FOREWORDThe applications of mathematics to physics (in particular, to mechanics) are well-known. We need only open a school text-book to find examples. The higher branches of mecha­ nics demand a complex and refined mathematical apparatus.There are, however, mathematical problems for whose solu­ tion we can successfully use the ideas and laws of physics. A number of problems of this kind soluble by methods drawn from mechanics (namely, by using the laws of equilibrium) were given by the author in his lecture "The solving of mathematical problems by the methods of mechanics", whichwas read to pupils in their seventh year of secondary school at the Moscow State University on 19 February 1956, this lecture, with very minor additions, makes up the contents of this article.Foreword vii1. Problem on a tangent toa circle 1 2. Problem on a tangent to an ellipse 5 3. Problems on tangents to parabolas and hyperbolas 11 4. Principle of least potential energy 18 5. Material points and the centre of gravity 23 6. The centre of gravity and a system of two material points 28 7. Theorems about the intersection of straight lines 30 8. The centre of gravity of a rod with many loads 35 9. A problem in the theory of numbers (formulation) 3910. A problem in the theory of numbers (solution) 43 11. The impossibility of perpetual motion 49Conclusion 51