MathDB

2

Part of 2019 New Zealand MO

Problems(2)

(x^2 + 3x + 1)^{x^2-x-6} = 1

Source: New Zealand MO 2019 Round 1 p2

9/20/2021
Find all real solutions to the equation (x2+3x+1)x2−x−6=1(x^2 + 3x + 1)^{x^2-x-6} = 1.
algebra
angle bisector wanted, 2 right angles given, intersections of diagonals of ABCD

Source: 2019 NZMO Round 2 p1

1/10/2021
Let XX be the intersection of the diagonals ACAC and BDBD of convex quadrilateral ABCDABCD. Let PP be the intersection of lines ABAB and CDCD, and let QQ be the intersection of lines PXPX and ADAD. Suppose that ∠ABX=∠XCD=90o\angle ABX = \angle XCD = 90^o. Prove that QPQP is the angle bisector of ∠BQC\angle BQC.
geometryangle bisectorright angle