Overview

This is a problem where you need to calculate the number of cooling packs required for each room to bring the temperature to the target temperature \(M\) degrees or below, and find the total.

Analysis

Key Observations

1. Each room can be considered independently: Since cooling packs used in one room don’t affect other rooms, we can calculate the number of cooling packs needed for each room individually and sum them up.

2. Rooms that already satisfy the condition: Rooms where the current temperature \(T_i\) is at or below the target temperature \(M\) don’t need any cooling packs.

3. Calculating the required number of packs: For rooms where \(T_i > M\), we need to lower the temperature by at least \((T_i - M)\) degrees. Since one pack lowers the temperature by \(D\) degrees, the required number of packs is \((T_i - M) \div D\) rounded up.

Concrete Example

• If \(T_i = 30\) degrees, \(M = 25\) degrees, \(D = 3\) degrees
• Temperature that needs to be lowered: \(30 - 25 = 5\) degrees
• Required number of packs: \(\lceil 5 \div 3 \rceil = \lceil 1.67 \rceil = 2\) packs
• (With 1 pack, it only drops 3 degrees to 27 degrees; with 2 packs, it drops 6 degrees to 24 degrees, achieving the target)

Why Rounding Up is Necessary

We don’t need to make the temperature exactly \(M\) degrees; we just need to make it \(M\) degrees or below. However, when \((T_i - M)\) is not evenly divisible by \(D\), we still need to cover the fractional part, so we calculate the number of packs by rounding up.

Algorithm

1. For each room \(i\), do the following:
   • If \(T_i \leq M\), no cooling packs are needed for that room (0 packs)
   • If \(T_i > M\), calculate the required number of packs \(= \lceil (T_i - M) / D \rceil\)
2. Sum up the required number of packs for all rooms and output the result

How to Calculate Ceiling

In Python, you can easily calculate the ceiling of a decimal using the math.ceil() function.

import math
packs_needed = math.ceil((t - M) / D)

If you want to use only integer arithmetic, you can use the following formula:

packs_needed = (t - M + D - 1) // D

Complexity

• Time complexity: \(O(N)\)
  • Because we perform constant-time calculations for each room
• Space complexity: \(O(N)\)
  • Because we store the list of temperatures (can be reduced to \(O(1)\) with sequential processing)

Implementation Notes

• Don’t forget the conditional check: Since no cooling packs are needed when \(T_i \leq M\), we only calculate when \(T_i > M\). Calculating negative values would result in an incorrect answer.
• Be careful about integer overflow: According to the constraints, \(T_i\) and \(M\) can be up to \(10^9\), but since Python doesn’t have integer overflow, no special handling is needed.
• Constraint that temperature doesn’t go below 0 degrees: The problem states “temperature will not go below 0 degrees,” but since the goal is to make it \(M\) degrees or below, this constraint doesn’t affect the calculation of the minimum number of packs.

Source Code

import math

def solve():
    N, M, D = map(int, input().split())
    T = list(map(int, input().split()))
    
    total_packs = 0
    for t in T:
        if t > M:
            # Number of packs needed to bring temperature to M or below
            # Ceiling of (t - M) / D
            packs_needed = math.ceil((t - M) / D)
            total_packs += packs_needed
    
    print(total_packs)

solve()

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