Update Automated MNLP evaluation report (2026-05-19)

EVAL_REPORT.md

@@ -2,7 +2,7 @@
 
 - **Model repo:** [`cs-552-2026-bilko/math_model`](https://huggingface.co/cs-552-2026-bilko/math_model)
 - **Owner(s):** group **bilko**
-- **Generated at:** 2026-05-18T02:30:25+00:00 (UTC)
+- **Generated at:** 2026-05-19T03:22:52+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._
@@ -11,7 +11,7 @@ _This PR is opened automatically by the course CI. It is **non-blocking** — yo
 
 | Benchmark | Accuracy | Status |
 |---|---:|---|
-| Math | 0.3600 | ok |
+| Math | 0.3100 | ok |
 | Knowledge | — | not run |
 | Multilingual | — | not run |
 | Safety | — | not run |
@@ -24,82 +24,114 @@ _Prompts are intentionally omitted to avoid revealing benchmark contents. For mu
 
 **Correct** (1 shown)
 
-- **reference**: `194`
-- **overall** (3/8 completions correct)
-- **extracted** (✓): `194`
+- **reference**: `4`
+- **overall** (1/8 completions correct)
+- **extracted** (✓): `4`
 - **completion**:
 
     ```text
-    <think>
-    1. **Understanding the problem**: We need to find the smallest positive integer \( k \) such that we can select \( k \) cells on a \( 194 \times 194 \) board, ensuring that the board can be tiled completely by \( 1 \times 2 \) and \( 2 \times 1 \) dominoes, with each domino containing at most one marked cell.
+    To solve the problem, we start by simplifying the given equation and analyzing the expression we need to maximize.
     
-    2. **Observation**: The problem resembles a \( 2 \times 2 \) chessboard where ea
-    
-    ... [truncated 17 chars] ...
-    
-    er two cells. If we mark one of the four cells, the other three cells must be covered by dominoes without any marked cell in the same domino. This can be visualized as a chessboard where the diagonals are marked.
-    
-    3. **Pattern for smaller boards**: For a \( 2 \times 2 \) board, marking one of the four cells allows the other three to be covered by dominoes without any marked cell in the same domino. This pattern extends to larger boards.
-    
-    4. **Generalizing to \( 194 \times 194 \) board**: We need to extend this pattern to a \( 194 \times 194 \) board. If we mark cells in a diagonal pattern, each domino will cover cells from different diagonals, ensuring that no two marked cells are in the same domino.
+    Given the equation:
+    \[
+    (x^3-3x^2+3x)+(y^3-3y^2+3y)+(z^3-3z^2+3z)=4,
+    \]
+    we notice that the terms can be rewritten using the binomial theorem, which states that \(a^3 - 3a^2 + 3a\) can be factored as \((a-1)^3 + (a-1)\). Applying this to each term, we get:
+    \[
+    (x-1)^3 + (x-1) + (y-1)^3
     
-    5. **Counting the marked cells**: The diagonal pattern on a \( 194 \times 194 \) board consists of 194 cells. Each domino covers exactly two cells, and since the board is tiled completely, the number of dominoes is \( \frac{194 \times 194}{2} \).
+    ... [truncated 225 chars] ...
     
-    6. **Conclusion**: Therefore, marking 194 cells in a diagonal pattern ensures that the board can be tiled completely with \( 1 \times 2 \) and \( 2 \times 1 \) dominoes, with each domino containing at most one marked cell.
+    erms, we get:
+    \[
+    (x-1)^3 + (y-1)^3 + (z-1)^3 + 3(x+y+z-3) = 4,
+    \]
+    which simplifies to:
+    \[
+    (x-1)^3 + (y-1)^3 + (z-1)^3 + 3(x+y+z-3) - 4 = 0.
+    \]
+    This can be further rearranged as:
+    \[
+    (x-1)^3 + (y-1)^3 + (z-1)^3 + 3(x+y+z-3) - 4 = 0.
+    \]
+    To maximize the expression \(x^2+y^2+z^2-x-y-z\), we consider the cases where two of \(x, y, z\) are equal and the third is different. Without loss of generality, let's assume \(x = y\) and \(z\) is different. This assumption simplifies our analysis.
     
