docs(docs): add ProblemBadges component, layout slot, and enrich pages with topics/video frontmatter, plus guide placeholders and topic‑landing map.

apps/docs/content/(05-decision-space)/1d-dynamic-programming/198-house-robber.mdx

@@ -2,13 +2,14 @@
 title: '198. House Robber'
 description: You are a professional robber planning to rob houses along a street. Each house has a certain amount of money stashed, the only constraint stopping you from robbing each of them is that adjacent houses have security systems connected and it will automatically contact the police if two adjacent houses were broken into on the same night
 icon: dot
+topics:
+  - { name: "Array", slug: "array" }
+  - { name: "Dynamic Programming", slug: "dynamic-programming" }
 sidebar:
   label: 'House Robber'
   badge: 'Medium'
 ---
 
-<Badge variant="accent">Array</Badge>
-
 ### Example 1:
 - Input: `nums = [1,2,3,1]`
 - Output: `4`

apps/docs/content/(05-decision-space)/1d-dynamic-programming/322-coin-change.mdx

@@ -2,13 +2,17 @@
 title: '322. Coin Change'
 description: You are given an integer array coins representing coins of different denominations and an integer amount representing a total amount of money
 icon: dot
+topics:
+  - { name: "Array", slug: "array" }
+  - { name: "Dynamic Programming", slug: "dynamic-programming" }
+  - { name: "Breadth-First Search", slug: "breadth-first-search" }
+  - { name: "Knapsack Problem", slug: "knapsack-problem" }
+  - { name: "Complete Knapsack", slug: "complete-knapsack" }
 sidebar:
   label: 'Coin Change'
   badge: 'Medium'
 ---
 
-<Badge variant="accent">Array</Badge>
-
 ### Example 1:
 - Input: `coins = [1,2,5], amount = 11`
 - Output: `3`

apps/docs/content/(05-decision-space)/1d-dynamic-programming/70-climbing-stairs.mdx

@@ -2,13 +2,15 @@
 title: '70. Climbing Stairs'
 description: You are climbing a staircase. It takes n steps to reach the top
 icon: dot
+topics:
+  - { name: "Math", slug: "math" }
+  - { name: "Dynamic Programming", slug: "dynamic-programming" }
+  - { name: "Memoization", slug: "memoization" }
 sidebar:
   label: 'Climbing Stairs'
   badge: 'Easy'
 ---
 
-<Badge variant="accent">Math</Badge>
-
 ### Example 1:
 - Input: `n = 2`
 - Output: `2`

apps/docs/content/(05-decision-space)/backtracking/39-combination-sum.mdx

@@ -2,13 +2,14 @@
 title: '39. Combination Sum'
 description: Given an array of distinct integers candidates and a target integer target, return a list of all unique combinations of candidates where the chosen numbers sum to target. You may return the combinations in any order
 icon: dot
+topics:
+  - { name: "Array", slug: "array" }
+  - { name: "Backtracking", slug: "backtracking" }
 sidebar:
   label: 'Combination Sum'
   badge: 'Medium'
 ---
 
-<Badge variant="accent">Array</Badge>
-
 ### Example 1:
 - Input: `candidates = [2,3,6,7], target = 7`
 - Output: `[[2,2,3],[7]]`

apps/docs/content/(05-decision-space)/backtracking/46-permutations.mdx

@@ -2,13 +2,14 @@
 title: '46. Permutations'
 description: Given an array nums of distinct integers, return all the possible permutations. You can return the answer in any order
 icon: dot
+topics:
+  - { name: "Array", slug: "array" }
+  - { name: "Backtracking", slug: "backtracking" }
 sidebar:
   label: 'Permutations'
   badge: 'Medium'
 ---
 
-<Badge variant="accent">Array</Badge>
-
 ### Example 1:
 - Input: `nums = [1,2,3]`
 - Output: `[[1,2,3],[1,3,2],[2,1,3],[2,3,1],[3,1,2],[3,2,1]]`

apps/docs/content/(05-decision-space)/backtracking/78-subsets.mdx

@@ -2,13 +2,15 @@
 title: '78. Subsets'
 description: Given an integer array nums of unique elements, return all possible subsets (the power set)
 icon: dot
+topics:
+  - { name: "Array", slug: "array" }
+  - { name: "Backtracking", slug: "backtracking" }
+  - { name: "Bit Manipulation", slug: "bit-manipulation" }
 sidebar:
   label: 'Subsets'
   badge: 'Medium'
 ---
 
-<Badge variant="accent">Array</Badge>
-
 ### Example 1:
 - Input: `nums = [1,2,3]`
 - Output: `[[],[1],[2],[1,2],[3],[1,3],[2,3],[1,2,3]]`