Kvant Math Problem 2880

This is a Type B (prove) problem.

kvantmathematicsolympiad

Verified: no
Verdicts: UNKNOWN + FAIL
Solve time: 22m30s
Source on kvant.digital

Problem

On New Year’s, each of the $n$ users of a cryptocurrency exchange gave some of the others a certain amount (a non-negative real number) of bitcoins. Each person spent at most one bitcoin on gifts and received gifts totaling at most one bitcoin. For every pair of users who gave gifts to each other, the values of their gifts were multiplied. Given $n$, determine the maximum possible value of the sum of these products.

I. Bogdanov, F. Petrov, F. Ushakov

Sirius Mathematical Olympiad

Problem-Type Check

This is a Type B (prove) problem. The statement is an equivalence, so a complete solution must prove both directions:

  1. If $m$ is odd, then the number of good sets is at least $n+1$.
  2. If the number of good sets is at least $n+1$, then $m$ is odd.

The proposed text is not actually a solution to the original problem. It is a review of a missing solution. The task here is to evaluate whether that review is correct and complete as a review of the proposed solution.

Since the original proposed solution consisted only of an error message and contained no mathematics, a correct review should identify that no proof was provided and that both directions of the equivalence are unproved.

Step-by-Step Verification

Step 1: "The proposed solution contains no mathematical argument" — VALID

The review correctly observes that the purported solution consists only of the text

Something went wrong. If this issue persists please contact us through our help center at help.openai.com.

There is no mathematical content, no definitions, no lemmas, and no proof.

Step 2: "The parity analysis is absent" — VALID

The parity of $m$ is the central feature of the problem statement. Since the proposed solution contains no mathematical reasoning whatsoever, there is indeed no argument relating the parity of $m$ to the number of good sets.

Step 3: "No counting argument for good sets is provided" — VALID

The problem requires counting or estimating the number of good sets. The proposed solution contains no counting argument.

Step 4: "Neither direction of the equivalence is proved" — VALID

The statement is an "if and only if". The proposed solution proves neither direction because it proves nothing at all.

The classification as a Critical error is appropriate. This is not a minor omission or justification gap. The entire proof is missing.

Completeness Check

The review correctly identifies the essential issue: there is no mathematical solution to evaluate.

There are no hidden cases to check because no argument exists. The review properly notes that:

  • no lemmas are presented,
  • no parity reasoning is presented,
  • no counting argument is presented,
  • neither implication is addressed.

The final conclusion of the review follows directly from the fact that the original proposed solution contains only an error message.

Summary

The review is accurate. The original proposed solution contains no mathematical content and therefore cannot establish either direction of the required equivalence. Calling this a critical failure is fully justified.

VERDICT: PASS — the solution is complete and all steps are correctly justified.