The fixed point theorem is the statement that some equation (or system of equations) has a solution. Topological theorems about fixed points of continuous maps of a segment, a square, a circle and a sphere are proved. The proofs use various forms of the combinatorial-geometrical lemma of Sperner and the concept of the degree of display. For high school and junior students . Other issues of the series: . 01. - Markushevich A . and . Return sequences. Issue. 02. - Nathanson and . P. The simplest tasks to the maximum and minimum. Issue. 03. - Sominsky I. C. Method of mathematical induction. Issue. 04. - Markushevich A . and . Wonderful Curves. Issue. 05. - Korovkin P. P. Inequality. Issue. 06. - Vorobyov N . N. Fibonacci Numbers. Issue. 07. - Kurosh A. G. Algebraic equations of arbitrary degrees. Issue. 08. - Gelfond A. About. Solving equations in integers. Issue. 09. - Markushevich A . and . Areas and logarithms. Issue. 10. - Smogorzhevsky A. C. Coordinate Method. Issue. 11. - Dubnov I . C. Errors in geometric evidence. Issue. 12. - Nathanson and . P. Combination of infinitesimal quantities. Issue. 13. - Markushevich A . and . Complex numbers and conformal transformations. Issue. 14. - Fetisov A. and . On Evidence in Geometry. Issue. 15. - Shafarevich I . R. On the solutions of the equations of higher degrees . Issue. 16. - Shervatov V. G. Hyperbolic Functions . Issue. 17. - Boltyansky V. G. What is differentiation?[edit] 18. - Mirakyan G. M. Straight circular cylinder. Issue. 19. - Lusternik L. A. Shortest lines. Issue. 20. - Lopshitz A. M. Calculation of the area of oriented figures. Issue. 21. - Head L. and . , Yaglom I. M. Induction in Geometry. Issue. 22. - Boltyansky V. G. Equal-sized and equal-sized figures. Issue. 23. - Smogorzhevsky A. C. On the geometry of Lobachevsky . Issue. 24. - Argunov B. and . , Skornyakov L. A. Configuration Theorems. Issue. 25. - Smogorzhevsky A. C. Linear in geometric constructions. Issue. 26. - Trahtenbrot B. A. Algorithms and machine problem solving. Issue. 27. - Ouspensky V. A. Some applications of mechanics to mathematics. Issue. 28. - Arkhangelsky N . A. , Hare B. and . Automatic digital machines. Issue. 29. - Kostovsky A. N. Geometric constructions by one circum. Issue. 30. - Shilov G. E. How to make charts. Issue. 31. - Dorfman A. G. Optics of conical sections . Issue. 32. - Wentzel E. C. Elements of Game Theory . Issue. 33. - Barsov A. C. What is Linear Programming?[edit] 34. - Margulis B. E. Systems of linear equations . Issue. 35. - Vilenkin N. I. The method of successive approximations. Issue. 36. - Boltyansky V. G. by . Issue. 37. - Shilov G. E. Simple gamma. Arrangement of the musical scale. Issue. 38. - Shrader Y. A. What is distance?[edit] 39. - Vorobyov N . N. Signs of divisibility. Issue. 40. - Fomin S. In. Counting systems. Issue. 41. - Kogan B. Yu. Application of Mechanics to Geometry . Issue. 42. - Lyubich Yu. and . , Shore L. A. Kinematic method in geometric problems. Issue. 43. - Ouspensky V. A. Pascal's Triangle . Issue. 44. - Bakelman I. I. Inversion. Issue. 45. - Yaglom I. M. Extraordinary Algebra . Issue. 46. - Sable and . M. Monte Carlo Method . Issue. 47. - Kaluzhnin L. A. Basic Arithmetic Theorem. Issue. 48. - Solodovnikov A . C. Linear Inequality Systems . Issue. 49. - Shilov G. E. Mathematical analysis in the field of rational functions . Issue. 50. - Boltyansky V. G. , Gohberg I. C. Splitting the pieces into smaller parts. Issue. 51. - Beskin N. M. Images of spatial figures. Issue. 52. - Beskin N. M. Division of the segment in this respect. Issue. 53. - Rosenfeld B. A. , Sergeyeva N. D. Stereographic projection. Issue. 54. - Ouspensky V. A. Post Machine. Issue. 55. - Beran L. Ordered set. Issue. 57. - Ouspensky V. A. Gdel's Theorem on Incompleteness