diff --git a/doc/src/SUMMARY.md b/doc/src/SUMMARY.md
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+++ b/doc/src/SUMMARY.md
@@ -46,3 +46,4 @@
   - [27: Verify atomically reference-counted Cell implementation](./challenges/0027-arc.md)
   - [28: Verify float to decimal conversion module](./challenges/0028-flt2dec.md)
   - [29: Safety of `boxed`](./challenges/0029-boxed.md)
+  - [31: Verify the safety of `BinaryHeap` functions](./challenges/0031-binary-heap.md)
diff --git a/doc/src/challenges/0031-binary-heap.md b/doc/src/challenges/0031-binary-heap.md
new file mode 100644
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@@ -0,0 +1,92 @@
+# Challenge 31: Verify the safety of `BinaryHeap` functions
+
+- **Status:** *Open*
+- **Solution:** *Option field to point to the PR that solved this challenge.*
+- **Tracking Issue:** [#746](https://github.com/model-checking/verify-rust-std/issues/746)
+- **Start date:** *2026/11/03*
+- **Reward:** *TBD*
+
+-------------------
+
+
+## Goal
+
+Verify the safety of `BinaryHeap` functions in [binary_heap/mod.rs](https://github.com/model-checking/verify-rust-std/blob/main/library/alloc/src/collections/binary_heap/mod.rs).
+
+## Motivation
+
+`BinaryHeap` is the priority queue of the standard library. To keep its elements in heap order, it moves them with raw pointer reads and copies, at positions chosen by comparing elements with `Ord`. `Ord` is safe code written by users and can return any result, so these moves must be safe even when the comparisons are inconsistent.
+
+## Description
+
+This challenge verifies the safety of `BinaryHeap`'s sift functions, `PeekMut`, the `TrustedLen` sorted iterators, and the safe methods built on them. The sifts move elements through `Hole`, which reads one element out with `ptr::read`, copies others into the gap with `ptr::copy_nonoverlapping` at positions chosen by `Ord`, and writes the element back in `Drop`. `PeekMut` shrinks the length to 1 while a `&mut T` to the top element is out.
+
+The other `unsafe impl`s on `IntoIter` (`TrustedFused`, `SourceIter`, `InPlaceIterable`, `AsVecIntoIter`) delegate to `vec::IntoIter` and are out of scope for this challenge.
+
+### Assumptions
+
+* `Vec`'s implementation may be assumed correct, and its existing verified contracts may be reused.
+* A panicking `Ord` is out of scope: solutions need not verify unwinding through `Drop for Hole`.
+
+### Success Criteria
+
+Write and prove the contract for the safety of the following unsafe functions:
+
+| Function |
+|---------|
+|Hole::new|
+|Hole::get|
+|Hole::move_to|
+|sift_up|
+|sift_down_range|
+|sift_down|
+|sift_down_to_bottom|
+
+Prove that the following implementations of the unsafe trait `TrustedLen` are correct, that is, that `size_hint` returns the exact number of remaining elements:
+
+| Implementation |
+|---------|
+|TrustedLen for IntoIterSorted|
+|TrustedLen for DrainSorted|
+
+Prove the absence of undefined behavior for the following safe abstractions:
+
+| Function |
+|---------|
+|push|
+|pop|
+|peek_mut|
+|PeekMut::deref|
+|PeekMut::deref_mut|
+|PeekMut::pop|
+|PeekMut::refresh|
+|Drop for PeekMut|
+|rebuild|
+|rebuild_tail|
+|append|
+|into_sorted_vec|
+|From<Vec<T>>|
+|IntoIterSorted::next|
+|DrainSorted::next|
+|Drop for Hole|
+
+The unsafe functions `from_raw_vec` and `as_mut_slice` require callers to keep the contents a max-heap. This is a correctness requirement, not a safety one, so the functions above must be safe even when the contents are not a max-heap.
+
+The verification must be unbounded---it must hold for heaps of arbitrary length.
+
+The verification must hold for generic type `T` (what satisfies this clause is to be decided in [issue #734](https://github.com/model-checking/verify-rust-std/issues/734)).
+
+The verification must hold for any implementation of `Ord` for `T`, including ones that return inconsistent results.
+
+### List of UBs
+
+All proofs must automatically ensure the absence of the following undefined behaviors [ref](https://github.com/rust-lang/reference/blob/142b2ed77d33f37a9973772bd95e6144ed9dce43/src/behavior-considered-undefined.md):
+
+* Accessing (loading from or storing to) a place that is dangling or based on a misaligned pointer.
+* Invoking undefined behavior via compiler intrinsics.
+* Reading from uninitialized memory except for padding or unions.
+* Producing an invalid value
+
+
+Note: All solutions to verification challenges need to satisfy the criteria established in the [challenge book](../general-rules.md)
+in addition to the ones listed above.