MathDB

1

Part of 2009 Turkey MO (2nd round)

Problems(2)

Turkey NMO 2009 Q1

Source:

8/31/2010
Find all prime numbers pp for which p3−4p+9p^3-4p+9 is a perfect square.
modular arithmeticnumber theory proposednumber theory
Turkey NMO 2009 Q4

Source:

8/31/2010
Let HH be the orthocenter of an acute triangle ABC,ABC, and let A1, B1, C1A_1, \: B_1, \: C_1 be the feet of the altitudes belonging to the vertices A, B, C,A, \: B, \: C, respectively. Let KK be a point on the smaller AB1AB_1 arc of the circle with diameter ABAB satisfying the condition ∠HKB=∠C1KB.\angle HKB = \angle C_1KB. Let MM be the point of intersection of the line segment AA1AA_1 and the circle with center CC and radius CLCL where KB∩CC1={L}.KB \cap CC_1=\{L\}. Let PP and QQ be the points of intersection of the line CC1CC_1 and the circle with center BB and radius BM.BM. Show that A, K, P, QA, \: K, \: P, \: Q are concyclic.
geometrygeometry proposed