Overview

This problem asks you to determine whether, when planting \(N\) flowers at equal intervals in a flower bed, the spacing between adjacent flowers is at least \(K\) centimeters.

Analysis

Key Insight

The key point of this problem is correctly calculating the spacing when \(N\) flowers are planted at equal intervals.

Let’s consider some concrete examples.

Example 1: Planting \(N = 3\) flowers in a flower bed with width \(W = 10\) centimeters

Flower bed: [0]--------[5]--------[10]
            Flower1    Flower2    Flower3

• Flowers are planted at positions \(0, 5, 10\)
• The number of gaps is 2 (between Flower1 and Flower2, between Flower2 and Flower3)
• Each gap is \(10 \div 2 = 5\) centimeters

Example 2: Planting \(N = 5\) flowers in a flower bed with width \(W = 20\) centimeters

Flower bed: [0]---[5]---[10]---[15]---[20]

• The number of gaps is 4
• Each gap is \(20 \div 4 = 5\) centimeters

Generalization

When planting \(N\) flowers, the number of spaces (gaps) between them is \(N - 1\).

Since the flower bed width \(W\) is divided equally into these \(N - 1\) gaps, each gap is:

\[\text{interval} = \frac{W}{N - 1}\]

If this interval is at least \(K\), then the condition for flowers to grow healthily is satisfied.

Decision Condition

\[\frac{W}{N - 1} \geq K\]

If this holds, output Yes; otherwise, output No.

Algorithm

1. Read \(N\), \(W\), and \(K\) as input
2. Calculate the interval when planted at equal spacing: \(\text{interval} = \frac{W}{N-1}\)
3. If \(\text{interval} \geq K\), output Yes; otherwise, output No

Complexity

• Time complexity: \(O(1)\) (only arithmetic operations and comparison)
• Space complexity: \(O(1)\) (only using a few variables)

Implementation Notes

• The number of gaps is \(N - 1\): It’s important to correctly understand the number of spaces between \(N\) flowers. This follows the same logic as “the number of line segments connecting \(N\) points.”
• Handling division: In Python, using the / operator performs floating-point division, so comparisons work correctly even when \(W\) or \(N-1\) don’t divide evenly.
• Integer-only method: The inequality \(\frac{W}{N-1} \geq K\) can be transformed by multiplying both sides by \((N-1)\) to get \(W \geq K \times (N-1)\), allowing the check to be done using only integers.

Source Code

N, W, K = map(int, input().split())

# When planting N flowers at equal intervals, the number of gaps is N-1
# The condition is satisfied if the interval (width W divided into N-1 equal parts) is at least K
interval = W / (N - 1)

if interval >= K:
    print("Yes")
else:
    print("No")

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