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Difference between revisions of "1960 IMO Problems/Problem 5"

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==Problem==
 
==Problem==
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Consider the cube <math>ABCDA'B'C'D'</math> (with face <math>ABCD</math> directly above face <math>A'B'C'D'</math>).
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a) Find the locus of the midpoints of the segments <math>XY</math>, where <math>X</math> is any point of <math>AC</math> and <math>Y</math> is any point of <math>B'D'</math>;
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b) Find the locus of points <math>Z</math> which lie on the segment <math>XY</math> of part a) with <math>ZY = 2XZ</math>.
  
 
==Solution==
 
==Solution==
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{{solution}}
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Let <math>A=(0,0,2)</math>, <math>B=(2,0,2)</math>, <math>C=(2,0,0)</math>, <math>D=(0,0,0)</math>, <math>A'=(0,2,2)</math>, <math>B'=(2,2,2)</math>, <math>C'=(2,2,0)</math>, and <math>D'=(0,2,0)</math>. Then there exist real <math>x</math> and <math>y</math> in the closed interval <math>[0,2]</math> such that <math>X=(x,0,2-x)</math> and <math>Y=(y,2,y)</math>.
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The midpoint of <math>XY</math> has coordinates <math>((x+y)/2, 1, (2-x+y)/2)</math>. Let <math>a</math> and <math>b</math> be the <math>x</math>- and <math>z</math>-coordinates of the midpoint of <math>XY</math>, respectively. We then have that <math>a+b=y+1</math> and <math>a-b=x-1</math>, so <math>a+b\in [1,3]</math> and <math>a-b\in [-1,1]</math>. The region of points that satisfy these inequalities is the closed square with vertices at <math>(1,1,2)</math>, <math>(2,1,1)</math>, <math>(1,1,0)</math>, and <math>(0,1,1)</math>. For every point <math>P</math> in this region, there exist unique points <math>X</math> and <math>Y</math> such that <math>P</math> is the midpoint of <math>XY</math>.
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If <math>Z\in XY</math> and <math>ZY=2XZ</math>, then <math>Z</math> has coordinates <math>((2x+y)/3, 2/3, (4-2x+y)/3)</math>. Let <math>a</math> and <math>b</math> be the <math>x</math>- and <math>z</math>- coordinates of <math>Z</math>. We then have that <math>a+b=(4/3)+(2/3)y</math> and <math>a-b=(4x-4)/3</math>, and <math>a\in (4/3,8/3)</math> and <math>b\in (-4/3,4/3)</math>. The region of points that satisfy these inequalities is the closed rectangle with vertices at <math>(0,2/3,4/3)</math>, <math>(2/3,2/3,1)</math>, <math>(1,2/3,2/3)</math>, and <math>(4/3,2/3,0)</math>. For every point <math>Z</math> in this region, there exist unique points <math>X</math> and <math>Y</math> such that <math>Z\in XY</math> and <math>ZY=2XZ</math>.
  
 
==See Also==
 
==See Also==
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{{IMO7 box|year=1960|num-b=4|num-a=6}}
  
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{{IMO box|year=1960|num-b=4|num-a=6}}
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[[Category:Olympiad Geometry Problems]]
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[[Category:3D Geometry Problems]]

Latest revision as of 22:28, 18 July 2016

Problem

Consider the cube $ABCDA'B'C'D'$ (with face $ABCD$ directly above face $A'B'C'D'$).

a) Find the locus of the midpoints of the segments $XY$, where $X$ is any point of $AC$ and $Y$ is any point of $B'D'$;

b) Find the locus of points $Z$ which lie on the segment $XY$ of part a) with $ZY = 2XZ$.

Solution

Let $A=(0,0,2)$, $B=(2,0,2)$, $C=(2,0,0)$, $D=(0,0,0)$, $A'=(0,2,2)$, $B'=(2,2,2)$, $C'=(2,2,0)$, and $D'=(0,2,0)$. Then there exist real $x$ and $y$ in the closed interval $[0,2]$ such that $X=(x,0,2-x)$ and $Y=(y,2,y)$.

The midpoint of $XY$ has coordinates $((x+y)/2, 1, (2-x+y)/2)$. Let $a$ and $b$ be the $x$- and $z$-coordinates of the midpoint of $XY$, respectively. We then have that $a+b=y+1$ and $a-b=x-1$, so $a+b\in [1,3]$ and $a-b\in [-1,1]$. The region of points that satisfy these inequalities is the closed square with vertices at $(1,1,2)$, $(2,1,1)$, $(1,1,0)$, and $(0,1,1)$. For every point $P$ in this region, there exist unique points $X$ and $Y$ such that $P$ is the midpoint of $XY$.

If $Z\in XY$ and $ZY=2XZ$, then $Z$ has coordinates $((2x+y)/3, 2/3, (4-2x+y)/3)$. Let $a$ and $b$ be the $x$- and $z$- coordinates of $Z$. We then have that $a+b=(4/3)+(2/3)y$ and $a-b=(4x-4)/3$, and $a\in (4/3,8/3)$ and $b\in (-4/3,4/3)$. The region of points that satisfy these inequalities is the closed rectangle with vertices at $(0,2/3,4/3)$, $(2/3,2/3,1)$, $(1,2/3,2/3)$, and $(4/3,2/3,0)$. For every point $Z$ in this region, there exist unique points $X$ and $Y$ such that $Z\in XY$ and $ZY=2XZ$.

See Also

1960 IMO (Problems)
Preceded by
Problem 4 		1 • 2 • 3 • 4 • 5 • 6 • 7 Followed by
Problem 6
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