Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2006 Romanian NMO Problems/Grade 8/Problem 3"

 
(→‎Problem)
 
(2 intermediate revisions by 2 users not shown)
Line 1: Line 1:
 
==Problem==
 
==Problem==
 
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.  
 
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.  
 +
 +
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>.
 +
 +
b) Find the minimal value of the angle between the lines <math>MN</math> and <math>BC'</math>.
 +
 
==Solution==
 
==Solution==
 +
{{solution}}
 
==See also==
 
==See also==
 
*[[2006 Romanian NMO Problems]]
 
*[[2006 Romanian NMO Problems]]
 +
[[Category:Olympiad Geometry Problems]]

Latest revision as of 10:24, 10 October 2007

Problem

Let $ABCDA_1B_1C_1D_1$ be a cube and $P$ a variable point on the side $[AB]$. The perpendicular plane on $AB$ which passes through $P$ intersects the line $AC'$ in $Q$. Let $M$ and $N$ be the midpoints of the segments $A'P$ and $BQ$ respectively.

a) Prove that the lines $MN$ and $BC'$ are perpendicular if and only if $P$ is the midpoint of $AB$.

b) Find the minimal value of the angle between the lines $MN$ and $BC'$.

Solution

This problem needs a solution. If you have a solution for it, please help us out by adding it.

See also

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2006_Romanian_NMO_Problems/Grade_8/Problem_3&oldid=17787"