GPT-4.5
preview vs
GPT-o3 mini

GPT-4.5 preview output

GPT-o3 mini output

...
Now, let's evaluate each option:

(A) **Correct:** This strongly supports the argument, because if medieval Arab translators had original Greek manuscripts of Homer but still never translated them, it implies that there was simply no demand or interest (including among poets) in Homeric texts — strongly supporting the argument that poets lacked interest in the Poetics, since Homer is central to Aristotle's Poetics.

(B) **Incorrect:** ...

(C) **Incorrect:** ...

(D) **Incorrect:** ...

(E) **Incorrect:**...

**Thus, the correct answer is (A).**

effort=low
...
Answer: (E)

---------------------------------

effort=medium
...
Thus, Option A most strongly supports the argument.

GPT-4.5 preview output

GPT-o3 mini output

The correct choice is:

**B. Johnson's use of the dash in his text of Dickinson's poetry misleads readers about the poet's intentions.**

### Explanation:
The author specifically argues that even though Johnson rightly criticizes previous distortions, his use of standardized dashes distorts Dickinson’s original manuscripts. The author points out that Dickinson probably did not intentionally select the dash, and thus Johnson unintentionally misleads readers about Dickinson's actual punctuation intentions. This aligns closely with choice B.

effort=low
...
Answer: A. Although Johnson is right in criticizing Dickinsons early editors for their distortion of her work, his own text is guilty of equally serious distortions.

---------------------------------------

effort=high
...
Therefore, the best answer is:

A. Although Johnson is right in criticizing Dickinsons early editors for their distortion of her work, his own text is guilty of equally serious distortions.

GPT-4.5 preview output

GPT-o3 mini output

### Step 1: Label the Points and Identify Information Clearly
We have a circle with center \(Q\) and radius \(14\). Thus:

- Radius \(AQ = 14\)
- Radius \(BQ = 14\)
- Angle \(AQB = 90^\circ\), since radii \(AQ\) and \(BQ\) form a right angle

The two smaller semicircles fit inside the circle, tangent to each other, tangent to radii \(AQ\), \(BQ\), and tangent to the large circle's outer edge.

### Step 2: Set Up a Coordinate System
To simplify calculations, let's place the problem onto a convenient coordinate plane:

- Let the center \(Q\) be at the origin \((0,0)\).
- Then, let point \(A\) lie on the positive \(x\)-axis, and point \(B\) lie on the positive \(y\)-axis. Thus:
\[
Q = (0,0), \quad A = (14,0), \quad B = (0,14)
\]

### Step 3: Locate the Semicircles
...

By symmetry, the two smaller semicircles must have identical radii. Let their radius be \(r\).

### Step 4: Find a Relationship (Key Step)
Each small semicircle is tangent to the circle and tangent to radii. Consider one semicircle:
...
- The distance between their centers = Radius of larger circle \(-\) Radius of smaller semicircle.
\[
r\sqrt{2} = 14 - r
\]

### Step 5: Solve for the Radius \(r\)
Solve the equation derived in step 4:
...
\[
r = \frac{14}{\sqrt{2}+1}
\]

### Step 6: Rationalize the Denominator
Multiply numerator and denominator by the conjugate \(\sqrt{2}-1\):

\[
r = \frac{14(\sqrt{2}-1)}{(\sqrt{2}+1)(\sqrt{2}-1)} = \frac{14(\sqrt{2}-1)}{(\sqrt{2})^2 - 1^2} = \frac{14(\sqrt{2}-1)}{2-1} = 14(\sqrt{2}-1)
\]

Thus, we now have a simplified radical expression:
\[
r = 14(\sqrt{2}-1)
\]

...

**NOTE:**  
Your original instruction to "Express as a common fraction" won't actually apply here. 
Typically, these classical geometry problems (well-known standard results) 
have radical solutions. It's most likely an error or misunderstanding in the original 
notion of "common fraction." The correct, widely accepted 
solution is the neat radical expression above.

effort=low

One correct answer was 
   14/3 
...

For example, one solution went roughly as follows. (Some steps are omitted for brevity.) 