+    Given \(x = y\), the equation becomes:
+    \[
+    2(x-1)^3 + (z-1)^3 + 3(2x+z-3) - 4 = 0.
+    \]
+    Substituting \(x = y\) into our original equation, we have:
+    \[
+    (x^3-3x^2+3x)+2(y^3-3y^2+3y)+(z^3-3z^2+3z)=4,
+    \]
+    which simplifies to:
+    \[
+    2(x^3-3x^2+3x) + (z^3-3z^2+3z) = 4.
+    \]
+    This can be rewritten as:
+    \[
+    2((x-1)^3 + (x-1)) + ((z-1)^3 + (z-1)) = 4.
+    \]
+    Further simplification gives us:
+    \[
+    (x-1)^3 + (y-1)^3 + (z-1)^3 + 3(x+y+z-3) = 4,
+    \]
+    which is the same equation we derived earlier. To maximize \(x^2+y^2+z^2-x-y-z\), we consider the case where \(x = y = 0\) and \(z = 2\). This gives us:
     \[
-    \boxed{194}
+    x^2+y^2+z^2-x-y-z = 0^2+0^2+2^2-0-0-2 = 4.
     \]
+    Therefore, the largest possible value of \(x^2+y^2+z^2-x-y-z\) is \(
+    </think>
+    
+    \boxed{4}\).
     ```
 
 **Incorrect** (1 shown)
 
-- **reference**: `$g(x)=c, g(x)=\lceil x \rceil, g(x)=\lfloor x \rfloor$`
+- **reference**: `504`
 - **overall** (0/8 completions correct)
-- **extracted** (✗): `h(x) = 0 \text{ or } h(x) = x + 1`
+- **extracted** (✗): `\text{Error in the problem setup or calculations}`
 - **completion**:
 
     ```text
-    <think>
-    Okay, let's try to solve this problem step by step. The goal is to determine all mappings \( h: \mathbb{Q} \to \mathbb{Z} \) that satisfy the given condition for every \( x \in \mathbb{Q} \), every \( a \in \mathbb{Z} \), and every positive integer \( b \):
-    
-    \[
-    h(x) = h\left(\frac{h(bx - a) + a}{b}\right).
-    \]
+    1. **Identify the properties of the trapezoid and the circle**: 
+       - The circle is inscribed in the trapezoid, touching all four sides.
+       - The trapezoid is isosceles, meaning the non-parallel sides are equal in length.
+       - The radius of the circle is \(3\).
     
-    First, let's understand the condition better. We need to find a function \( h \)
+    2. **Use the properties of the circle to find the lengths of the parallel sides**:
+       - The diameter of the circle is \(2 \times 3 = 6\).
     
     ... [truncated 823 chars] ...
     
-    tion.
-    
-    5. **Checking for non-zero solutions:**
-    
-       Suppose \( h(x) = x + 1 \) for some \( x \). Then:
-       \[
-       h\left(\frac{h(bx - a) + a}{b}\right) = \frac{h(bx - a) + a}{b} + 1.
-       \]
-       Substituting \( h \):
-       \[
-       \frac{(bx - a) + a}{b} + 1 = x + 1.
-       \]
-       Simplifying:
-       \[
-       x + 1 = x + 1.
-       \]
-       This is true, so \( h(x) = x + 1 \) is a valid solution.
-    
-    6. **Conclusion:**
-    
-       We have found that \( h(x) = 0 \) and \( h(x) = x + 1 \) are solutions. To ensure these are the only solutions, we need to check if there can be other functions that satisfy the condition.
-    
-       Suppose there exists another function \( h \) that satisfies the condition. Then, for any \( x \in \mathbb{Q} \), \( h(x) \) must be an integer. Moreover, for any \( a \in \mathbb{Z} \), \( h(bx - a) \) must also be an integer. This implies that \( h \) is a linear function, as previously shown.
-    
-       Therefore, \( h(x) \) must be of the form \( h(x) = kx + c \). We have already shown that \( k = 0 \) or \( k = 1 \) are the only possibilities, leading to \( h(x) = 0 \) or \( h(x) = x + 1 \).
-    
-       Hence, the only solutions are \( h(x) = 0 \) and \( h(x) = x + 1 \).
+    (radius of the circle).
+       - By the Pythagorean theorem:
+         \[
+         3^2 = 24^2 + d^2
+         \]
+         \[
+         9 = 576 + d^2 \implies d^2 = -567
+         \]
+         This is a contradiction, indicating an error in the setup or calculation.
+    
+    5. **Re-evaluate the problem setup**:
+       - The correct setup should ensure that \(r\) and \(s\) are distinct and valid lengths. Given the area and the diameter, we need to re-evaluate the relationship between \(r\), \(s\), and the height.
+       - The correct calculation for the height should be:
+         \[
+         72 = \frac{1}{2} \times 6 \times h \implies h = 24
+         \]
+       - Using the Pythagorean theorem correctly, we find:
+         \[
+         3^2 = 24^2 + d^2 \implies 9 = 576 + d^2 \implies d^2 = -567
+         \]
+         This indicates an error in the problem setup or an incorrect interpretation of the circle's properties.
+    
+    6. **Conclusion**:
+       - Given the constraints and calculations, the problem as stated contains inconsistencies or errors. The correct relationship and solution require a re-evaluation of the problem's parameters or a different approach to ensure the trapezoid's properties hold true.
     
     \[
-    \boxed{h(x) = 0 \text{ or } h(x) = x + 1}
-    \
+    </think>
+    
+    \boxed{\text{Error in the problem setup or calculations}}
+    \]
     ```