Problem Analysis #common_12

Motif 3 · Run 1

Python ● execution succeeded
Correct
97.43s
10,107 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

등비수열 {an}\{a_n\}이 2(a1+a4+a7)=a4+a7+a10=62(a_1+a_4+a_7)=a_4+a_7+a_{10}=6을 만족시킬 때, a10a_{10}의 값은?

① 227\frac{22}7 ② 247\frac{24}7 ③ 267\frac{26}7 ④ 307\frac{30}7 ⑤ 327\frac{32}7

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

Ground Truth
2
Recorded model response / solution
Reasoning Process

Recorded Python call:

from sympy import symbols, solve, simplify

a, r = symbols('a r')
S = 1 + r**3 + r**6

# Conditions:
# a*S = 3
# a*r^3*S = 6
solutions = solve([a*S - 3, a*r**3*S - 6], [a, r], dict=True)

real_solutions = [sol for sol in solutions if sol[r].is_real]

a10 = simplify(a*r**9)
for sol in solutions:
    print("a =", sol[a], "; r =", sol[r], "; a10 =", a10.subs(sol))
print()
for sol in real_solutions:
    print("real: a =", sol[a], "; r =", sol[r], "; a10 =", a10.subs(sol))
print("a10 value:", simplify(a10.subs({a: Rational(3,7), r**3: 2})))
Tool Output
Execution failed a = 3/7 ; r = 2**(1/3) ; a10 = 24/7 a = 3/7 ; r = -2**(1/3)/2 - 2**(1/3)*sqrt(3)*I/2 ; a10 = 3*(-2**(1/3)/2 - 2**(1/3)*sqrt(3)*I/2)**9/7 a = 3/7 ; r = -2**(1/3)/2 + 2**(1/3)*sqrt(3)*I/2 ; a10 = 3*(-2**(1/3)/2 + 2**(1/3)*sqrt(3)*I/2)**9/7 real: a = 3/7 ; r = 2**(1/3) ; a10 = 24/7 name 'Rational' is not defined
Reasoning Process

Recorded Python call:

from sympy import Rational
from sympy import symbols, solve, simplify

a, r = symbols('a r')
S = 1 + r**3 + r**6

solutions = solve([a*S - 3, a*r**3*S - 6], [a, r], dict=True)

# All solutions should give a10 = (3/7)*(r^3)^3 = (3/7)*8 = 24/7
a10_all = [simplify(a*r**9).subs(sol) for sol in solutions]
print(a10_all)
print(24/7)
Tool Output
Execution succeeded [24/7, 3*(-2**(1/3)/2 - 2**(1/3)*sqrt(3)*I/2)**9/7, 3*(-2**(1/3)/2 + 2**(1/3)*sqrt(3)*I/2)**9/7] 3.4285714285714284
Reasoning Process

Recorded Python call:

from sympy import Rational, symbols, simplify, root, cbrt
from sympy import I, sqrt

# All solutions satisfy r^3 = 2, so r^9 = (r^3)^3 = 8
a_val = Rational(3,7)
a10_val = a_val * (2**3)
print(a10_val)
Tool Output
Execution succeeded 24/7
Reasoning Process

등비수열의 일반항을 an=arn−1a_n=ar^{n-1}라 하자.

S=1+r3+r6S=1+r^3+r^6로 두면,

a1+a4+a7=aS,a4+a7+a10=ar3S.a_1+a_4+a_7=aS,\quad a_4+a_7+a_{10}=ar^3S.

주어진 조건에서

2aS=6⇒aS=3,2aS=6 \Rightarrow aS=3, ar3S=6.ar^3S=6.

두 식을 나누면

r3=2.r^3=2.

따라서

S=1+2+4=7,a=37.S=1+2+4=7,\quad a=\frac{3}{7}.

이제

a10=ar9=a(r3)3=37⋅23=247.a_{10}=ar^9=a(r^3)^3=\frac{3}{7}\cdot 2^3=\frac{24}{7}.

따라서 정답은 선택지 2이다.

2\boxed{2}
Final Model Output
2

Comparative Analysis (Pcommon_12) (Average across runs)

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

Avg Time Taken

Solar Pro 4
55.7
Union Alpha
50.7
Motif 3
46.2
MiMo V2.6 Pro
45.9
Grok 4.7
25.8
Muse Spark 1.3
12.0
DeepSeek V4.1 Flash
10.0
Solar Mini 4 Preview
7.2

Avg Token Usage

Motif 3
7,491
Grok 4.7
7,221
Solar Pro 4
7,167
MiMo V2.6 Pro
6,354
DeepSeek V4.1 Flash
5,101
Solar Mini 4 Preview
4,853
Muse Spark 1.3
4,348
Union Alpha
2,039
    CSAT 2026 · Problem common_12 · Motif 3 | EntropyMath