Update Automated MNLP evaluation report (2026-06-10)

EVAL_REPORT.md

@@ -2,7 +2,7 @@
 
 - **Model repo:** [`cs-552-2026-llmfao/math_model`](https://huggingface.co/cs-552-2026-llmfao/math_model)
 - **Owner(s):** group **llmfao**
-- **Generated at:** 2026-06-09T08:56:40+00:00 (UTC)
+- **Generated at:** 2026-06-10T21:20:55+00:00 (UTC)
 - **Pipeline:** [mnlp-project-ci](https://github.com/eric11eca/mnlp-project-ci)
 
 _This PR is opened automatically by the course CI. It is **non-blocking** — you do not need to merge it. The next nightly run will refresh this file._
@@ -17,7 +17,7 @@ _This PR is opened automatically by the course CI. It is **non-blocking** — yo
 
 | Benchmark | Accuracy | Status |
 |---|---:|---|
-| Math | 0.4100 | ok |
+| Math | 0.4000 | ok |
 | Knowledge | — | not run |
 | Multilingual | — | not run |
 | Safety | — | not run |
@@ -30,81 +30,85 @@ _Prompts are intentionally omitted to avoid revealing benchmark contents. For mu
 
 **Correct** (1 shown)
 
-- **reference**: `(3,2,5)`
-- **overall** (8/8 completions correct)
-- **extracted** (✓): `(3, 2, 5)`
+- **reference**: `501`
+- **overall** (2/8 completions correct)
+- **extracted** (✓): `501`
 - **completion**:
 
     ```text
     <think>
-    Okay, so I need to find all ordered triples (p, m, n) where p is a prime greater than 2, m and n are positive integers with m ≤ n, and both m^{p-1} + n and n^{p-1} + m are powers of p. Hmm, let's break this down step by step.
+    Okay, so I need to figure out how many integers k between -500 and 500 inclusive make the equation log(kx) = 2 log(x + 2) have exactly one real solution. Hmm, let's start by recalling some logarithm properties. 
     
-    First, let's note that p is a prime greater than 2, so p is an odd prime. That might be useful because powers of p are also odd when p is odd. Also, m and n are posi
+    First, the equation given is log(kx) = 2 log(x + 2). I know that 2 log(x + 2) can be rewritten using logarithm rules as log((x + 2)^2). So the equation becomes log(kx) = log((x +
     
-    ... [truncated 823 chars] ...
+    ... [truncated 822 chars] ...
+    
+    2) \implies kx = (x + 2)^2
+    \]
+    
+    This quadratic equation is \(x^2 + (4 - k)x + 4 = 0\). For this equation to have exactly one real solution, the discriminant must be zero. The discriminant \(D\) is given by:
+    
+    \[
+    D = (4 - k)^2 - 16
+    \]
     
-    nd positive integers \( m \) and \( n \) with \( m \leq n \), we need to find all ordered triples \((p, m, n)\) such that \( m^{p-1} + n \) and \( n^{p-1} + m \) are both powers of \( p \).
+    Setting \(D = 0\), we solve:
     
-    1. **Prime Consideration**: \( p \) is a prime greater than 2, so \( p \) is odd. This implies that \( p^a \) is also odd.
-    2. **Initial Checks**: We start by checking small values of \( m \) and \( n \). For \( m = 2 \) and \( n = 5 \) with \( p = 3 \):
-       - \( m^{p-1} + n = 2^2 + 5 = 4 + 5 = 9 = 3^2 \)
-       - \( n^{p-1} + m = 5^2 + 2 = 25 + 2 = 27 = 3^3 \)
-       - Both results are powers of 3, and \( m = 2 \leq n = 5 \).
+    \[
+    (4 - k)^2 - 16 = 0 \implies (4 - k)^2 = 16 \implies 4 - k = \pm 4 \implies k = 0 \text{ or } k = 8
+    \]
+    
+    However, \(k = 0\) is invalid because \(\log(0x)\) is undefined. Thus, \(k = 8\) is valid. 
+    
+    Next, we check for \(k < 0\) values. For \(k < 0\), the quadratic equation \(x^2 + (4 - k)x + 4 = 0\) has two real roots. Since \(k < 0\), \(x\) must be in \((-2, 0)\). Evaluating the quadratic function at \(x = -2\) and \(x = 0\) confirms the function changes sign, ensuring exactly one root in \((-2, 0)\).
     
-    3. **Verification for Other Primes**:
-       - For \( p = 5 \), checking \( m = 2 \) and \( n = 109 \) (resulting in \( 109^4 + 2 \) which is not a power of 5).
-       - For \( p = 7 \), checking \( m = 2 \) and \( n = 2337 \) (resulting in \( 2337^6 + 2 \) which is not a power of 7).
-       - Checking other values of \( m \) and \( n \) for primes \( p \geq 5 \) also fails to produce valid solutions.
+    For \(k > 8\), the quadratic equation has two positive roots, leading to two solutions, which are both valid. Thus, these values of \(k\) are excluded.
     
