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293 lines
8.2 KiB
Python
Executable file
293 lines
8.2 KiB
Python
Executable file
"""#
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### 谜题描述
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Vasya has got three integers n, m and k. He'd like to find three integer points (x_1, y_1), (x_2, y_2), (x_3, y_3), such that 0 ≤ x_1, x_2, x_3 ≤ n, 0 ≤ y_1, y_2, y_3 ≤ m and the area of the triangle formed by these points is equal to nm/k.
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Help Vasya! Find such points (if it's possible). If there are multiple solutions, print any of them.
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Input
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The single line contains three integers n, m, k (1≤ n, m ≤ 10^9, 2 ≤ k ≤ 10^9).
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Output
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If there are no such points, print \"NO\".
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Otherwise print \"YES\" in the first line. The next three lines should contain integers x_i, y_i — coordinates of the points, one point per line. If there are multiple solutions, print any of them.
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You can print each letter in any case (upper or lower).
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Examples
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Input
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4 3 3
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Output
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YES
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1 0
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2 3
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4 1
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Input
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4 4 7
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Output
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NO
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Note
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In the first example area of the triangle should be equal to nm/k = 4. The triangle mentioned in the output is pictured below:
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<image>
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In the second example there is no triangle with area nm/k = 16/7.
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Here is a reference code to solve this task. You can use this to help you genereate cases or validate the solution.
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```python
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def gcd(a, b):
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if b == 0:
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return a
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else:
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return gcd(b, a % b)
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n,m,k = map(int,raw_input().split())
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was_even = False
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if (2*n*m)%k == 0:
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if k%2 == 0:
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k =k/2
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was_even = True
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g = gcd(k,n)
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k_j = k/g
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a = n/g
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b = m/k_j
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if was_even == False:
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if a*2<n:
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a = a*2
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else:
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b = b*2
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print 'YES'
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print 0,0
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print a,0
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print 0,b
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else:
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print 'NO'
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```
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请完成上述谜题的训练场环境类实现，包括所有必要的方法。
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"""
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from bootcamp import Basebootcamp
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import re
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import random
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from math import gcd
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from bootcamp import Basebootcamp
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class Avasyaandtrianglebootcamp(Basebootcamp):
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def __init__(self, min_n=1, max_n=10**3, min_m=1, max_m=10**3, min_k=2, max_k=10**6, ensure_solvable=None):
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self.min_n = min_n
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self.max_n = max_n
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self.min_m = min_m
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self.max_m = max_m
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self.min_k = min_k
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self.max_k = max_k
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self.ensure_solvable = ensure_solvable
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def case_generator(self):
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max_attempts = 1000
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for _ in range(max_attempts):
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# 生成参数时根据 ensure_solvable 调整策略
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if self.ensure_solvable:
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n = random.randint(self.min_n, self.max_n)
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m = random.randint(self.min_m, self.max_m)
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a = random.randint(1, n)
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b = random.randint(1, m)
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numerator = 2 * n * m
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denominator = a * b
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if denominator == 0:
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continue
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if numerator % denominator != 0:
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continue
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k = numerator // denominator
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if k < self.min_k or k > self.max_k or k < 2:
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continue
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points = [(0, 0), (a, 0), (0, b)]
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valid = all(0 <= x <= n and 0 <= y <= m for x, y in points)
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if valid:
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return {
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'n': n,
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'm': m,
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'k': k,
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'solvable': True,
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'points': points
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}
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else:
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n = random.randint(self.min_n, self.max_n)
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m = random.randint(self.min_m, self.max_m)
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k = random.randint(self.min_k, self.max_k)
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solvable = (2 * n * m) % k == 0
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if self.ensure_solvable is not None and solvable != self.ensure_solvable:
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continue
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if not solvable:
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return {
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'n': n,
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'm': m,
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'k': k,
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'solvable': False,
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'points': None
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}
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was_even = False
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current_k = k
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if current_k % 2 == 0:
