[tab] evaluation/gpt-5-verfict-easy-no-answer.out (plaintext)

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Processing file: evaluation/easy_problems.jsonl
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Entry 1:
Problem:
 Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.

The probability of snow on 1st January is 1/20. What is the probability that snow will fall on the next 1st January?
OpenAI GPT-5 Difficulty verdict: \boxed{\frac{1}{20}}

Difficulty: 1/10 — It directly uses the given probability for any Jan 1 with no additional computation or conditions.
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Entry 2:
Problem:
 Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.

When orange trees are planted to the acre, each tree yields 100 pounds of fruit. For each additional tree over 100, the yield per tree is reduced by 1 pound. How many trees should be planted so that the yield per acre is 4816 pounds?
OpenAI GPT-5 Difficulty verdict: \boxed{172}
Difficulty: 3/10 — It’s a straightforward quadratic setup from a yield model and selecting the feasible root.
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Entry 3:
Problem:
 Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.

Find the smallest prime number \( p \) such that \( p \) divides \( n^2 + 7n + 23 \).
OpenAI GPT-5 Difficulty verdict: \boxed{11}
Difficulty: 3/10 — Quick modular checks show 2,3,5,7 fail and 11 works (e.g., n=7 gives 49+49+23=121 divisible by 11).
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Entry 4:
Problem:
 Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.

Is there a geometric transformation that maps every point to exactly one point such that any triangle can be transformed into another triangle with the same area? (Alternatively, can any triangle be transformed into a regular triangle with the same area?)
OpenAI GPT-5 Difficulty verdict: Yes. Any two nondegenerate triangles are related by a unique affine transformation sending corresponding vertices to each other; its determinant equals the area ratio, so if the areas are equal the map is area-preserving (|det|=1). Thus any triangle can be mapped to any other triangle of the same area, in particular to an equilateral triangle of the same area.

\boxed{Yes}

Difficulty: 4/10
One-liner: It’s a direct application of affine geometry—map three vertices to three vertices and use det=1 to preserve area.
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Entry 5:
Problem:
 Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.

How many natural numbers greater than 6 but less than 60 are relatively prime to 15?
OpenAI GPT-5 Difficulty verdict: \boxed{29}
Difficulty: 3/10 — It’s a straightforward application of inclusion–exclusion to count numbers not divisible by 3 or 5 in a short interval.
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Entry 6:
Problem:
 Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.

Resolve the expression $\frac{x^5 + x^4 - x^3 + 2x^2 + x}{(x-1)^2 (x^2 + 1)^2}$.
OpenAI GPT-5 Difficulty verdict: \boxed{\frac{3}{4(x-1)}+\frac{1}{(x-1)^2}+\frac{x+5}{4(x^2+1)}-\frac{x+3}{2(x^2+1)^2}}

Difficulty: 5/10
Why: It’s a standard partial fraction decomposition with repeated linear and quadratic factors, requiring a routine but somewhat lengthy system of equations.
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Entry 7:
Problem:
 Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.

What is the total volume in cubic feet of three boxes if each box is a cube with edge length 4 feet?
OpenAI GPT-5 Difficulty verdict: \boxed{192}

Difficulty: 1/10 — straightforward cube volume (4^3) and simple multiplication by 3.
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