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IMO Shortlist 1975 problem 12

Consider on the first quadrant of the trigonometric circle the arcs $AM_1 = x_1,AM_2 = x_2,AM_3 = x_3, \ldots , AM_v = x_v$ , such that $x_1 < x_2 < x_3 < \cdots < x_v$ . Prove that
$\sum_{i=0}^{v-1} \sin 2x_i - \sum_{i=0}^{v-1} \sin (x_i- x_{i+1}) < \frac{\pi}{2} + \sum_{i=0}^{v-1} \sin (x_i + x_{i+1})$

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IMO Shortlist 1982 problem 8

A convex, closed figure lies inside a given circle. The figure is seen from every point of the circumference at a right angle (that is, the two rays drawn from the point and supporting the convex figure are perpendicular). Prove that the center of the circle is a center of symmetry of the figure.

IMO Shortlist 1981 problem 11

On a semicircle with unit radius four consecutive chords $AB,BC, CD,DE$ with lengths $a, b, c, d$ , respectively, are given. Prove that
$a^2 + b^2 + c^2 + d^2 + abc + bcd < 4.$

IMO Shortlist 1977 problem 4

Describe all closed bounded figures $\Phi$ in the plane any two points of which are connectable by a semicircle lying in $\Phi$ .

IMO Shortlist 1973 problem 5

A circle of radius 1 is located in a right-angled trihedron and touches all its faces. Find the locus of centers of such circles.

IMO Shortlist 1973 problem 2

Given a circle $K$ , find the locus of vertices $A$ of parallelograms $ABCD$ with diagonals $AC \leq BD$ , such that $BD$ is inside $K$ .

IMO Shortlist 1971 problem 4

We are given two mutually tangent circles in the plane, with radii $r_1, r_2$ . A line intersects these circles in four points, determining three segments of equal length. Find this length as a function of $r_1$ and $r_2$ and the condition for the solvability of the problem.