here I shall collect in one place links solutions to old problems from International Mathematical Olympiads, posted in this page all given by Kostas Dortsios [a Greek fond of 3D Geometry and New Technologies]
Soon there shall be a pdf collecting all the given solutions in one place
Two planes, P and Q; intersect along the line p: The point A is given in the plane P; and the point C in the plane Q; neither of these points lies on the straight line p: Construct an isosceles trapezoid ABCD (with AB parallel to CD) in which a circle can be inscribed, and with vertices B and D lying in the planes P and Q respectively.
proposed by Czechoslovakia
[variation by Kostas Dortsios, ABDC instead of ABCD]
Two planes, P and Q, intersect along the line p. The point A is given in the plane P, and the point C in the plane Q, neither of these points lies on the straight line p. Construct an isosceles trapezoid ABDC (with AB parallel to CD) in which a circle can be inscribed, and with vertices B and D lying in the planes P and Q respectively
1960 IMO Problem 5 (CZS) (here - Kostas Dortsios) [only in Greek at the moment]
Consider the cube ABCDA'B'C'D' (with face ABCD directly above face A'B'C'D').
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 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 segments XY of part (a) such that ΖΥ = 2 ΧΖ .
proposed by Czechoslovakia
by Gheorghe D. Simionescu
1962 IMO Problem 3 (CZS) (soon)
Consider the cube ABCDA΄B΄C΄D΄ (ABCD and A΄B΄C΄D΄ are the upper and lower bases, respectively, and edges AA΄, BB΄, CC΄, DD΄ are parallel). The point X moves at constant speed along the perimeter of the square ABCD in the direction ABCDA, and the point Y moves at the same rate along the perimeter of the square B΄C΄CB in the direction B΄C΄CB΄ Β. Points X and Y begin their motion at the same instant from the starting positions A and B΄, respectively. Determine and draw the locus of the midpoints of the segments XY.
1961 IMO Problem 6 (ROM) (soon)
Consider a plane ε and three non-collinear points A, B, C on the same side of ε, suppose the plane determined by these three points is not parallel to ε. In plane a take three arbitrary points A΄, B΄, C΄. Let L, M, N be the midpoints of segments AA΄, BB΄, CC΄΄, let G be the centroid of triangle LMN. (We will not consider positions of the points A΄, B΄, C΄such that the points L, M, N do not form a triangle.) What is the locus of point G as A΄, B΄, C΄. range independently over the plane ε ?
by Gheorghe D. Simionescu
1962 IMO Problem 3 (CZS) (soon)
Consider the cube ABCDA΄B΄C΄D΄ (ABCD and A΄B΄C΄D΄ are the upper and lower bases, respectively, and edges AA΄, BB΄, CC΄, DD΄ are parallel). The point X moves at constant speed along the perimeter of the square ABCD in the direction ABCDA, and the point Y moves at the same rate along the perimeter of the square B΄C΄CB in the direction B΄C΄CB΄ Β. Points X and Y begin their motion at the same instant from the starting positions A and B΄, respectively. Determine and draw the locus of the midpoints of the segments XY.
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