MathDB

1.2

Part of 2016 Saudi Arabia Pre-TST

Problems(2)

similar triangles nad perpendicularity wanted, inscribed isosceles

Source: 2016 Saudi Arabia Pre-TST Level 4 1.2

9/13/2020
Let ABCABC be a non isosceles triangle inscribed in a circle (O)(O) and BE,CFBE, CF are two angle bisectors intersect at II with EE belongs to segment ACAC and FF belongs to segment ABAB. Suppose that BE,CFBE, CF intersect (O)(O) at M,NM,N respectively. The line d1d_1 passes through MM and perpendicular to BMBM intersects (O)(O) at the second point P,P, the line d2d_2 passes through NN and perpendicular to CNCN intersect (O)(O) at the second point QQ. Denote H,KH, K are two midpoints of MPMP and NQNQ respectively. 1. Prove that triangles IEFIEF and OKHOKH are similar. 2. Suppose that S is the intersection of two lines d1d_1 and d2d_2. Prove that SOSO is perpendicular to EFEF.
geometrysimilar trianglesperimeterisosceles
(a + b)/c +(b + c)/a +(c + a)/b >= a + b + c +1/a+1/b+1/c

Source: 2016 Saudi Arabia Pre-TST Level 4+ 1.2

9/13/2020
Let a,b,ca, b, c be positive numbers such that a2+b2+c2+abc=4a^2+b^2+c^2+abc = 4. Prove that a+bc+b+ca+c+ab≥a+b+c+1a+1b+1c\frac{a + b}{c} +\frac{b + c}{a} +\frac{c + a}{b} \ge a + b + c + \frac{1}{a} + \frac{1}{b} +\frac{1}{c}
algebrainequalities