-    4. **Conclusion**: After extensive checks, the only solution that satisfies all conditions is when \( p = 3 \), \( m = 2 \), and \( n = 5 \).
+    Therefore, the valid integers \(k\) are all negative integers from \(-500\) to \(-1\) inclusive, plus \(k = 8\). The count of negative integers from \(-500\) to \(-1\) is 500, and adding \(k = 8\) gives:
     
-    Thus, the ordered triple is \(\boxed{(3, 2, 5)}\).
+    \[
+    500 + 1 = 501
+    \]
+    
+    The number of integers \(k\) is \(\boxed{501}\).
     ```
 
 **Incorrect** (1 shown)
 
-- **reference**: `$g(x)=c, g(x)=\lceil x \rceil, g(x)=\lfloor x \rfloor$`
+- **reference**: `756`
 - **overall** (0/8 completions correct)
-- **extracted** (✗): `h(x) = c`
+- **extracted** (✗): `240`
 - **completion**:
 
     ```text
     <think>
-    Okay, let's try to figure out this problem. We need to find all functions h from the rationals to the integers that satisfy the equation for every x, a, and b:
-    
-    h(x) = h((h(bx - a) + a)/b)
+    Okay, so I need to solve this problem about three spherical balloons with radii 11, 13, and 19. Each pair of balloons is externally tangent, which means the distance between their centers is equal to the sum of their radii. Then there's a plane that cuts each balloon, creating three congruent circular cross-sections. The centers of these cross-sections are points X, Y, and Z, which are all
     
-    Hmm, that looks a bit complicated. Let me break it down. The function h is defined on the rationals, and for any rational x, any integer a, and any positive integer b, the value of h at x must equal h e
-    
-    ... [truncated 822 chars] ...
-    
-    ermine all mappings \( h: \mathbb{Q} \to \mathbb{Z} \) that satisfy the given equation for every \( x \in \mathbb{Q} \), \( a \in \mathbb{Z} \), and every positive integer \( b \):
-    
-    \[
-    h(x) = h\left( \frac{h(bx - a) + a}{b} \right)
-    \]
-    
-    we start by analyzing the equation. First, we test specific values for \( a \) and \( b \). Setting \( a = 1 \) and \( b = 1 \), the equation becomes:
-    
-    \[
-    h(x) = h(h(x - 1) + 1)
-    \]
-    
-    This recursive relation suggests a potential structure for \( h \). We then consider the recursive relation for \( h(x + 1) \):
+    ... [truncated 823 chars] ...
     
-    \[
-    h(x + 1) = h(h(x) + 1)
-    \]
+    ^2 - k^2} = \sqrt{19^2 - m^2}\).
     
-    Assuming \( h \) is non-constant leads to contradictions. For instance, if \( h \) is non-constant, there exist \( x_0 \) and \( x_1 \) such that \( h(x_0) \neq h(x_1) \). This implies \( h(h(x_0) + 1) \neq h(x_1) \), contradicting the original equation. Therefore, \( h \) must be constant.
+    3. **Set up equations for the radii:**
+       - \(11^2 - h^2 = 13^2 - k^2\) implies \(k^2 - h^2 = 48\).
+       - \(13^2 - k^2 = 19^2 - m^2\) implies \(m^2 - k^2 = 192\).
     
-    Testing constant functions \( h(x) = c \) (where \( c \in \mathbb{Z} \)) satisfies the equation trivially. Non-constant functions fail to satisfy the equation due to contradictions in the recursive relations.
+    4. **Solve for \(m^2\) in terms of \(h^2\):**
+       - From \(k^2 - h^2 = 48\), we get \(k^2 = h^2 + 48\).
+       - Substitute \(k^2\) into \(m^2 - k^2 = 192\): \(m^2 = h^2 + 48 + 192 = h^2 + 240\).
     
-    Thus, the only solutions are constant functions. The final answer is:
+    5. **Calculate the distance between the centers of the balloons:**
+       - The distance between \(A\) and \(B\) is 24, which is the hypotenuse of a right triangle with legs \(h - k\) and 560. This gives:
+         \[
+         (h - k)^2 + 560^2 = 24^2
+         \]
+       - This equation is impossible because \((h - k)^2\) must be non-negative, leading to a contradiction. However, considering the problem's constraints, we proceed with the equations derived from the radii.
     
-    \[
-    \boxed{h(x) = c}
-    \]
+    6. **Determine \(XZ^2\) using the distances:**
+       - Using the distances \(h\), \(k\), and \(m\) and the congruent cross-sections, we find \(XZ^2\) by solving the system of equations:
+         \[
+         (h - m)^2 + D^2 = 32^2
+         \]
+       - Substituting \(m^2 = h^2 + 240\) and solving for \(D^2\), we find \(D^2 = 240\).
     
-    for some integer \( c \).
+    Thus, the final answer is \(\boxed{240}\).
     ```