Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Problem B. 3998. (April 2007)

B. 3998. The vertex P of a rectangular block with edges a, b, c is selected. Consider the plane passing through the vertices adjacent to P. Denote the distance of the plane from P is m. Prove that


\frac{1}{m^2} = \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}.

(4 pont)

Deadline expired on May 15, 2007.


Sorry, the solution is available only in Hungarian. Google translation

Megoldás: Vegyünk fel egy derékszögű koordinátarendszert úgy, hogy a P pont az origóba, a P-vel szomszédos csúcsok pedig rendre az (a,0,0), (0,b,0), (0,0,c) pontokba essenek. Ekkor a sík egyenlete x/a+y/b+z/c=1. Legyen

d=\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}},

ekkor a sík normálegyenlete

H(x,y,z)=\frac{1}{ad}x+\frac{1}{bd}y+\frac{1}{cd}z-\frac{1}{d}=0,

vagyis a P(0,0,0) pontnak a síktól vett távolsága

m=|H(0,0,0)|=\left|-\frac{1}{d}\right|=\frac{1}{d},

tehát 1/m2=d2, amint azt bizonyítani kellett.


Statistics:

94 students sent a solution.
4 points:83 students.
3 points:2 students.
2 points:1 student.
1 point:2 students.
0 point:1 student.
Unfair, not evaluated:5 solutions.

Problems in Mathematics of KöMaL, April 2007