Exercises in orthodox geometry

Here is a collection of interesting problems in geometry. Solution of each problem requires only one non-trivial idea. The project is oriented to graduate students who want to learn fast simple tricks in geometry. For problem solvers. Easy problems are marked by “◦” and hard problems by “∗”. If the solution requires knowledge of some not quite standard theorem it is marked by “+”, in this case the needed theorem mentioned in the beginning of the section. The problems which known to have solutions based on different ideas are marked by “♯”. To get a hint, send an e-mail to the above address with number and name of the problem. For problem makers. This collection is under permanent development. If you have suitable problems or corrections please e-mail it to me. Many thanks. I want to thank everyone sharing these problems. Also I want to thank R. Matveyev, P. Petersen, S. Tabachnikov and number of students in my classes for their interest in this list and for correcting number of mistakes. 1 Curves and surfaces 1.1. Kneser’s spiral. Let γ be a plane curve with strictly monotonic curvature function. Prove that γ has no self-intersections. [Ovsienko–Tabachnikov] 1.2. Closed curve. A smooth closed simple curve with curvature at most 1 bounds a rigion Ω in a plane. Prove that Ω contains a disc or radius 1. ??? 1.3 . A curve in a sphere. Let γ be a closed curve in a unit sphere which intersects each equator, prove that its length is at least 2π. N. Nadirashvili 1.4 . A spring in a tin. Let α be a closed smooth immersed curve inside a unit disc. Prove that average absolute curvature of α is at least 1, with equality if and only if α is the unit circle possibly traversed more than once. ??? If instead of disc we have a rigion bounded by closed convex curve γ then it is still true that average absolute curvature of α is at least as big as average absolute curvature of γ. The proof is not that simple, see ???.