MathDB

1

Part of 2010 China Team Selection Test

Problems(8)

China 2010 quiz1 Problem 1

Source:

9/5/2010
Assume real numbers ai,bi (i=0,1,⋯ ,2n)a_i,b_i\,(i=0,1,\cdots,2n) satisfy the following conditions: (1) for i=0,1,⋯ ,2n−1i=0,1,\cdots,2n-1, we have ai+ai+1≥0a_i+a_{i+1}\geq 0; (2) for j=0,1,⋯ ,n−1j=0,1,\cdots,n-1, we have a2j+1≤0a_{2j+1}\leq 0; (2) for any integer p,qp,q, 0≤p≤q≤n0\leq p\leq q\leq n, we have ∑k=2p2qbk>0\sum_{k=2p}^{2q}b_k>0. Prove that ∑i=02n(−1)iaibi≥0\sum_{i=0}^{2n}(-1)^i a_i b_i\geq 0, and determine when the equality holds.
inductionstrong inductioninequalities unsolvedinequalities
China 2010 quiz2 problem 1

Source:

9/8/2010
Let △ABC\triangle ABC be an acute triangle with AB>ACAB>AC, let II be the center of the incircle. Let M,NM,N be the midpoint of ACAC and ABAB respectively. D,ED,E are on ACAC and ABAB respectively such that BD∥IMBD\parallel IM and CE∥INCE\parallel IN. A line through II parallel to DEDE intersects BCBC in PP. Let QQ be the projection of PP on line AIAI. Prove that QQ is on the circumcircle of △ABC\triangle ABC.
geometrycircumcirclegeometric transformationreflectiongeometry unsolved
China 2010 quiz3 problem 1

Source:

9/12/2010
Let ABCDABCD be a convex quadrilateral with A,B,C,DA,B,C,D concyclic. Assume ∠ADC\angle ADC is acute and ABBC=DACD\frac{AB}{BC}=\frac{DA}{CD}. Let Γ\Gamma be a circle through AA and DD, tangent to ABAB, and let EE be a point on Γ\Gamma and inside ABCDABCD. Prove that AE⊥ECAE\perp EC if and only if AEAB−EDAD=1\frac{AE}{AB}-\frac{ED}{AD}=1.
geometrysymmetrygeometric transformationangle bisector
China 2010 quiz4 problem 1

Source:

9/13/2010
Let △ABC\triangle ABC be an acute triangle, and let DD be the projection of AA on BCBC. Let M,NM,N be the midpoints of ABAB and ACAC respectively. Let Γ1\Gamma_1 and Γ2\Gamma_2 be the circumcircles of △BDM\triangle BDM and △CDN\triangle CDN respectively, and let KK be the other intersection point of Γ1\Gamma_1 and Γ2\Gamma_2. Let PP be an arbitrary point on BCBC and E,FE,F are on ACAC and ABAB respectively such that PEAFPEAF is a parallelogram. Prove that if MNMN is a common tangent line of Γ1\Gamma_1 and Γ2\Gamma_2, then K,E,A,FK,E,A,F are concyclic.
geometrycircumcircleratiosimilar trianglesgeometry unsolved
China 2010 quiz5 Problem 1

Source:

9/13/2010
Given integer n≥2n\geq 2 and positive real number aa, find the smallest real number M=M(n,a)M=M(n,a), such that for any positive real numbers x1,x2,⋯ ,xnx_1,x_2,\cdots,x_n with x1x2⋯xn=1x_1 x_2\cdots x_n=1, the following inequality holds: ∑i=1n1a+S−xi≤M\sum_{i=1}^n \frac {1}{a+S-x_i}\leq M where S=∑i=1nxiS=\sum_{i=1}^n x_i.
inequalitiesinequalities unsolved
China 2010 quiz6 Problem 1

Source:

9/14/2010
Let ω\omega be a semicircle and ABAB its diameter. ω1\omega_1 and ω2\omega_2 are two different circles, both tangent to ω\omega and to ABAB, and ω1\omega_1 is also tangent to ω2\omega_2. Let P,QP,Q be the tangent points of ω1\omega_1 and ω2\omega_2 to ABAB respectively, and PP is between AA and QQ. Let CC be the tangent point of ω1\omega_1 and ω\omega. Find tan⁡∠ACQ\tan\angle ACQ.
trigonometrygeometrycircumcircleincenterradical axispower of a pointgeometry unsolved
China TST 2010, Problem 1

Source:

8/28/2010
Given acute triangle ABCABC with AB>ACAB>AC, let MM be the midpoint of BCBC. PP is a point in triangle AMCAMC such that ∠MAB=∠PAC\angle MAB=\angle PAC. Let O,O1,O2O,O_1,O_2 be the circumcenters of △ABC,△ABP,△ACP\triangle ABC,\triangle ABP,\triangle ACP respectively. Prove that line AOAO passes through the midpoint of O1O2O_1 O_2.
geometrycircumcircletrigonometryparallelogramgeometric transformationgeometry unsolved
China TST 2010, Problem 4

Source:

8/28/2010
Let G=G(V,E)G=G(V,E) be a simple graph with vertex set VV and edge set EE. Suppose ∣V∣=n|V|=n. A map f: V→Zf:\,V\rightarrow\mathbb{Z} is called good, if ff satisfies the followings: (1) ∑v∈Vf(v)=∣E∣\sum_{v\in V} f(v)=|E|; (2) color arbitarily some vertices into red, one can always find a red vertex vv such that f(v)f(v) is no more than the number of uncolored vertices adjacent to vv. Let m(G)m(G) be the number of good maps. Prove that if every vertex in GG is adjacent to at least one another vertex, then n≤m(G)≤n!n\leq m(G)\leq n!.
inequalitiesinductionfunctioncombinatorics unsolvedcombinatorics