Editing Openai/6926a9e8-c270-8004-89f7-c49d6ee67c80 (section)

=== Below is a point-by-point mapping. ===

==== Prompt term: ====

aiai2+ai+12+⋯+ai+k2\frac{a_i}{\sqrt{a_i^2 + a_{i+1}^2+\cdots + a_{i+k}^2}}ai2+ai+12+⋯+ai+k2ai
Model used this directly to say:

=11+∑j=1k(ai+j/ai)2.= \frac{1}{\sqrt{1 + \sum_{j=1}^k (a_{i+j}/a_i)^2}}.=1+∑j=1k(ai+j/ai)21.
How:
* Uses exact formula from the prompt
* Performs an algebraic transformation to simplify and analyze maxima

==== Prompt: ====

: 

Model:
* repeatedly invokes this to justify searching for the maximum of S over all possible inputs
* correctly interprets λ as supremum of S

How:
By analyzing worst-case sequences, trying to push the sum toward 1 per term, discussing scaling and normalization.

==== Prompt: ====

an+i=ai.a_{n+i} = a_i.an+i=ai.
Model:
* discusses how “the problem is cyclic”
* analyzes how a small value in one place affects many denominators due to cyclic shifts
* uses cyclicity to argue you cannot make all terms ≈ 1 because indices wrap around

How:
In the text:

: 

This is a direct use of the prompt’s modulo-n definition.

==== Prompt: ====

: 

Model:
* acknowledges this inequality
* uses it to comment on size relations between n and √k
* but does not use it in a deep extremal argument

How:
In the passage:

: 

So yes, it uses the information, though not strategically.

==== The partial response sets up: ====
* Di=…D_i = \dotsDi=…
* S=∑ai/DiS = \sum a_i/\sqrt{D_i}S=∑ai/Di
* “We attempt to maximize S”.

The model continues exactly this approach—trying to maximize S using inequalities, normalization, extremal sequences.

How:
It explicitly continues the same line of thought — analyzing special sequences, trying equal sequences, exploring extremal behavior.