China Team Selection Test 1986

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China Team Selection Test 1986

ChinaTeam Selection Test1986 * I.Day 1[~T~| Let ABCD be a cyclic quadrilateral. Provo that the incenters of the triangles ABC. BCD. CDA and DAB form a

China Team Selection Test 1986 a rectangle.|~2] Let Of , 1 < X < n , and tti , 1 < i < n be 2• n real numbers. Prove that for Xi , 1 < i < n satisfying X\ < x-2 < xn the following s

tatemenst are equivalent:nni) 52®**** <^bkXk,fc=lJb—1ASnnii.) 57 flit < 57 bfi for .9 = 1.2,.... JI - 1 and 57 nk = 57 &*•fc=l Jt=l*=1k=l|~3~| Given a China Team Selection Test 1986

positive integer .4 written in decimal expansion: {an.On-i,... ,«o) and let /(/1) denote 5~^ ~n k • <*k- Define .11 = f(A), A-2 = f(Ai). Prove that:I

China Team Selection Test 1986

. There exists positive integer jfc for which A+I = Aff II- Find such Alt for 19SG.[T] Given a triangle ABC for which c = 90 degrees, prove that given

ChinaTeam Selection Test1986 * I.Day 1[~T~| Let ABCD be a cyclic quadrilateral. Provo that the incenters of the triangles ABC. BCD. CDA and DAB form a

China Team Selection Test 1986 ts from 1 up to*=1n-1).ChinaTeam Selection Test 1986Day 2ỊTỊ Given a square ABCi) whose side length is 1, /’ and Q are points on the sides AB and AD.

If the perimeter of ABQ is 2 find the angle BCQ.[~2~| Given a tetrahedron A1ỈCD. E, E. c, are on the respectively on the segments AB, AC and AD. Prove China Team Selection Test 1986

that:i) area EEC < maxarea ylBC,area ABD,íưí'í\ ACD.ÍÌXCÍI BCD. ii) The same as above replacing "area” for "perimeter”.|~3~| Lot Xi, 1 < i < n 1)0 re

China Team Selection Test 1986

al numbers with n > 3. Let p and <1 be their symmet ric sum of degree 1 and 2 respectively. Prove that:i)p2 • -—- - 2q > 0ii)|-G< ựp2 — "y • ~—- for e

ChinaTeam Selection Test1986 * I.Day 1[~T~| Let ABCD be a cyclic quadrilateral. Provo that the incenters of the triangles ABC. BCD. CDA and DAB form a

China Team Selection Test 1986 nts, also, the endpoints of these chords should be among the 4 - k points.I. Prove that 2 ■ k pairwisely non-intersecting chords can be drawn for each

of whom its endpoints differ in at most 3 ■ k - 1. II. Prove that the 3 ■ k - 1 cannot be improved.ChinaTeam Selection Test1987Day 1[ I I I a.) For a China Team Selection Test 1986

ll positive integer k find the smallest positive integer f(k) such that 5 sets «1,: *5 exist, sat isfying:T. each has k elements: IT. X,- and Xj+1 are

China Team Selection Test 1986

disjoint for X — 1,2, ...,5 («6 — -**1) HI. the union of t he 5 sets has exact ly f(k) elements.

ChinaTeam Selection Test1986 * I.Day 1[~T~| Let ABCD be a cyclic quadrilateral. Provo that the incenters of the triangles ABC. BCD. CDA and DAB form a