IMO Shortlist 1964 problem 5
Dodao/la:
arhiva2. travnja 2012. In tetrahedron
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
, vertex
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
is connected with
![D_0](/media/m/c/6/3/c63683659fedff4c95bb7ef19417a40d.png)
, the centrod if
![\triangle ABC](/media/m/1/f/3/1f3c3c0f3e134a169655f9511ba6ea82.png)
. Line parallel to
![DD_0](/media/m/c/d/c/cdc19826ba6e880ef2396b142a8bb075.png)
are drawn through
![A,B](/media/m/7/1/7/7174f8a9f33236ee137c01b144237389.png)
and
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
. These lines intersect the planes
![BCD, CAD](/media/m/f/8/9/f89608bf63bf746815c6db6c21e6d386.png)
and
![ABD](/media/m/a/5/4/a548bc577543629d304ecba1a042f910.png)
in points
![A_2, B_1,](/media/m/c/0/7/c07c85e5a0e97ce5699a9daeb6db3eb7.png)
and
![C_1](/media/m/b/0/b/b0b10dc32c3e01824e0f0b6753ac2537.png)
, respectively. Prove that the volume of
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
is one third the volume of
![A_1B_1C_1D_0](/media/m/9/7/d/97dade3d60b4ae1d12c1fb03c9fa287e.png)
. Is the result if point
![D_o](/media/m/6/4/f/64f62898ca2f7b12c3c78f9be98e9bb5.png)
is selected anywhere within
![\triangle ABC](/media/m/1/f/3/1f3c3c0f3e134a169655f9511ba6ea82.png)
?
%V0
In tetrahedron $ABCD$, vertex $D$ is connected with $D_0$, the centrod if $\triangle ABC$. Line parallel to $DD_0$ are drawn through $A,B$ and $C$. These lines intersect the planes $BCD, CAD$ and $ABD$ in points $A_2, B_1,$ and $C_1$, respectively. Prove that the volume of $ABCD$ is one third the volume of $A_1B_1C_1D_0$. Is the result if point $D_o$ is selected anywhere within $\triangle ABC$?
Izvor: Međunarodna matematička olimpijada, shortlist 1964