Problem Analysis #common_06

Motif 3 · Run 1

Python ● execution succeeded

Correct

21.57s

6,635 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

1보다 큰 두 실수 $a,b$ 가 $\log_a b=3$ , $\log_3\frac ba=\frac12$ 를 만족시킬 때, $\log_9(ab)$ 의 값은?

① $\frac38$ ② $\frac12$ ③ $\frac58$ ④ $\frac34$ ⑤ $\frac78$

최종 답은 선택지 번호 1~5 중 하나를 $\boxed{}$ 안에 쓰세요.

Ground Truth

Recorded model response / solution

Reasoning Process

Recorded Python call:

import sympy as sp

a = sp.symbols('a', positive=True, real=True)
# b = a^3 from log_a b = 3
# b/a = sqrt(3) from log_3(b/a)=1/2
# => a^3/a = a^2 = sqrt(3)
sol_a = sp.solve(a**2 - sp.sqrt(3), a)
print("possible a:", sol_a)

a_val = sol_a[1]  # positive root > 1
b_val = a_val**3
ab = a_val * b_val
print("a:", a_val)
print("b:", b_val)
print("ab:", sp.simplify(ab))
print("log_9(ab):", sp.simplify(sp.log(ab, 9)))

Tool Output

Execution failed possible a: [3**(1/4)] list index out of range

Reasoning Process

Recorded Python call:

a_val = 3**(sp.Rational(1,4))
b_val = a_val**3
ab = a_val * b_val
print("ab simplified:", sp.simplify(ab))
print("log_9(ab):", sp.simplify(sp.log(ab, 9)))

Tool Output

Execution failed name 'sp' is not defined

Reasoning Process

Recorded Python call:

import sympy as sp
a_val = 3**(sp.Rational(1,4))
b_val = a_val**3
ab = a_val * b_val
print("ab simplified:", sp.simplify(ab))
print("log_9(ab):", sp.simplify(sp.log(ab, 9)))

Tool Output

Execution succeeded ab simplified: 3 log_9(ab): 1/2

Reasoning Process

주어진 조건에서

\log_a b=3 \implies b=a^3

또

\log_3\frac ba=\frac12 \implies \frac ba=3^{1/2}=\sqrt3

이므로

\frac{a^3}{a}=a^2=\sqrt3

따라서

a=3^{1/4},\qquad b=a^3=3^{3/4}

이고,

ab=a^4=(3^{1/4})^4=3

따라서

\log_9(ab)=\log_9 3=\frac12

Python 검증:

import sympy as sp
a = 3**sp.Rational(1,4)
b = a**3
sp.simplify(sp.log(a*b, 9))  # 1/2

정답은 ②번이다.

\boxed{2}

Final Model Output

Comparative Analysis (Pcommon_06) (Average across runs)

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

Avg Time Taken

Union Alpha

87.6

Motif 3

59.6

Grok 4.7

38.9

MiMo V2.6 Pro

19.9

Muse Spark 1.3

11.5

Solar Pro 4

8.7

DeepSeek V4.1 Flash

4.2

Solar Mini 4 Preview

3.4

Avg Token Usage

Muse Spark 1.3

6,137

Grok 4.7

6,067

Union Alpha

5,169

Motif 3

4,942

MiMo V2.6 Pro

2,646

DeepSeek V4.1 Flash

2,595

Solar Mini 4 Preview

2,496

Solar Pro 4

1,372