9. In 4ABC with circumcircle Γ, points M, N are on sides AB, AC respectively such that M C bisects ∠C and BN bisects ∠B. Suppose that the line M N meets Γ at one of the two points, say D. Prove that
− = .
DB DC AD
Olympiad Geometry Problems III IMO 2011 Preparation June 28, 2011 1. Point D lies on side BC of triangle ABC such that BD bisects ∠BAC. Points M, N lie on sides AB, AC respectively such that ∠ADM = ∠ABC and ∠ADN = ∠ACB. Segments M N and AD intersect at P . Prove that
10. In 4ABC with orthocenter H, E, F are the feet of the altitudes from B and C respectively. Points U, V lie on segments BF, CE respectively such that H lies on segment U V . Points X, Y are on segments BE, CF respectively such that XU ⊥ U V and Y U ⊥ U V . Prove that XY k BC.
AD3 = AB · AC · AP . 2. In acute 4ABC, D, E lie on sides BC, CA respectively such that AD bisects ∠A and BE bisects ∠B. AD meets BE at the incenter I. The line OH (O circumcenter, H orthocenter) meets lines AC and BC at P and Q respectively. Prove that if C, D, I, E are concyclic, then P Q = AP + BQ.
11. In a cyclic quadrilateral ABCD with circumcircle ω, DD intersects line AB at M (DD denotes the tangent line to ω at D), CC intersects AB at N , and lines CD and AB intersect at P . Prove that
3. The incircle of 4ABC touches the sides BC, CA, AB at points D, E, F respectively. Points K, L are on segments DF, EF respectively such that AK k BL k DE. Prove that
(a) A, E, F, K are concyclic; B, D, F, L are concyclic. (b) C, K, L are collinear.
AM · AN = BM · BN
MA · MB = NA · NB
4. In an isosceles triangle ABC with AB = AC, incenter I and circumcenter O, D lies on side AC such that ID k AB. Prove that CI ⊥ OD.
12. In 4ABC with ∠B > 90◦ , orthocenter H and circumcenter O, the altitude of A meets the line BC at D. E is a point on line BC different from D such that DE = DC. If F = AB ∩ EH, prove that DO ⊥ DF .
5. In triangle ABC, points M, D are on side BC such that
DB M B
− = .
DC M C AC 2
13. In 4ABC with orthocenter H, the altitude of A meets side BC at D. Points M, N NA DA MA = = . AD intersects lie on sides AC, AB respectively such that MC NB DH BM at E, and AD intersects CN at F . Prove that ∠ABF = ∠ACE.
Point N lies on segment AM such that ∠BN M = ∠A. Prove that ∠CN M = ∠CAD. 6. M is the midpoint of side BC of an acute triangle ABC, and D, E, F are the feet of the altitudes feom vertices A, B, C respectively. Denote H as the orthocenter, and S as the midpoint of AH. F E intersects AH at G, and AM intersects the circumcircle of triangle BCH at N which is on the same side of BC as H. Prove that ∠HN A = ∠GN S.
14. In an acute, nonisosceles 4ABC, M is the midpoint of BC. Points D, E lie on DA EA median AM such that = . BD meets CE at F . Prove that ∠AF B = DB EC ∠AF C.
7. In triangle ABC with BC > AC, M lie son side BC such that M A = M B, Point D lies on side BC for which ∠DAB = ∠DCA, and E lies on side AB such that ∠EDA = ∠EBC. Point X lies on segment AM and denote Z = ED ∩ BX, L = AZ ∩ CX. Prove that quadrilateral ALDC is cyclic. 8. In 4ABC with ∠A = 90◦ , the altitude from A intersects BC at H. M, N are the midpoints of AC, AB, respectively. Segments CN and AH meet at E, and segments BM and AH meet at D. Prove that ∠ACD = ∠ABE. 1