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Difference between revisions of "1965 IMO Problems/Problem 3"

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== Problem ==
 
== Problem ==
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Given the tetrahedron <math>ABCD</math> whose edges <math>AB</math> and <math>CD</math> have lengths <math>a</math> and <math>b</math> respectively. The distance between the skew lines <math>AB</math> and <math>CD</math> is <math>d</math>, and the angle between them is <math>\omega </math>. Tetrahedron <math>ABCD</math> is divided into two solids by plane <math>\varepsilon </math>, parallel to lines <math>AB</math> and <math>CD</math>. The ratio of the distances of <math>\varepsilon </math> from <math>AB</math> and <math>CD</math> is equal to <math>k</math>. Compute the ratio of the volumes of the two solids obtained.
+
 
 +
Given the tetrahedron <math>ABCD</math> whose edges <math>AB</math> and <math>CD</math> have lengths <math>a</math> and <math>b</math> respectively. The distance between the skew lines <math>AB</math> and <math>CD</math> is <math>d</math>, and the angle between them is <math>\omega</math>. Tetrahedron <math>ABCD</math> is divided into two solids by plane <math>\varepsilon</math>, parallel to lines <math>AB</math> and <math>CD</math>. The ratio of the distances of <math>\varepsilon</math> from <math>AB</math> and <math>CD</math> is equal to <math>k</math>. Compute the ratio of the volumes of the two solids obtained.
 +
 
  
 
== Solution ==
 
== Solution ==
  
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Let the plane meet AD at X, BD at Y, BC at Z and AC at W. Take plane parallel to BCD through WX and let it meet AB in P.
+
Let the plane meet <math>AD</math> at <math>X</math>, <math>BD</math> at <math>Y</math>, <math>BC</math> at <math>Z</math> and <math>AC</math> at <math>W</math>.
 +
Take a plane parallel to <math>BCD</math> through <math>WX</math> and let it meet <math>AB</math> in <math>P</math>.
 +
 
 +
[[File:Prob_1965_3.png|600px]]
 +
 
 +
Since the distance of <math>AB</math> from <math>WXYZ</math> is <math>k</math> times the distance of <math>CD</math>,
 +
we have that <math>AX = k \cdot XD</math> and hence <math>AX/AD = k/(k+1).</math> Similarly
 +
<math>AP/AB = AW/AC = AX/AD.</math> <math>XY</math> is parallel to <math>AB</math>, so also
 +
<math>AX/AD = BY/BD = BZ/BC.</math>
 +
 
 +
vol <math>ABWXYZ =</math> vol <math>APWX +</math> vol <math>WXPBYZ.</math>  <math>APWX</math> is similar to the
 +
tetrahedron <math>ABCD.</math> The sides are <math>k/(k + 1)</math> times smaller, so
 +
vol <math>APWX = k^3/(k + 1)^3</math> vol <math>ABCD.</math>
 +
 
 +
The base of the prism <math>WXPBYZ</math> is <math>BYZ</math> which is similar to <math>BCD</math> with
 +
sides <math>k/(k + 1)</math> times smaller and hence area <math>k^2/(k + 1)^2</math> times
 +
smaller.  Its height is <math>1/(k + 1)</math> times the height of <math>A</math> above
 +
<math>ABCD,</math> so vol prism <math>= 3 k^2/(k + 1)^3</math> vol <math>ABCD.</math>
 +
 
 +
Thus vol <math>ABWXYZ = (k^3 + 3k^2)/(k + 1)^3</math> vol <math>ABCD.</math>
 +
 
 +
We get the volume of the other piece as vol <math>ABCD\ -</math> vol <math>ABWXYZ,</math> and
 +
hence the ratio is (after a little manipulation) <math>k^2(k + 3)/(3k + 1).</math>
 +
 
 +
 
 +
== Remark (added by pf02, November 2024) ==
 +
 
 +
Note that the problem is untypically sloppy or misleading.
 +
It mentions the sizes <math>a, b, d, \omega</math> as if they are needed.
 +
In fact, as the solution above shows, they are not needed either
 +
in expressing the result, or in solving the problem.  A thorough
 +
problem solver might worry about not having used all the data in
 +
the problem.
  
