C'_i for i from 3 to N.
    for i in range(3, N + 1):
        
        # Calculate the denominator term A_{i-1} * A_i.
        # Using 0-based indexing: A_list[i-2] * A_list[i-1].
        # Check if indices i-2 and i-1 are valid for A_list. Given the loop range and N >= 2, these indices are valid.
        if i-2 >= len(A_list) or i-1 >= len(A_list):
             # This situation implies N doesn't match the length of A, or an off-by-one error.
             # Given problem constraints, this check is mainly defensive.
             pass
        
        Ai_minus_1 = A_list[i-2]
        Ai = A_list[i-1]
        
        # Calculate the value of the denominator. A_i >= 1 guarantees denom_val >= 1.
        denom_val = Ai_minus_1 * Ai
        
        # Check for division by zero, although constraints A_i >= 1 make this impossible.
        if denom_val == 0:
             # This case is impossible under problem constraints.
             # If it could happen, proper error handling would be needed.
             pass 

        # Compute C'_i = (A_1 * A_2) / (A_{i-1} * A_i) using Fraction.
        # Pass integer numerator and denominator to the Fraction constructor for accuracy.
        C_prime_i = Fraction(A1A2_num, denom_val)

        # Add the computed term C'_i to the running sum.
        current_sum += C_prime_i
        
        # If the current index i matches the starting position K,
        # store the current partial sum. This value represents \sum_{j=2}^K C'_j.
        if i == K:
            sum_K = current_sum

    # After the loop completes, current_sum holds the total sum \sum_{j=2}^N C'_j.
    sum_N = current_sum

    # Check if the total sum sum_N is zero. This check is mostly defensive,
    # as sum_N >= C'_2 = 1 because all terms C'_i are non-negative (actually positive due to A_i >= 1).
    if sum_N == 0:
         # This case indicates an issue, likely impossible under constraints.
         pass # It's guaranteed sum_N >= 1.

    # Calculate the final probability p_K = sum_K / sum_N.
    # sum_K represents \sum_{j=2}^K C'_j and sum_N represents \sum_{j=2}^N C'_j.
    final_prob = sum_K / sum_N
    
    # Print the final probability as an irreducible fraction p/q.
    # The Fraction class automatically simplifies the fraction and ensures the denominator is positive.
    print(f"{final_prob.numerator}/{final_prob.denominator}")

# Execute the solve function to run the program logic
solve()
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