Overview

This problem requires checking the temperature of each room and, for rooms with temperatures higher than \(37\) degrees, calculating the total number of operations needed to lower them to \(37\) degrees (where each operation decreases the temperature by \(0.1\) degrees).

Analysis

Key Observations

1. Understanding the conditions: Rooms with temperatures at or below \(37\) degrees require no action. We only need to process rooms with temperatures above \(37\) degrees.

2. Calculating the number of operations: For a room with temperature \(T_i\) degrees that needs to be lowered to \(37\) degrees, the temperature difference is \(T_i - 37\) degrees. Since each operation decreases the temperature by \(0.1\) degrees, the required number of operations is: $\(\text{operations} = \frac{T_i - 37}{0.1} = (T_i - 37) \times 10\)$

3. Concrete example: For a room with temperature \(38.5\) degrees

   • Temperature difference: \(38.5 - 37 = 1.5\) degrees
   • Number of operations: \(1.5 \times 10 = 15\) times

Issues with the Naive Approach

For this problem, simply calculating the number of operations for each room and summing them is sufficient. Since \(N \leq 2 \times 10^5\), an \(O(N)\) algorithm is fast enough.

However, handling floating-point numbers requires caution. For example, 0.1 * 3 might become 0.30000000000000004, and simple calculations may introduce errors.

Algorithm

1. Process each room’s temperature \(T_i\) in order
2. Only when \(T_i > 37\):
   • Calculate the temperature difference \(\text{diff} = T_i - 37\)
   • Calculate the number of operations \(= \text{diff} \times 10\) (round to integer)
   • Add to the total
3. Output the final total

Complexity

• Time complexity: \(O(N)\) (constant time calculation for each room)
• Space complexity: \(O(N)\) (for storing the list of temperatures)

Implementation Notes

Handling Floating-Point Errors

Floating-point calculations may introduce errors. For example:

>>> 38.5 - 37
1.4999999999999964  # We actually want 1.5

Because of this, simply using int(diff * 10) might result in 14 instead of the correct value.

Solution: Use the round() function to absorb floating-point errors through rounding.

operations = round(diff * 10)  # 1.4999... * 10 = 14.999... → rounds to 15

Alternative Solution: Using Integer Arithmetic

To completely avoid floating-point errors, you can treat the input as integers multiplied by \(10\).

# Example: "38.5" → treat as 385
# 37 degrees → 370
# Number of operations = 385 - 370 = 15

Source Code

N = int(input())
T = list(map(float, input().split()))

total_operations = 0
for t in T:
    if t > 37.0:
        # Calculate the temperature difference and determine the number of operations in 0.1-degree units
        diff = t - 37.0
        # Round to account for floating-point errors
        operations = round(diff * 10)
        total_operations += operations

print(total_operations)

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