usaco plat 2020 us open problem 2 exercise solution

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12-10-2024

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USACO 2020 US Open, Platinum Problem 2: Exercise Solution

This document provides a detailed solution for the USACO 2020 US Open, Platinum Problem 2: Exercise. The problem revolves around finding the minimum number of exercises a cow needs to complete to achieve a desired strength level.

Problem Description

Farmer John has a bunch of cows who are training for a competition. There are N cows, each with an initial strength S_i. To increase their strength, they must do exercises. Each exercise increases a cow's strength by X, but only if the cow's current strength is less than or equal to Y.

Farmer John wants to figure out the minimum number of exercises each cow needs to complete to reach a target strength T.

Approach

The key to solving this problem lies in understanding that each cow's progress towards the target strength is determined by the number of exercises they can perform that actually increase their strength.

Here's how we can solve the problem:

1. Calculate the "effective" strength increases: For each cow, we need to figure out how many times they can benefit from an exercise that increases strength by X before reaching the target strength.

   • Example: If a cow has strength S = 5, X = 3, Y = 10, and T = 15, the cow can perform a maximum of 3 effective exercises (5 + 3 + 3 + 3 = 14, exceeding T).
   • Formula: The number of effective exercises for a cow can be calculated as: min(floor((T - S) / X), floor((Y - S) / X))

2. Determine the minimum number of exercises: The minimum number of exercises a cow needs to complete is simply the number of effective exercises. This is because any additional exercise beyond the effective ones will not increase their strength further.

Code Example

def min_exercises(s, x, y, t):
    """Calculates the minimum number of exercises needed for a cow to reach target strength.

    Args:
        s: Initial strength of the cow.
        x: Strength increase per exercise.
        y: Maximum strength for exercise effectiveness.
        t: Target strength.

    Returns:
        The minimum number of exercises needed.
    """
    effective_exercises = min((t - s) // x, (y - s) // x)
    return effective_exercises

# Example usage
n = int(input())
s = []
x, y, t = map(int, input().split())
for i in range(n):
    s.append(int(input()))

for i in range(n):
    print(min_exercises(s[i], x, y, t))

Conclusion

This solution provides a clear and efficient way to determine the minimum number of exercises needed for each cow to reach their target strength. It emphasizes the importance of identifying the "effective" exercises that actually contribute to strength gain.