MathDB

7

Part of 2017 QEDMO 15th

Problems(2)

x +(3x-y)/(x^2 + y^2) = 3, y - (x + 3y/(x^2 + y^2) = 0

Source: 15th -a QEDMO problem 7 (19. - 22. 10. 2017) https://artofproblemsolving.com/community/c1512515_qedmo_2005

5/29/2021
Find all real solutions x,yx, y of the system of equations {x+3x−yx2+y2=3y−x+3yx2+y2=0\begin{cases} x + \dfrac{3x-y}{x^2 + y^2} = 3 \\ \\ y-\dfrac{x + 3y}{x^2 + y^2} = 0 \end{cases}
algebrasystem of equationsSystem
integer matrix BA^kB^{-1} , det (A) = 1, det (B) \ne 0

Source: 15th -b QEDMO problem 7 (19. - 22. 10. 2017) https://artofproblemsolving.com/community/c1512515_qedmo_2005

5/30/2021
Let A,BA, B be integer n×nn\times n-matrices, where det(A)=1det (A) = 1 and det(B)≠0det (B) \ne 0. Show that there is a k∈Nk \in N, for which BAkB−1BA^kB^{-1} is a matrix with integer entries.