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Problem #PRU-110157

Problems Geometry Plane geometry Plane transformations Similarity Homothetic transformations, rotations of homothetic transformations Circles Connection between the arc length, the angle, and the chord Calculus Derivative Derivative and multiple roots

14–16 5.0

The circles $σ_{1}$ and $σ_{2}$ intersect at points $A$ and $B$ . At the point $A$ to $σ_{1}$ and $σ_{2}$ , respectively, the tangents $l_{1}$ and $l_{2}$ are drawn. The points $T_{1}$ and $T_{2}$ are chosen respectively on the circles $σ_{1}$ and $σ_{2}$ so that the angular measures of the arcs $T_{1} A$ and $A T_{2}$ are equal (the arc value of the circle is considered in the clockwise direction). The tangent $t_{1}$ at the point $T_{1}$ to the circle $σ_{1}$ intersects $l_{2}$ at the point $M_{1}$ . Similarly, the tangent $t_{2}$ at the point $T_{2}$ to the circle $σ_{2}$ intersects $l_{1}$ at the point $M_{2}$ . Prove that the midpoints of the segments $M_{1} M_{2}$ are on the same line, independent of the positions of the points $T_{1}, T_{2}$ .

Problem #PRU-56572

Problems Geometry Plane geometry Circles Connection between the arc length, the angle, and the chord

13–14 2.0

The isosceles trapeziums $A B C D$ and $A_{1} B_{1} C_{1} D_{1}$ with corresponding parallel sides are inscribed in a circle. Prove that $A C = A_{1} C_{1}$ .

Problem #PRU-56573

Problems Geometry Plane geometry Circles Connection between the arc length, the angle, and the chord

13–14 2.0

From the point $M$ , moving along a circle the perpendiculars $M P$ and $M Q$ are dropped onto the diameters $A B$ and $C D$ . Prove that the length of the segment $P Q$ does not depend on the position of the point $M .$

Problem #PRU-58094

Problems Geometry Plane geometry Circles Central angle. Arc length and circumference Connection between the arc length, the angle, and the chord Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–15 4.0

Several chords are drawn through a unit circle. Prove that if each diameter intersects with no more than $k$ chords, then the total length of all the chords is less than $π k$ .