MathDB

1

Part of 2008 Turkey MO (2nd round)

Problems(2)

O,H,H_0 collinear

Source: Turkey NMO 2008 Problem 1

12/2/2008
Given an acute angled triangle ABC ABC , O O is the circumcenter and H H is the orthocenter.Let A1 A_1,B1 B_1,C1 C_1 be the midpoints of the sides BC BC,AC AC and AB AB respectively. Rays [HA1 [HA_1,[HB1 [HB_1,[HC1 [HC_1 cut the circumcircle of ABC ABC at A0 A_0,B0 B_0 and C0 C_0 respectively.Prove that O O,H H and H0 H_0 are collinear if H0 H_0 is the orthocenter of A0B0C0 A_0B_0C_0
geometrycircumcirclesymmetrygeometric transformationhomothetyEulerratio
Find the sum of the functions

Source: Turkey NMO 2008 Problem 4

12/2/2008
f:N×Z→Z f: \mathbb N \times \mathbb Z \rightarrow \mathbb Z satisfy the given conditions a) a) f(0,0)\equal{}1 , f(0,1)\equal{}1 , b) b) ∀k∉{0,1} \forall k \notin \left\{0,1\right\} f(0,k)\equal{}0 and c) c) ∀n≥1 \forall n \geq 1 and k k , f(n,k)\equal{}f(n\minus{}1,k)\plus{}f(n\minus{}1,k\minus{}2n) find the sum \displaystyle\sum_{k\equal{}0}^{\binom{2009}{2}}f(2008,k)
functionalgebra unsolvedalgebra