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Question #41733

Difficulty: 8/10 | 🟢 Free

The Question

Let $P$ be a variable plane $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ (where $a, b, c > 0$ ) such that the volume of the tetrahedron formed by $P$ and the coordinate planes is $\frac{9}{2}$ . Let $d$ be the perpendicular distance from the origin to the plane $P$ . Find the maximum value of $\lfloor d^2 + \{a\} + \{b\} + \{c\} \rfloor$ , where $\lfloor \cdot \rfloor$ denotes the greatest integer function and $\{ \cdot \}$ denotes the fractional part function.

Options

A.
3
B.
4
C.
5
D.
6

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Distance from point to plane|Variable plane intercepts|Integral Part and Fractional Part

Why This Question Is Tricky

Common Mistakes

This question tests optimization under constraints while navigating the non-differentiable Greatest Integer and Fractional Part functions.

The lethal trap lies in assuming the maximum of the sum occurs near the critical point of the continuous function $d^2$ (where $a=b=c=3$ and $d^2=3$ ).

Near this point, the sum of fractional parts approaches 2, leading students to incorrectly conclude the maximum GIF is 4.

However, by moving away from the symmetric maximum to asymmetric boundaries (e.g., $a \to 2^-, b \to 5^-$ ), the fractional parts can sum to 2.7 while $d^2$ only drops to 2.34, pushing the total sum above 5.

This punishes students who rely solely on local derivatives without checking boundary behaviors of the fractional part.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Distance from point to plane
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free