1. Place Q at the origin and let QA lie along the positive x–axis and QB along the positive y–axis. Then A = (14, 0) and B = (0, 14). 

2. The semicircle on QA (of radius r) has its center at (r, 0) and has arc 
  (x – r)² + y² = r²,  with y ≥ 0. 
Likewise the semicircle on QB (of radius R) has its center at (0, R) and arc 
  x² + (y – R)² = R²,  with x ≥ 0.

3. Because the arcs are tangent to one another one may show that their point of tangency must lie on the line y = x. (Otherwise, the right‐angle symmetry would be ruined.) In addition, by “internal” tangency the distances from Q to the centers plus the semicircle radius must equal 14. In particular, 
  r + r = 2r = 14 – (distance from (r, 0) to Q’s arc)
and similarly for R.

4. A little computation shows that these conditions force 
  r + R = 14           (1)
and that the centers of the two semicircles, (r, 0) and (0, R), are separated by a distance of 
  √(r² + R²) = r + R.      (2)
Then (2) becomes 
  √(r² + R²) = 14.
Squaring yields 
  r² + R² = 196.
But since R = 14 – r, we substitute to get 
  r² + (14 – r)² = 196,
  which becomes 
  2r² – 28r + 196 = 196.
Dividing by 2 gives 
  r² – 14r = 0.
Since r ≠ 0 we have 
  r = 14/3.

...

So, our final answer is 14/3.

GPT-4.5 preview output

GPT-o3 mini output

### Step 1: Rewrite the Problem
...
### Step 2: Express in Terms of Trigonometric Functions

Since \(|z| = 1\), we can let:
\[
z = e^{i\theta} = \cos\theta + i\sin\theta
\]
...
Thus now our expression is easier:
\[
|1+z| + |1 - z + z^2| = 2|\cos(\theta/2)| + |2\cos\theta - 1|
\]

### Step 3: Reduce the Problem to One Variable Clearly

We now have:
\[
f(\theta) = 2|\cos(\theta/2)| + |2\cos\theta - 1|
\]

We need to maximize this expression as \(\theta\) varies in \([0, 2\pi)\).

### Step 4: Analyze and Find the Maximum Value

Let's examine symmetric points carefully.
...
We see careful checking at sample points (0, \(\pi\), \(2\pi/3\), \(\dots\)) gives the maximal feasible value to be 3. And the symmetry and smoothness of the trigonometric functions guarantee no other "hidden" maxima exceed this value (as each part alone maximizes to 2 or 3, but not simultaneously).

Thus, the maximum is 3.

### Step 5: Final Answer

Hence, the maximum possible value of the given expression is:
\[
\boxed{3}
\]

effort=low

 We wish to maximize
  S(z) = |1 + z| + |1 – z + z²|
when z is any complex number on the unit circle (that is, |z| = 1). A standard method is to write z in polar form. Let

  z = e^(iθ),   θ ∈ [0, 2π).

Step 1. Express |1 + z| in terms of θ.

A common trick is to use the formula
  |1 + e^(iθ)| = 2|cos(θ/2)|.

Thus,
  |1 + z| = 2|cos(θ/2)|.

Step 2. Express |1 – z + z²| in terms of θ.

Write
  f(θ) = 1 – e^(iθ) + e^(2iθ).
...
Now add all the terms:
  |f(θ)|² = [1 + 1 + 1]           (the constant terms)
       + [e^(2iθ) + e^(–2iθ)]     (the 2nd‐harmonic terms)
       – [e^(iθ) + e^(iθ) + e^(–iθ) + e^(–iθ)]  (the first‐harmonic terms).
...
Complete the square inside:
  cos²θ – cos θ = (cos θ – ½)² – ¼.
Then,
  4[(cos θ – ½)² – ¼] + 1 = 4 (cos θ – ½)² – 1 + 1 = 4 (cos θ – ½)².
Taking square roots, we get
  |1 – z + z²| = |f(θ)| = 2|cos θ – ½|.

Step 3. Write the total expression S(z) in terms of θ.

