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question:In an isosceles triangle with sides measuring 3 units and a base of 5 units, find the coordinates of the points where the altitude, cevian, median, and symmedian lines intersect the base.
answer:The points are represented in the coordinate system as follows: - Symmedian: left( begin{array}{cc} 5 & 0 frac{125}{68} & frac{25 sqrt{11}}{68} end{array} right) - Cevian (which is also the median in an isosceles triangle): left( begin{array}{cc} 5 & 0 frac{5}{4} & frac{sqrt{11}}{4} end{array} right) - Altitude: left( begin{array}{cc} 5 & 0 frac{125}{36} & frac{25 sqrt{11}}{36} end{array} right) These points define the lines that intersect the base of the triangle at the specified coordinates.
question:Multiply the following two matrices: left( begin{array}{ccccc} 3 & 1 & -2 & 0 & 0 1 & -2 & 1 & -1 & -2 3 & 0 & -1 & 3 & -1 end{array} right) and left( begin{array}{cccc} -3 & 2 & -1 & 3 3 & -2 & -1 & 1 -1 & -1 & -3 & 2 1 & -2 & 2 & -3 -2 & -2 & 0 & -2 end{array} right)
answer:The product of the two matrices is: left( begin{array}{cccc} -4 & 6 & 2 & 6 -7 & 11 & -4 & 10 -3 & 3 & 6 & 0 end{array} right)
question:Prove that if x and y are coprime non-square natural numbers, and R is a coprime integer, then sqrt{x} + sqrt{y} neq R and sqrt{xy} neq R.
answer:We have already proven that sqrt{xy} neq R for coprime non-square natural numbers x and y, and a coprime integer R. Now, to prove that sqrt{x} + sqrt{y} neq R, assume for contradiction that sqrt{x} + sqrt{y} = R, where R is rational. We can express this as: x + y + 2sqrt{xy} = R^2 Since x and y are coprime non-square integers, sqrt{xy} is irrational. This means that x + y + 2sqrt{xy}, which contains an irrational term, is also irrational. However, R^2 is a perfect square and hence rational. This contradiction implies that our assumption must be false, and therefore sqrt{x} + sqrt{y} neq R.
question:Let x_i,y_iinmathbb{R}, a_iin[0,1], with y_1>y_2 and y_3>y_4. We also have (y_1-y_3)^2 + (y_2 -y_4)^2 <varepsilon . I need to solve the following constrained optimization problem min_{x_1,x_2,x_3,x_4}sum_{i = 1}^4 a_i(x_i-y_i)^2/2 text{subject to } x_2geq x_1, quad quad quad ; x_4 geq x_3 quad quad quad ;quad quad quad ;quad quad quad ; (x_1-x_3)^2 + (x_2-x_4)^2leq varepsilon
answer:To solve this constrained optimization problem, we can use the method of Lagrange multipliers. The Lagrangian is given by L(lambda,x) = sum_{i = 1}^4 a_i(x_i-y_i)^2/2 + lambda_1(x_1-x_2)+ lambda_2(x_3-x_4) + lambda_3( (x_1-x_3)^2 + (x_2-x_4)^2-varepsilon) Differentiating and assuming binding constraints (x_1 = x_2, x_3 = x_4, varepsilon = (x_1-x_3)^2 + (x_2-x_4)^2) we get x_k = frac{sum_i{a_i y_i}mp sqrt{varepsilon/2}}{sum_i a_i }, text{ for } k = 1,2 and x_k = frac{sum_i{a_i y_i}pm sqrt{varepsilon/2}}{sum_i a_i }, text{ for } k = 3,4