MathDB

2

Part of 2024 OMpD

Problems(3)

calculate the perimeter of triangle MNP

Source: 2024 5th OMpD L2 P2 - Brazil - Olimpíada Matemáticos por Diversão

10/16/2024
Let ABCDABCD be a convex quadrilateral, and MM, NN, and PP be the midpoints of diagonals ACAC and BDBD, and side ADAD, respectively. Also, suppose that ∠ABC+∠DCB=90\angle{ABC} + \angle{DCB} = 90 and that AB=6AB = 6, CD=8CD = 8. Calculate the perimeter of triangle MNPMNP.
geometryperimeter
Let ABCDE be a convex pentagon whose vertices lie on a circle

Source: 2024 5th OMpD L3 P2 - Brazil - Olimpíada Matemáticos por Diversão

10/16/2024
Let ABCDE ABCDE be a convex pentagon whose vertices lie on a circle Γ \Gamma . The tangents to Γ \Gamma at C C and E E intersect at X X , and the segments CE CE and AD AD intersect at Y Y . Given that CE CE is perpendicular to BD BD , that XY XY is parallel to BD BD , that AY=BD AY = BD , and that ∠BAD=30∘ \angle BAD = 30^\circ , what is the measure of the angle ∠BDA \angle BDA ?
Proposed by João Gilberti Alves Tavares
geometry
matrices A, B such that ABBA - BAAB = A - B.

Source: 2024 5th OMpD LU P2 - Brazil - Olimpíada Matemáticos por Diversão

10/16/2024
Let n n be a positive integer, and let A A and B B be n×n n \times n matrices with real coefficients such that
ABBA−BAAB=A−B. ABBA - BAAB = A - B.
(a) Prove that Tr(A)=Tr(B) \text{Tr}(A) = \text{Tr}(B) and that Tr(A2)=Tr(B2) \text{Tr}(A^2) = \text{Tr}(B^2) .
(b) Prove that det⁡A=det⁡B \det A = \det B .
Note: Tr(X) \text{Tr}(X) denotes the trace of X X , which is the sum of the elements on its main diagonal, and det⁡X \det X denotes the determinant of X X .
Matricesalgebralinear algebra