Problem Analysis #common_21

Motif 3 · Run 1

Python ● execution succeeded

Correct

620.02s

128,029 toks

Infron · Motif 3 · 46문항 × 3회 · Python TAR. 131,072 output tokens · 기본 1,800초 · 6+1 turns. ACC는 전체 138회, Pass@1은 각 문항의 첫 유효 시도, Pass@3는 3회 내 정답으로 계산합니다.

Problem Statement

최고차항의 계수가 양수인 삼차함수 $f(x)$ 와 실수 t에 대하여 함수 $g(x)=\begin{cases}-f(x)&(x<t)\\f(x)&(x\ge t)\end{cases}$ 는 실수 전체의 집합에서 연속이고 다음 조건을 만족시킨다. (가) 모든 실수 a에 대하여 $\lim_{x\to a+}\frac{g(x)}{x(x-2)}$ 의 값이 존재한다. (나) $\lim_{x\to m+}\frac{g(x)}{x(x-2)}$ 의 값이 음수가 되도록 하는 자연수 m의 집합은 $\{g(-1),-\frac72g(1)\}$ 이다. $g(-5)$ 의 값을 구하시오. (단, $g(-1)\ne-\frac72g(1)$ )

최종 정수 답을 $\boxed{}$ 안에 쓰세요.

Ground Truth

Recorded model response / solution

Reasoning Process

Recorded Python call:

import sympy as sp

# Define symbols
a, s = sp.symbols('a s', positive=True, real=True)
# We have a = 3/14, s = 11/3
# Let's verify the conditions and compute g(-5)

a_val = sp.Rational(3,14)
s_val = sp.Rational(11,3)
t_val = 2

# Define f(x)
x = sp.symbols('x')
f = a_val * x * (x-2) * (x - s_val)
sp.simplify(f)

Tool Output

Execution succeeded

Reasoning Process

Recorded Python call:

a_val = 3/14
s_val = 11/3

def f(x):
    return a_val * x * (x-2) * (x - s_val)

def g(x):
    if x < 2:
        return -f(x)
    else:
        return f(x)

# Check conditions
print("f(-5) =", f(-5))
print("g(-5) =", g(-5))
print("g(-1) =", g(-1))
print("g(1) =", g(1))
print("-7/2 g(1) =", -7/2 * g(1))

# Check limits
import math
def h(x):
    return g(x) / (x * (x-2))

# Check limits at m=1,2,3,4
for m in [1,2,3,4,5]:
    # right-hand limit
    eps = 1e-6
    if m == 0:
        pass
    else:
        val = h(m + eps)
        print(f"L_{m} = {val}")

Tool Output

Execution succeeded f(-5) = -65.0 g(-5) = 65.0 g(-1) = 2.999999999999999 g(1) = -0.5714285714285714 -7/2 g(1) = 2.0 L_1 = 0.5714283571428571 L_2 = -0.3571426428571428 L_3 = -0.1428569285714285 L_4 = 0.07142878571428576 L_5 = 0.2857145

Reasoning Process

Recorded Python call:

# Exact verification using rational arithmetic
from fractions import Fraction

a = Fraction(3,14)
s = Fraction(11,3)
t = 2

def f_exact(x):
    return a * x * (x-2) * (x - s)

def g_exact(x):
    if x < t:
        return -f_exact(x)
    else:
        return f_exact(x)

print("f(-5) =", f_exact(-5))
print("g(-5) =", g_exact(-5))
print("g(-1) =", g_exact(-1))
print("g(1) =", g_exact(1))
print("-7/2 g(1) =", -Fraction(7,2) * g_exact(1))

# Check limits exactly for m=1,2,3,4
for m in [1,2,3,4]:
    # h(m) = g(m)/(m(m-2))
    if m == 0 or m == 2:
        # limit from right
        # For m=2, limit is a*(2-s) since t=2 and x>2 => g(x)=f(x)
        val = a * (m - s)
    else:
        val = g_exact(m) / (m * (m-2))
    print(f"L_{m} = {val}")

Tool Output

Execution succeeded f(-5) = -65 g(-5) = 65 g(-1) = 3 g(1) = -4/7 -7/2 g(1) = 2 L_1 = 4/7 L_2 = -5/14 L_3 = -1/7 L_4 = 1/14

Reasoning Process

The solution is unique and verified. The function $f(x)=\frac{3}{14}x(x-2)(x-\frac{11}{3})$ with $t=2$ satisfies all conditions, giving $g(-5)=65$ .

$\boxed{65}$

Final Model Output

Comparative Analysis (Pcommon_21) (Average across runs)

Averages use reported measurements only. Missing usage is not treated as zero.

Avg Time Taken

Motif 3

1140.1

MiMo V2.6 Pro

651.3

Solar Pro 4

583.2

Grok 4.7

207.1

DeepSeek V4.1 Flash

84.4

Muse Spark 1.3

81.4

Solar Mini 4 Preview

78.2

Union Alpha

0.5

Avg Token Usage

Motif 3

126,933

MiMo V2.6 Pro

75,681

Solar Pro 4

53,446

Grok 4.7

48,743

DeepSeek V4.1 Flash

45,275

Muse Spark 1.3

30,851

Solar Mini 4 Preview

16,968