IMO Shortlist 1966 problem 17
Kvaliteta:
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Avg: 0,0 Let
and
be two arbitrary parallelograms in the space, and let
be points dividing the segments
in equal ratios.
a.) Prove that the quadrilateral
is a parallelogram.
b.) What is the locus of the center of the parallelogram
when the point
moves on the segment
?
(Consecutive vertices of the parallelograms are labelled in alphabetical order.
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
![A^{\prime }B^{\prime}C^{\prime }D^{\prime }](/media/m/4/b/2/4b24d7eb03c501b8a082b9ec3f66a2fd.png)
![M,](/media/m/7/4/8/74823070e1f7ea0d7a4ab48e207bc31b.png)
![N,](/media/m/7/9/a/79a4f22a4522fc99b6f195180241b267.png)
![P,](/media/m/a/1/4/a14de1403fe9b971067bc4aac029b8a8.png)
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
![AA^{\prime },](/media/m/a/2/d/a2de4c2b63529dc2d4f5540d3b113cdf.png)
![BB^{\prime },](/media/m/1/5/6/15623b27f68e829c157da7e755153257.png)
![CC^{\prime },](/media/m/e/d/3/ed31a5678c737130dfaa381b02a69310.png)
![DD^{\prime }](/media/m/9/a/4/9a4082a44aa3681deeb1ff90a31fc99e.png)
a.) Prove that the quadrilateral
![MNPQ](/media/m/7/4/a/74aafacb841213f440a2ccfc75aad569.png)
b.) What is the locus of the center of the parallelogram
![MNPQ,](/media/m/9/9/4/9949edaa8f030ec4d7e693c43250f559.png)
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
![AA^{\prime }](/media/m/7/4/a/74a0b04574d469aa777673e3b4a271ff.png)
(Consecutive vertices of the parallelograms are labelled in alphabetical order.
Izvor: Međunarodna matematička olimpijada, shortlist 1966