docs(content): add new problem pages, improve binary‑search docs, fix image syntax, update appendix badge

apps/docs/content/(01-linear-structures)/prefix-sum/index.mdx

@@ -26,7 +26,10 @@ The leading `prefix[0] = 0` is the sum of zero elements. That extra slot is what
 
 
 <Frame caption="Each bar is the running total up to that index. Every step keeps the previous total (blue) and adds one new element (amber)" hint="">
-  <img src="../../../public/prefix-sum_1.png" alt="" />
+
+![](../../../public/prefix-sum_1.png)
+
+
 </Frame>
 
 ## Any range sum is one subtraction