Add Mana Collection to CHT module (#4983)

content/5_Plat/Convex_Hull_Trick.problems.json

@@ -150,6 +150,18 @@
         "usacoId": "998"
       }
     },
+    {
+      "uniqueId": "usaco-1285",
+      "name": "Mana Collection",
+      "url": "http://www.usaco.org/index.php?page=viewproblem2&cpid=1285",
+      "source": "Platinum",
+      "difficulty": "Hard",
+      "isStarred": false,
+      "tags": ["Bitmask DP", "Convex"],
+      "solutionMetadata": {
+        "kind": "internal"
+      }
+    },
     {
       "uniqueId": "joi-17-LongDistanceCoach",
       "name": "2017 - Long-Distance Coach",

content/extraProblems.json

@@ -1682,19 +1682,6 @@
         "usacoId": "1284"
       }
     },
-    {
-      "uniqueId": "usaco-1285",
-      "name": "Mana Collection",
-      "url": "http://usaco.org/index.php?page=viewproblem2&cpid=1285",
-      "source": "Platinum",
-      "difficulty": "N/A",
-      "isStarred": false,
-      "tags": ["DP", "Bitmask", "Convex"],
-      "solutionMetadata": {
-        "kind": "USACO",
-        "usacoId": "1285"
-      }
-    },
     {
       "uniqueId": "usaco-1286",
       "name": "Subtree Activation",

solutions/platinum/usaco-1285.mdx

@@ -0,0 +1,28 @@
+---
+id: usaco-1285
+source: USACO Platinum 2023 January
+title: Mana Collection
+author: Daniel Zhu
+---
+
+The [official editorial](https://usaco.org/current/data/sol_prob2_platinum_jan23.html)
+provides a thorough solution for this problem, but here are a few details to pay attention to.
+
+Doing a Floyd-Warshall at the start of the algorithm is necessary because,
+although the bitmask DP enumerates the next *unvisited* node,
+it may be optimal or even necessary to go through a node we've already visited in order to get there.
+
+For instance, consider the following adjacency list:
+```
+1 -> 2
+2 -> 1
+1 -> 3
+```
+To get from node 2 to node 3, we **have** to pass through node 1 no matter what.
+
+Also, it may seem more intuitive to define the bitmask DP as `dp[mask][i]`,
+where `mask` represents the nodes we've currently visited (in reverse order),
+and `i` represents the node we're currently at.
+
+The issue with this definition, however, is that our state has no idea what the final set of visited nodes is,
+meaning we have no way of accurately computing the effect of traversing an edge.