Three points x1, x2, x3 are given on a plane. Determine a point x0 such that the sum of the squares of distances from the point x0 to the points x1, x2, x3 is the smallest.106 106) In the space ℝn there are given N points x1, … , xN and N positive numbers m1, … , mN. Determine a point x0, such that the sum with the coefficients mi of the squares of distances from the point x0 to X1, … , xN is the smallest.
107 107) Solve the previous problem, provided that the point x0 lies on the sphere of unit radius.
108 108) Solve the previous problem, provided that the point x0 belongs to the ball of unit radius.
109 109) Find the distance from a point to the ellipse. How many normals can be drawn from a point to the ellipse (Apollonius’s problem)?
110 110) Find the distance from a point x0 to a parabola.
111 111) Find the distance from a point x0 to a hyperbole.
112 112) Find the distance from a point x0 in the space ℝn to the hyperplane H = {x ∈ ℝn|〈a, x〉 = β}.
113 113) Find the distance from a point x0 to the hyperplane in a Hilbert space.
114 114) Find the distance from a point x0 in the space ℝn to a line.
115 115) Find the minimum of a linear function in the space ℝn on a unit ball.
116 116) In the ellipse x2/a2 + y2/b2 = 1 insert a rectangle of the largest area with sides parallel to the coordinate axes.
117 117) In the ellipsoid x2/a2 + y2/b2 + z2/c2 = 1 insert a rectangular parallelepiped of the largest volume with edges parallel to the axes of coordinates.
118 118) Prove the inequality between the power averagessolving the problem
119 119) Prove the inequality
120 120) Prove the inequality
121 121) Prove the Hölder inequalityMake sure that for y = (y1, … , yn) = 0, the equality holds only when |xi| = λ|yi|, i = 1, … , n.
122 122) Prove the Minkowski inequalityMake sure that for y = (y1, … , yn) = 0, the equality holds only when xi = λyi, λ > 0,i = 1, … , n.
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