−
Since the distance of AB from WXYZ is k times the distance of CD, we have that AX = k·XD and hence that AX/AD = k/(k+1). Similarly AP/AB = AW/AC = AX/AD. XY is parallel to AB, so also AX/AD = BY/BD = BZ/BC.
+
However, one can imagine other solutions, where these quantities
 +
would be used in the process of solving the problem.  For example
 +
one could break up the doubly truncated triangular prism <math>ABWXYZ</math>
 +
into pyramids <math>APWX, PXYZW, BPZY</math>. Computing the volume of each
 +
of these pyramids would require all the data in the problem.
 +
The end result should of course be the same, but the thorough
 +
problem solver would not have the uneasy feeling of not having
 +
used all the data in the problem.
  
−
vol ABWXYZ = vol APWX + vol WXPBYZ. APWX is similar to the tetrahedron ABCD. The sides are k/(k+1) times smaller, so vol APWX = k3(k+1)3 vol ABCD. The base of the prism WXPBYZ is BYZ which is similar to BCD with sides k/(k+1) times smaller and hence area k2(k+1)2 times smaller. Its height is 1/(k+1) times the height of A above ABCD, so vol prism = 3 k2(k+1)3 vol ABCD. Thus vol ABWXYZ = (k3 + 3k2)/(k+1)3 vol ABCD. We get the vol of the other piece as vol ABCD - vol ABWXYZ and hence the ratio is (after a little manipulation) k2(k+3)/(3k+1).
 
  
 
== See Also ==  
 
== See Also ==  

Latest revision as of 18:09, 10 November 2024

Problem

Given the tetrahedron $ABCD$ whose edges $AB$ and $CD$ have lengths $a$ and $b$ respectively. The distance between the skew lines $AB$ and $CD$ is $d$, and the angle between them is $\omega$. Tetrahedron $ABCD$ is divided into two solids by plane $\varepsilon$, parallel to lines $AB$ and $CD$. The ratio of the distances of $\varepsilon$ from $AB$ and $CD$ is equal to $k$. Compute the ratio of the volumes of the two solids obtained.


Solution

Let the plane meet $AD$ at $X$, $BD$ at $Y$, $BC$ at $Z$ and $AC$ at $W$. Take a plane parallel to $BCD$ through $WX$ and let it meet $AB$ in $P$.

Prob 1965 3.png

Since the distance of $AB$ from $WXYZ$ is $k$ times the distance of $CD$, we have that $AX = k \cdot XD$ and hence $AX/AD = k/(k+1).$ Similarly $AP/AB = AW/AC = AX/AD.$ $XY$ is parallel to $AB$, so also $AX/AD = BY/BD = BZ/BC.$

vol $ABWXYZ =$ vol $APWX +$ vol $WXPBYZ.$ $APWX$ is similar to the tetrahedron $ABCD.$ The sides are $k/(k + 1)$ times smaller, so vol $APWX = k^3/(k + 1)^3$ vol $ABCD.$

The base of the prism $WXPBYZ$ is $BYZ$ which is similar to $BCD$ with sides $k/(k + 1)$ times smaller and hence area $k^2/(k + 1)^2$ times smaller. Its height is $1/(k + 1)$ times the height of $A$ above $ABCD,$ so vol prism $= 3 k^2/(k + 1)^3$ vol $ABCD.$

Thus vol $ABWXYZ = (k^3 + 3k^2)/(k + 1)^3$ vol $ABCD.$

We get the volume of the other piece as vol $ABCD\ -$ vol $ABWXYZ,$ and hence the ratio is (after a little manipulation) $k^2(k + 3)/(3k + 1).$


Remark (added by pf02, November 2024)

Note that the problem is untypically sloppy or misleading. It mentions the sizes $a, b, d, \omega$ as if they are needed. In fact, as the solution above shows, they are not needed either in expressing the result, or in solving the problem. A thorough problem solver might worry about not having used all the data in the problem.

However, one can imagine other solutions, where these quantities would be used in the process of solving the problem. For example one could break up the doubly truncated triangular prism $ABWXYZ$ into pyramids $APWX, PXYZW, BPZY$. Computing the volume of each of these pyramids would require all the data in the problem. The end result should of course be the same, but the thorough problem solver would not have the uneasy feeling of not having used all the data in the problem.


See Also

1965 IMO (Problems) • Resources
Preceded by
Problem 2 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Problem 4
All IMO Problems and Solutions
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