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IMO Shortlist 1966 problem 20

Given three congruent rectangles in the space. Their centers coincide, but the planes they lie in are mutually perpendicular. For any two of the three rectangles, the line of intersection of the planes of these two rectangles contains one midparallel of one rectangle and one midparallel of the other rectangle, and these two midparallels have different lengths. Consider the convex polyhedron whose vertices are the vertices of the rectangles.

a.) What is the volume of this polyhedron ?

b.) Can this polyhedron turn out to be a regular polyhedron ? If yes, what is the condition for this polyhedron to be regular ?

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IMO Shortlist 1977 problem 14

Let $E$ be a finite set of points such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that at least one of the following alternatives holds:

(i) $E$ contains five points that are vertices of a convex pyramid having no other points in common with $E;$

(ii) some plane contains exactly three points from $E.$

IMO Shortlist 1977 problem 8

Let $S$ be a convex quadrilateral $ABCD$ and $O$ a point inside it. The feet of the perpendiculars from $O$ to $AB, BC, CD, DA$ are $A_1, B_1, C_1, D_1$ respectively. The feet of the perpendiculars from $O$ to the sides of $S_i$ , the quadrilateral $A_iB_iC_iD_i$ , are $A_{i+1}B_{i+1}C_{i+1}D_{i+1}$ , where $i = 1, 2, 3.$ Prove that $S_4$ is similar to S.

IMO Shortlist 1975 problem 13

Let $A_0,A_1, \ldots , A_n$ be points in a plane such that
(i) $A_0A_1 \leq \frac{1}{ 2} A_1A_2 \leq \cdots \leq \frac{1}{2^{n-1} } A_{n-1}A_n$ and
(ii) $0 < \measuredangle A_{0}A_{1}A_{2} < \measuredangle A_{1}A_{2}A_{3} < \cdots < \measuredangle A_{n-2}A_{n-1}A_{n} < 180^\circ$ ,
where all these angles have the same orientation. Prove that the segments $A_kA_{k+1},A_mA_{m+1}$ do not intersect for each $k$ and $n$ such that $0 \leq k \leq m - 2 < n- 2.$

IMO Shortlist 1966 problem 49

Two mirror walls are placed to form an angle of measure $\alpha$ . There is a candle inside the angle. How many reflections of the candle can an observer see?

IMO Shortlist 1966 problem 22

Let $P$ and $P^{\prime }$ be two parallelograms with equal area, and let their sidelengths be $a,$ $b$ and $a^{\prime },$ $b^{\prime }.$ Assume that $a^{\prime }\leq a\leq b\leq b^{\prime },$ and moreover, it is possible to place the segment $b^{\prime }$ such that it completely lies in the interior of the parallelogram $P.$

Show that the parallelogram $P$ can be partitioned into four polygons such that these four polygons can be composed again to form the parallelogram $P^{\prime }$ .

IMO Shortlist 1966 problem 4

Given $5$ points in the plane, no three of them being collinear. Show that among these $5$ points, we can always find $4$ points forming a convex quadrilateral.