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Mathematical olympiad 15.05.2010 for students Faculty of Mechanics and Mathematics, Moscow State University 1. Let f : [0, 1] [0, 1] and g : [0, 1] [0, 1] be continuous functions such that f (g (x)) = g (f (x)) for all x [0, 1]. Suppose also that g is continuously differentiable in (0, 1) and g (x) 1 if x (0, 1). Prove that there exists a point t [0, 1] such that f (t) = g (t) = t. 2. Fix vectors v1 , . . . , vn Rk . Denote by (v1 , . . . , vn ) the k в n-matrix formed by the coordinates of v1 , . . . , vn . Prove that there exist m and w1 , . . . , wn Rm with the following two properties: (i) (w1 , . . . , wn )(v1 , . . . , vn )T = 0 Rmk ; (ii) for every l and any elements u1 , . . . , un Rl such that (u1 , . . . , un )(v1 , . . . , vn )T = 0 Rlk , there exists a linear map T : Rm Rl such that T (wi ) = ui for all i = 1, . . . , n. 3. Let A = (aij ) be a square n в n-matrix with real entries. Suppose that for each subset I {1, . . . , n} and the matrix AI := (aks )k,sI we have An = 0. Prove that one I can make the matrix A upper-triangular with zeroes at the principal diagonal by finitely many transformations of the following type: one transposes any two lines in the matrix and then transposes the columns with the same numbers. 4. Let S AB C be a pyramid. Consider spheres X inscribed into the trihedral corner with the vertex S . Describe the set of the intersection points of the three planes tangent to X passing through lines AB , B C , C A and not containing the pyramid faces. (S, A, B , C are fixed while X varies.) 5. For which (a) odd n (b) even n can one color the vertices of the n-dimensional cube into n colors so that for each vertex all n neighboring vertices have different colors? Mathematical olympiad 15.05.2010 for students Faculty of Mechanics and Mathematics, Moscow State University 1. Let f : [0, 1] [0, 1] and g : [0, 1] [0, 1] be continuous functions such that f (g (x)) = g (f (x)) for all x [0, 1]. Suppose also that g is continuously differentiable in (0, 1) and g (x) 1 if x (0, 1). Prove that there exists a point t [0, 1] such that f (t) = g (t) = t. 2. Fix vectors v1 , . . . , vn Rk . Denote by (v1 , . . . , vn ) the k в n-matrix formed by the coordinates of v1 , . . . , vn . Prove that there exist m and w1 , . . . , wn Rm with the following two properties: (i) (w1 , . . . , wn )(v1 , . . . , vn )T = 0 Rmk ; (ii) for every l and any elements u1 , . . . , un Rl such that (u1 , . . . , un )(v1 , . . . , vn )T = 0 Rlk , there exists a linear map T : Rm Rl such that T (wi ) = ui for all i = 1, . . . , n. 3. Let A = (aij ) be a square n в n-matrix with real entries. Suppose that for each subset I {1, . . . , n} and the matrix AI := (aks )k,sI we have An = 0. Prove that one I can make the matrix A upper-triangular with zeroes at the principal diagonal by finitely many transformations of the following type: one transposes any two lines in the matrix and then transposes the columns with the same numbers. 4. Let S AB C be a pyramid. Consider spheres X inscribed into the trihedral corner with the vertex S . Describe the set of the intersection points of the three planes tangent to X passing through lines AB , B C , C A and not containing the pyramid faces. (S, A, B , C are fixed while X varies.) 5. For which (a) odd n (b) even n can one color the vertices of the n-dimensional cube into n colors so that for each vertex all n neighboring vertices have different colors?