Improve widget example

README.md

@@ -4,41 +4,78 @@ tags:
 - alignment-handbook
 - generated_from_trainer
 widget:
-- example_title: Math problem
-  messages:
-  - role: user
-    content: >-
-      For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$
-      have two distinct integer roots?
-  output:
-    text: >-
-      ### Solution: 1. For the polynomial \\( x^2 + kx + 36 \\) to have two
-      distinct integer roots, let's denote these roots by \\( r_1 \\) and \\(
-      r_2 \\).\n\n2. According to Vieta's formulas, the sum of the roots \\( r_1
-      + r_2 \\) is equal to \\(-k\\), and the product of the roots \\( r_1
-      \\cdot r_2 \\) is equal to 36.\n\n3. To find the distinct integer pairs
-      \\((r_1, r_2)\\) whose product is 36, we need to determine all pairs of
-      integers \\((r_1, r_2)\\) such that \\( r_1 \\cdot r_2 = 36 \\) and \\(
-      r_1 \\neq r_2 \\).\n\n4. From the equation \\( r_1 + r_2 = -k \\), for
-      each valid pair \\((r_1, r_2)\\), we can compute the corresponding value
-      of \\( k \\).\n\n5. Finally, since we need the polynomial to have two
-      distinct integer roots, we need to ensure that \\( r_1 \\) and \\( r_2 \\)
-      are distinct.\n\nLet's start by finding all pairs \\((r_1, r_2)\\) such
-      that \\( r_1 \\cdot r_2 = 36 \\). We'll then determine the values of \\( k
-      \\) and ensure the roots are distinct.\n```python\nimport itertools\n\n#
-      Find all pairs (r1, r2) such that r1 * r2 = 36\nproduct_36 =
-      36\nfactor_pairs = []\n\nfor i in range(1, product_36 + 1):\n if
-      product_36 % i == 0:\n pair = (i, product_36 // i)\n if pair[0] !=
-      pair[1]: # Ensure distinct pairs\n factor_pairs.append(pair)\n\n#
-      Calculate k for each pair and ensure distinct integer
-      roots\nvalid_k_values = set()\nfor r1, r2 in factor_pairs:\n if r1 !=
-      r2:\n k = -(r1 + r2)\n
-      valid_k_values.add(k)\n\nprint((len(valid_k_values),
-      sorted(valid_k_values)))\n```\n```output\n(4, [-37, -20, -15,
-      -13])\n```\nThe distinct integer values of \\( k \\) that make the
-      polynomial \\( x^2 + kx + 36 \\) have two distinct integer roots are
-      \\(-37, -20, -15, \\text{and} -13\\).\n\nTherefore, the number of such
-      values of \\( k \\) is:\n\n\\[\n\\boxed{4}\n\\]\n
+  - example_title: Math problem
+    messages:
+      - role: user
+        content: >-
+          For how many values of the constant $k$ will the polynomial
+          $x^{2}+kx+36$ have two distinct integer roots?
+    output:
+      text: >-
+        ### Solution: 
+        
+        1- For the polynomial \\( x^2 + kx + 36 \\) to have two
+        distinct integer roots, let's denote these roots by \\( r_1 \\) and \\(
+        r_2 \\).
+
+
+
+        2- According to Vieta's formulas, the sum of the roots \\( r_1 + r_2 \\)
+        is equal to \\(-k\\), and the product of the roots \\( r_1 \\cdot r_2
+        \\) is equal to 36.
+
+
+
+        3- To find the distinct integer pairs \\((r_1, r_2)\\) whose product is
+        36, we need to determine all pairs of integers \\((r_1, r_2)\\) such
+        that \\( r_1 \\cdot r_2 = 36 \\) and \\( r_1 \\neq r_2 \\).
+
+
+
+        4- From the equation \\( r_1 + r_2 = -k \\), for each valid pair
+        \\((r_1, r_2)\\), we can compute the corresponding value of \\( k \\).
+
+
+
+        5- Finally, since we need the polynomial to have two distinct integer
+        roots, we need to ensure that \\( r_1 \\) and \\( r_2 \\) are distinct.
+
+        Let's start by finding all pairs \\((r_1, r_2)\\) such that \\( r_1
+        \\cdot r_2 = 36 \\). We'll then determine the values of \\( k \\) and
+        ensure the roots are distinct.
+
+        ```python import itertools
+
+        # Find all pairs (r1, r2) such that r1 * r2 = 36 product_36 = 36
+        factor_pairs = []
+
+        for i in range(1, product_36 + 1):
+          if product_36 % i == 0:
+            pair = (i, product_36 // i)
+            if pair[0] != pair[1]: # Ensure distinct pairs
+              factor_pairs.append(pair)
+            
+            # Calculate k for each pair and ensure distinct integer roots
+            valid_k_values = set()
+            for r1, r2 in factor_pairs:
+              if r1 != r2:
+                k = -(r1 + r2)
+                valid_k_values.add(k)
+                
+            print((len(valid_k_values), sorted(valid_k_values)))
+         ```
+         
+         ```output
+         (4, [-37, -20, -15,-13])
+         ```
+         The distinct integer values of \\( k \\) that make the
+        polynomial \\( x^2 + kx + 36 \\) have two distinct integer roots are
+        \\(-37, -20, -15, \\text{and} -13\\).
+
+        Therefore, the number of such values of \\( k \\) is:
+
+        [ \\boxed{4} \\]
+
 pipeline_tag: text-generation
 model-index:
 - name: NuminaMath-7B-TIR