Problem Analysis #5

Gemini-3-Pro-Preview · Run 1

30.46s
5,708 toks
Problem Statement

Find

\na6+b6+c6+d6+e6\na^6 + b^6 + c^6 + d^6 + e^6

\nfor natural numbers a,b,c,d,ea,b,c,d,e satisfying

{\na+b+c+d+e=231\na2+b2+c2+d2+e2=14003\na3+b3+c3+d3+e3=997227\na4+b4+c4+d4+e4=77752499\na5+b5+c5+d5+e5=6386215731\begin{cases}\na + b + c + d + e = 231 \\\na^{2} + b^{2} + c^{2} + d^{2} + e^{2} = 14003 \\\na^{3} + b^{3} + c^{3} + d^{3} + e^{3} = 997227 \\\na^{4} + b^{4} + c^{4} + d^{4} + e^{4} = 77752499 \\\na^{5} + b^{5} + c^{5} + d^{5} + e^{5} = 6386215731 \end{cases}
Ground Truth
540937826363
Execution Trace
Reasoning Process

[PYTHON] S1 = 231 S2 = 14003 S3 = 997227 S4 = 77752499 S5 = 6386215731

Calculate elementary symmetric polynomials e1, e2, e3, e4, e5

e1 = S1

e1 = S1

2e2 = e1*S1 - S2

e2 = (e1 * S1 - S2) // 2

3e3 = e2S1 - e1S2 + S3

e3 = (e2 * S1 - e1 * S2 + S3) // 3

4e4 = e3S1 - e2S2 + e1*S3 - S4

e4 = (e3 * S1 - e2 * S2 + e1 * S3 - S4) // 4

5e5 = e4S1 - e3S2 + e2S3 - e1S4 + S5

e5 = (e4 * S1 - e3 * S2 + e2 * S3 - e1 * S4 + S5) // 5

print(f"{e1=}") print(f"{e2=}") print(f"{e3=}") print(f"{e4=}") print(f"{e5=}")

Calculate S6

S6 = e1S5 - e2S4 + e3S3 - e4S2 + e5*S1

S6 = e1 * S5 - e2 * S4 + e3 * S3 - e4 * S2 + e5 * S1 print(f"{S6=}") [/PYTHON]

Tool Output
[PYTHON OUTPUT] e1=231 e2=19679 e3=769461 e4=13696848 e5=87805620 S6=540937826363 [/PYTHON OUTPUT]
Reasoning Process

The elementary symmetric polynomials e1,e2,e3,e4,e5e_1, e_2, e_3, e_4, e_5 for the variables a,b,c,d,ea, b, c, d, e can be determined from the power sums Sk=ak+bk+ck+dk+ekS_k = a^k + b^k + c^k + d^k + e^k using Newton's sums. The relations are:

e_1 &= S_1 \\ 2e_2 &= e_1 S_1 - S_2 \\ 3e_3 &= e_2 S_1 - e_1 S_2 + S_3 \\ 4e_4 &= e_3 S_1 - e_2 S_2 + e_1 S_3 - S_4 \\ 5e_5 &= e_4 S_1 - e_3 S_2 + e_2 S_3 - e_1 S_4 + S_5 \end{align*}$$ Using the given values: $S_1 = 231$ $S_2 = 14003$ $S_3 = 997227$ $S_4 = 77752499$ $S_5 = 6386215731$ The calculated elementary symmetric polynomials are: $e_1 = 231$ $e_2 = 19679$ $e_3 = 769461$ $e_4 = 13696848$ $e_5 = 87805620$ For $n=5$ variables, the Newton sum for $S_6$ is given by:

S_6 - e_1 S_5 + e_2 S_4 - e_3 S_3 + e_4 S_2 - e_5 S_1 = 0

Thus, Thus,

S_6 = e_1 S_5 - e_2 S_4 + e_3 S_3 - e_4 S_2 + e_5 S_1

Substitutingthevaluesyields: Substituting the values yields:

S_6 = 540937826363

$$\boxed{540937826363}$$
Final Model Output
540937826363

Comparative Analysis (P5) (Average across runs)

Avg Time Taken

Solar-Open-100B
529.7
Deepseek-V3.2
55.2
Gemini-3-Pro-Preview
47.5
Grok-4.1-fast
46.2
K-EXAONE-236B-A23B
42.5
Claude-Opus-4.5
18.4
GPT-5.1 (high)
15.4

Avg Token Usage

Solar-Open-100B
20,191
Gemini-3-Pro-Preview
11,089
K-EXAONE-236B-A23B
10,356
Grok-4.1-fast
8,582
Claude-Opus-4.5
5,049
Deepseek-V3.2
4,802
GPT-5.1 (high)
3,897
    Open · Problem 5 · Gemini-3-Pro-Preview | EntropyMath