Yusuf Sila: Math Olympiad : Training

Math Olympiad : Training

3:34 AM

Muhammad Yusuf , Posted in Olympiad , 0 Comments

1. In a quadrilateral ACGE, H is the intersection of AG and CE, the lines AE and CG meet at I and the lines AC and EG meet at D. Let B be the intersection of the line IH and AC. Prove that AB/BC = AD/DC, or equivalently, DB is the harmonic mean of DA and DC.
2. The extensions of the chords QR and Q’R’ of a circle Γ intersect at a point P outside Γ. Tangents PA and PA’ are drawn from P to Γ. Prove that A, X, A’ are collinear where X is the intersection of QR’ and Q’R.
3. In a quadrilateral ABCD, E is a point on CD, BE intersects AC at F and the extension of DF meets BC at G. Suppose that AC bisects ∠BAD. Prove that ∠GAC = ∠EAC.
4. (Crux 2333) Points D and E are on the sides AC and AB of ∆ABC. Suppose F and G are points of BC and ED, respectively, such that BF : FC = EG : GD = BE : CD. Prove that GF is parallel to the angle bisector of ∠BAC.