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current_k //= 2
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was_even = True
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g = gcd(current_k, n)
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k_j = current_k // g
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a = n // g
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b = m // k_j
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if not was_even:
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if 2 * a <= n:
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a *= 2
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else:
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b *= 2
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if b > m:
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continue
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points = [(0, 0), (a, 0), (0, b)]
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valid = all(0 <= x <= n and 0 <= y <= m for x, y in points)
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if valid:
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return {
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'n': n,
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'm': m,
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'k': k,
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'solvable': True,
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'points': points
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}
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# Fallback for ensure_solvable=True
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if self.ensure_solvable:
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n, m = self.min_n, self.min_m
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a, b = n, m
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k = (2 * n * m) // (a * b)
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while k < 2 or (2 * n * m) % (a * b) != 0 or k < self.min_k or k > self.max_k:
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a = random.randint(1, n)
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b = random.randint(1, m)
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k = (2 * n * m) // (a * b)
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points = [(0, 0), (a, 0), (0, b)]
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return {
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'n': n,
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'm': m,
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'k': k,
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'solvable': True,
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'points': points
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}
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# Fallback for other cases
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n, m, k = self.max_n, self.max_m, self.max_k
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while (2 * n * m) % k == 0:
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k = random.randint(self.min_k, self.max_k)
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return {
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'n': n,
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'm': m,
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'k': k,
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'solvable': False,
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'points': None
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}
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@staticmethod
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def prompt_func(question_case) -> str:
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n = question_case['n']
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m = question_case['m']
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k = question_case['k']
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target_expr = f"{n}×{m}/{k} = {n*m}/{k}"
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prompt = f"""Vasya has three integers n={n}, m={m}, and k={k}. He wants to find three integer points (x1, y1), (x2, y2), (x3, y3) such that:
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- All coordinates satisfy 0 ≤ xi ≤ {n} and 0 ≤ yi ≤ {m} for i = 1, 2, 3.
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- The area of the triangle formed by these points is exactly (n×m)/k = {target_expr}.
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Determine if such points exist. If yes, output "YES" followed by the coordinates. Otherwise, output "NO".
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Format your answer as:
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[answer]
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YES
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x1 y1
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x2 y2
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x3 y3
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[/answer]
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or
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[answer]
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NO
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[/answer]
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Place your final answer within [answer] tags."""
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return prompt
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@staticmethod
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def extract_output(output):
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answer_blocks = re.findall(r'\[answer\](.*?)\[/answer\]', output, flags=re.DOTALL | re.IGNORECASE)
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if not answer_blocks:
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return None
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last_answer = answer_blocks[-1].strip()
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lines = [line.strip() for line in last_answer.split('\n') if line.strip()]
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if not lines:
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return None
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if lines[0].upper() == 'NO':
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return 'NO'
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elif lines[0].upper() == 'YES' and len(lines) == 4:
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points = []
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for line in lines[1:4]:
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parts = line.split()
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if len(parts) != 2:
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return None
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try:
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x, y = int(parts[0]), int(parts[1])
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points.append((x, y))
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except ValueError:
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return None
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return points
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return None
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@classmethod
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def _verify_correction(cls, solution, identity):
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n = identity['n']
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m = identity['m']
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k = identity['k']
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solvable = identity['solvable']
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if not solvable:
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return isinstance(solution, str) and solution.upper() == 'NO'
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if solution == 'NO':
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return False
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if not isinstance(solution, list) or len(solution) != 3:
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return False
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for x, y in solution:
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if not (0 <= x <= n and 0 <= y <= m):
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return False
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x1, y1 = solution[0]
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x2, y2 = solution[1]
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x3, y3 = solution[2]
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area_twice = abs((x2 - x1) * (y3 - y1) - (x3 - x1) * (y2 - y1))
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expected_twice_area = (2 * n * m) // k
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return area_twice == expected_twice_area
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