MathDB

2008 Turkey MO (2nd round)

Part of Turkey MO (2nd round)

Subcontests

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O,H,H_0 collinear

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

Find the sum of the functions

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)
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Turkey NMO 2008 Problem 3

Let a.b.c be positive reals such that their sum is 1. Prove that \frac{a^{2}b^{2}}{c^{3}(a^{2}\minus{}ab\plus{}b^{2})}\plus{}\frac{b^{2}c^{2}}{a^{3}(b^{2}\minus{}bc\plus{}c^{2})}\plus{}\frac{a^{2}c^{2}}{b^{3}(a^{2}\minus{}ac\plus{}c^{2})}\geq \frac{3}{ab\plus{}bc\plus{}ac}

Turkey 2008 National Mathematical Olympiad 6th Question

There is a connected network with 2008 2008 computers, in which any of the two cycles don't have any common vertex. A hacker and a administrator are playing a game in this network. On the 1st 1st move hacker selects one computer and hacks it, on the 2nd 2nd move administrator selects another computer and protects it. Then on every 2k\plus{}1th move hacker hacks one more computer(if he can) which wasn't protected by the administrator and is directly connected (with an edge) to a computer which was hacked by the hacker before and on every 2k\plus{}2th move administrator protects one more computer(if he can) which wasn't hacked by the hacker and is directly connected (with an edge) to a computer which was protected by the administrator before for every k>0 k>0. If both of them can't make move, the game ends. Determine the maximum number of computers which the hacker can guarantee to hack at the end of the game.