MathDB

3

Part of 2013 Bogdan Stan

Problems(4)

Engaging geometry collinearity

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7/5/2020
O O is the center of a parallelogram ABCD. ABCD. Let G G on the segment OB OB (excluding its endpoints), N N on the line DC DC and M M on the segment AD AD (excluding its endpoints) such that CN>ND,AM=6MD CN>ND, AM=6MD and so that there exists a natural number n≥3 n\ge 3 such that OB=nGO. OB=nGO. Show that G,M,N G,M,N are collinear if and only if (CNND−6)(n+1)=2. \left( \frac{CN}{ND} -6 \right) (n+1)=2.
geometryparallelogram
cos(a)+cos(b)+cos(c)=sin(a)+sin(b)+sin(c)=0

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7/5/2020
Let a,b,c a,b,c be three real numbers such that cos⁡a+cos⁡b+cos⁡c=sin⁡a+sin⁡b+sin⁡c=0. \cos a+\cos b+\cos c=\sin a+\sin b+\sin c=0. Prove that i) cos⁡6a+cos⁡6b+cos⁡6c=3cos⁡(2a+2b+2c) \cos 6a+\cos 6b+\cos 6c=3\cos (2a+2b+2c) ii) sin⁡6a+sin⁡6b+sin⁡6c=3sin⁡(2a+2b+2c) \sin 6a+\sin 6b+\sin 6c=3\sin (2a+2b+2c)
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trigonometryinequalitiesTrigonometric inequalityalgebra
Complex eigenvalues; 4 matrices

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7/5/2020
Let be four n×n n\times n real matrices A,B,C,D A,B,C,D having the property that C+D−1 C+D\sqrt{-1} is the inverse of A+B−1. A+B\sqrt{-1} . Show that ∣det⁡(A+B−1)∣2⋅∣det⁡C∣=det⁡A. \left| \det\left( A+B\sqrt{-1} \right) \right|^2\cdot\left| \det C \right| =\det A.
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eigenvaluematriesMatriceslinear algebra
A rational antiderivative

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7/5/2020
∫1+2x31+x2−2x3+x6dx \int \frac{1+2x^3}{1+x^2-2x^3+x^6} dx
Ion Nedelcu and Lucian Tutescu
integrationIndefinite integralantiderivativecalculus