MathDB

1

Part of 2014 JBMO TST - Turkey

Problems(2)

Show that the sum of the angles is equal to 180

Source: Turkey JBMO TST 2014 P1

5/30/2014
In a triangle ABCABC, the external bisector of ∠BAC\angle BAC intersects the ray BCBC at DD. The feet of the perpendiculars from BB and CC to line ADAD are EE and FF, respectively and the foot of the perpendicular from DD to ACAC is GG. Show that ∠DGE+∠DGF=180∘\angle DGE + \angle DGF = 180^{\circ}.
ratiogeometryangle bisectorgeometry proposed
x(x+1)^3=(2x+a)(x+a+1)

Source: Turkey JBMO TST 2014 P5

6/21/2014
Find all real values of aa for which the equation x(x+1)3=(2x+a)(x+a+1)x(x+1)^3=(2x+a)(x+a+1) has four distinct real roots.
quadraticsalgebra proposedalgebra