Difference between revisions of "2006 Romanian NMO Problems/Grade 8/Problem 3"
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==Problem== | ==Problem== | ||
| − | Let <math>ABCDA_1B_1C_1D_1</math> be a | + | Let <math>ABCDA_1B_1C_1D_1</math> be a cube and <math>P</math> a variable point on the side <math>[AB]</math>. The perpendicular plane on <math>AB</math> which passes through <math>P</math> intersects the line <math>AC'</math> in <math>Q</math>. Let <math>M</math> and <math>N</math> be the midpoints of the segments <math>A'P</math> and <math>BQ</math> respectively. |
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| + | a) Prove that the lines <math>MN</math> and <math>BC'</math> are perpendicular if and only if <math>P</math> is the midpoint of <math>AB</math>. | ||
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| + | b) Find the minimal value of the angle between the lines <math>MN</math> and <math>BC'</math>. | ||
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==Solution== | ==Solution== | ||
{{solution}} | {{solution}} | ||
Latest revision as of 10:24, 10 October 2007
Problem
Let 
be a cube and 
a variable point on the side ![$[AB]$](http://latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
. The perpendicular plane on 
which passes through intersects the line 
in 
. Let 
and 
be the midpoints of the segments 
and 
respectively.
a) Prove that the lines 
and 
are perpendicular if and only if is the midpoint of .
b) Find the minimal value of the angle between the lines and .
Solution
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