mechmat competition2003

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Open Competition for University Students of Mechanics and Mathematics Faculty of Kyiv State Taras Shevchenko University Problems 1-8 are for 1-2 years students, problems 5-12 are for 3-4 years students ∞ 9n + n(3n + 1)(3n + 2) n=1 √ √ Evaluate the limit lim N − max { n} , (A Kukush) Evaluate N →∞ 1≤n≤N where {x} denotes fractional part of x (D Mitin) For every n ∈ N find the minimum of k ∈ N for which there exist x1 , , xk ∈ Rn such k n that ∀ x ∈ R ∃ a1 , , ak > : x = xi (A Bondarenko) i=1 Find all n ∈ N for which there exist square n × n matrices A and B such that rankA + rankB ≤ n and every square real matrix X which commutes with A and B is of the form X = λI, λ ∈ R (A Bondarenko) √ (A Kukush) Prove the inequality 3 4 n n < 2, n ∞ n n x3 + x3 For every real x = find the sum of the series 3n+1 − x n=0 For every positive integers m ≤ n prove the inequality m n k m! k (−1)m+k Cm ≤ Cnm m m m k=0 (A Kukush) (D Mitin) A parabola with focus F and a triangle T are given at the plane Construct with the compass and the ruler a triangle similar to T such that one of its vertices is F and two other vertices lie on parabola (G Shevchenko) Do there exist a set A ⊂ R , measurable by Lebesgue such that for every set E with zero Lebesgue measure the set A\E is not Borelian? (A Bondarenko) 10 Given is a real symmetric matrix A = (aij )ni,j=1 with eigenvectors ek , k = 1, n and eigenvalues λk , k = 1, n respectively Construct a real symmetric nonnegatively definite matrix X = (xij )ni,j=1 which minimizes the distance d(X, A) = n (xij − aij )2 i,j=1 (A Kukush) 11 Let ϕ be a conform mapping from Ω = {Imz > 0}\T onto {Imz > 0}, where T is a triangle with vertices {1, −1, i} Prove that if z0 ∈ Ω and ϕ(z0 ) = z0 then |ϕ (z0 )| (T Androshchuk) 12 The vertices of a triangle are independent uniformly distributed at unit circle random points Find the expectation of the area of this triangle (A Kukush)
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