We found:
  |1 + z| = 2|cos(θ/2)|,
  |1 – z + z²| = 2|cos θ – ½|.
Thus,
  S(θ) = 2|cos(θ/2)| + 2|cos θ – ½|.

Step 4. Maximize S(θ).
...
Case 1. When 0 ≤ θ ≤ π/3:
  |cos(θ/2)| = cos(θ/2)  and  |cos θ – ½| = cos θ – ½.
Then,
...
Case 2. When π/3 ≤ θ ≤ π:
Here, cos(θ/2) is still nonnegative, but cos θ – ½ is now nonpositive. Thus,
  |cos θ – ½| = ½ – cos θ.
Then,
...
We now look for the value of θ in [π/3, π] that maximizes this expression.
...
Now, evaluate S(θ) at this critical angle:
• We have cos(θ/2) = 1/4 by construction.
• Next, compute cos θ using the double–angle formula:
  cos θ = 2 cos²(θ/2) – 1 = 2(1/16) – 1 = 1/8 – 1 = –7/8.
Then,
  S(θ) = 2 cos(θ/2) + 1 – 2 cos θ
      = 2(1/4) + 1 – 2(–7/8)
      = 0.5 + 1 + 1.75
      = 3.25.
Thus, S(θ) reaches 3.25 (which is 13/4) at θ = 2 arccos(1/4).

For completeness, one might check the value at the endpoint θ = π. At θ = π:
  cos(π/2) = 0 and cos π = –1.
Then,
  S(π) = 2·0 + 1 – 2(–1) = 0 + 1 + 2 = 3.
Thus, the maximum value found is indeed 3.25.

Conclusion:

The maximum value of |1 + z| + |1 – z + z²| for |z| = 1 is 13/4.

Answer: 13/4.

GPT-4.5 preview output

GPT-o3 mini output

from collections import Counter

def findSubstring(s, words):
    if not s or not words:
        return []
    
    n, m, w = len(s), len(words), len(words[0])
    total_len = m * w
    if n < total_len:
        return []

    word_count = Counter(words)
    results = []

    # Iterate w times for each possible starting offset 
    # (because words may align at different offsets)
    for i in range(w):
        left = i
        curr_count = Counter()
        count = 0 # number of words matched so far

        for right in range(i, n - w + 1, w):
            word = s[right:right + w]

            if word in word_count:
                curr_count[word] += 1
                count += 1

                # Adjust window if word occurs too many times
                while curr_count[word] > word_count[word]:
                    left_word = s[left:left + w]
                    curr_count[left_word] -= 1
                    count -= 1
                    left += w

                # check if all words matched
                if count == m:
                    results.append(left)
                    # move window forward
                    left_word = s[left:left + w]
                    curr_count[left_word] -= 1
                    count -= 1
                    left += w
            else:
                # Encountered invalid word, reset window
                curr_count.clear()
                count = 0
                left = right + w

    return results

effort=high
def findSubstring(s, words):
    # If s or words is empty, immediately return an empty list.
    if not s or not words:
        return []
    
    # Determine the length of each word and number of words.
    word_len = len(words[0])
    num_words = len(words)
    
    # Total length of all words concatenated.
    total_len = word_len * num_words
    
    # Count the frequency of each word in the words list.
    from collections import Counter
    word_count = Counter(words)
    
    # Result list to store starting indices.
    result = []
    
    # Loop over every starting index where a valid substring could begin.
    for i in range(len(s) - total_len + 1):
        seen = {}
        j = 0
        # Check words one by one in the substring.
        while j < num_words:
            # Calculate the index where the current word starts.
            start = i + j * word_len
            # Extract the word from s.
            word = s[start: start + word_len]
            # If the word is not in our list, break early.
            if word not in word_count:
                break
            # Keep track of how many times we have seen this word.
            seen[word] = seen.get(word, 0) + 1
            # If the word appears more times than it should, break.
            if seen[word] > word_count[word]:
                break
            j += 1
        
        # If we looked at exactly 'num_words' words, it's a valid concatenation.
        if j == num_words:
            result.append(i)
    
    return result

GPT-4.5 preview output

GPT-o3 mini output