Kvant Math Problem 1568
This is a Type B problem.
Verified: no
Verdicts: FAIL + FAIL
Solve time: 27m35s
Source on kvant.digital
Problem
Prove that for $n\ge5$, a cross-section of a pyramid whose base is a regular $n$-gon cannot be a regular $(n+1)$-gon.
N. Kh. Agakhanov, D. Teryoshin
All-Russian Mathematical Olympiad for School Students (1996)
Problem-Type Check
This is a Type B problem. The task is to prove an impossibility statement: that no plane section of a pyramid with a regular $n$-gon base for $n\ge 5$ can be a regular $(n+1)$-gon. The proposed solution attempts a proof by contradiction, which is appropriate for a Type B problem. The general approach—considering the number of vertices on lateral edges versus the base edge—is consistent with the problem type. The requirement to handle all $n\ge5$ is recognized, and no construction or optimization is needed beyond establishing the impossibility. The problem-type alignment is correct.
Step-by-Step Verification
Step 1: Identification of vertex placement — “Exactly one base edge contributes a vertex, the remaining $n$ vertices lie on lateral edges” — VALID. This follows from convexity of the pyramid and the pigeonhole principle, as $n+1>n$.
Step 2: Labeling of vertices $P_1,\dots,P_n$ and $Q$ and ordering of the polygon — VALID. This is consistent notation and correctly represents the assumed section.
Step 3: Parameterization along lateral edges with ratios $t_i$ and claim that equality of side lengths forces $t_1 = \dots = t_n$ — WRONG. The solution references a quadratic function of $t_i$ and $t_{i+1}$ and claims convexity implies a unique minimum at $t_i=t_{i+1}$, but no explicit computation or rigorous argument is provided. Convexity in this context is not obvious; the squared distance along triangular lateral faces depends on both horizontal and vertical coordinates in a nontrivial way. This is a critical error because the main contradiction relies on establishing that all $t_i$ must be equal.
Step 4: Plane through equal-ratio points forms a regular $n$-gon parallel to the base — UNJUSTIFIED. This step depends entirely on Step 3. Without Step 3, the claim that the lateral-edge intersections form a regular $n$-gon is unproved. Classification: Justification gap, conditional on Step 3.
Step 5: Metric comparison: distances from $Q$ to $P_1$ and $P_2$ are strictly less than $|P_1P_2|$ — UNJUSTIFIED. The solution invokes qualitative trapezoid geometry and convexity of angles to argue strict inequality, but no explicit calculation or inequality is provided. Equality in degenerate configurations is not ruled out rigorously. This is a critical error, as it is essential to the impossibility argument.
Step 6: Concluding impossibility of a regular $(n+1)$-gon — UNJUSTIFIED. Steps 3 and 5 are required to produce the contradiction. Since both steps are critically flawed, this final conclusion does not follow rigorously.
Completeness Check
The solution fails to rigorously justify two essential claims: the uniqueness of the lateral-edge ratios $t_1 = \dots = t_n$ for equal side lengths and the metric comparison between $Q$ and its neighbors. Both claims are indispensable to the argument. All other combinatorial and geometric considerations are correctly addressed. There are no unhandled values of $n\ge5$; the gaps are in the central geometric and algebraic reasoning.
Summary
The proposed solution correctly identifies the combinatorial structure of potential sections and the main geometric obstruction, but it does not rigorously establish the critical properties necessary for the contradiction. The most severe flaw is the unproved assertion that equality of side lengths along lateral edges forces all $t_i$ to be equal. Without this step, the impossibility proof is incomplete.
VERDICT: FAIL — the solution does not rigorously prove that equality of side lengths along lateral edges forces all intersection ratios to be equal, which is essential for the contradiction.