Olympiad Corner
~Oi1i~.~.fiJ c=)
1997 Chinese Mathematical °.zympiad:
!1RsJM
Part I (8:00-12:30, January 13, 1997) Problem 1~ Let ;XI, X2, ..., X1997be real
numbers &atisfying th~ following two conditions: 1 (1)
-
-
-~~Xi -v3
~J3
(i=
1, 2, ..., 1997);
+ ;\i!t*C!f;f , JIM:!:ill T -fft*~~ ~~tJi:(Euler) 0 f&~~':(£=+~~ !f;fB~§~aJI ' {g.~gj]!c:£*~ ~~~ ' ~ftRM 0 ~~J:~Ij)~ :@...i}A:fD.1J;l;t.i)~fm~:fJ1. ..majj ~ ~ ajj(JIg0 ~ ~ :fJ1. .:(£~f/ if?d(JIg ~ ~
(2) XI +X2+ ...+X1997 = -318J3~
tJi:,fj (EulerLine)-
Find the maximumvalue of
M~OO.m:~(JIgI5(J1g~iV&o
12
12
stfl.ii ' AH:fa A 'O51.8U~ffi {~~~ %ABC;f[]A'B'C'~J;J~~ ' mPj, AH:A'0=BC:B'C'=2:1 f'tiJ:5t1!?' :l;.JL\G -tl?reJt:l~ AA' 9f.fjX, AG:A'G=2:1
-~.~=-~
1E11I:t. ' =- ~ % HAG;f[] OA'G ffi ~ 0 8311: t.:Ift~
12
~-ff~~.:=.~~ ABC, ;;:m~E~ LHGA = LOGA' .:=...iif1'9 ~~ (mid-poinJS),A"B' ~D iR Sf,$t ~ mJ.-.10 ' G' H JjJ(.-~~ ' mm~lli Problem 2. Let AIBI C1D1 be an c' $t;8tl1~m~.:=. ~jj~.¥ (perpendicular bisectors) 0 ~1r}~rn : ~, ~ROG:GH= 1:2 0 arbitrary convexquadrilateral. Let P be XI +xz +"'+XI997'
~.:=...¥iRSf,:i1-~mx~~~I!i'
a point inside the quadrilateral such that 0 0 ~15 0 ~~.:=.~~ the segmentsfrom P to eachvertex form llPIII-~15 ABC ~'J}~IIJL\ (circumcentre)'rn acuteangles with the two sides through the vertex. Recursivelydefine Ak.Bk. Ck }J.mfg**B~;;Ho and Dkas the points symmetricto P with A respect to the lines Ak-1Bk-l. Bk-lCk-l. Ck-'IDk-l and Dk-lAk-l. respectively ", r'I
(k=2.3
/
).
B /,~
(continued onpag(! 4)
L-r
B~..."
' '0," :A" '"~'-+-:~. ~
c
A
B
C, (III.=.)
~m~OH~~l5m1'i¥-}L,'E~
(II-)
:I=::g~ fL IS iii (Nine-point circle)~
"~;,-~
IiI/L'
.
~-1iOO ~ .=-;:I1!J7f;:;i A'B'C';fDABC:;;p {gffifJ'), , ffij1i.f;JLI!.if~1'J 0 ~{I.if
~./j\~~rI"J.=-;:I1!J7f;:;iA'B'C' fim.=-
0 m~1L!!i1il~1~-OO;;m~=.
:;llj~ ABC ~ =.~ ~~!!i A', B', c', =.~~.¥.ff:. D' E' FPJ&=.m ~;fO.¥/L,r~ ~ ~ IS K' L' M ~ III (liJlZ9)
0
;:I1!J 7f;:;iABC rI"J~ I!i .=- ;:I1!J 7f;:;i(medial triangle)
0 E rI"J.=- f~;@j
(altitudes)
I;n
tlfi,t~.:t£ OA' , DB' :fa OC'L ' I!I JJ:t 0 l!it!?ffi~7 ~1!i.=-;:I1!J7f;:;i A'B'C' rI"J ¥;JL'
(orthocentre);:I1!J@
(111=)
0
([lJIZY)
:fiI~ ~fL 15m ~ ~ # 11/a';J::7f; ~ m aJj ~~¥~W.~tl (Poncelet) ~ 1821 ~§Jt.t(tIili a';J, f&~ A', B', C', K, L, M ~155}J3X".R:fiIm:m:lZYl!im <111-=: 1i!i ' ~ f& m aJj~ 'fIm m 1i!ia';JIZY I!i # iii , ~;fIJffl~.=. 'fImm-g-a';Jm:m:11maJj ~ 1ft ~ ~J1. ff: 1iiJ.=. :frj ~ ~ 7i- ~ 1;1 ~.=. 'fIm fil. 'N L ~ ~ -'fIm fil ' JWi: 1:& IL\ (0) ..m/L\ (G) :fD~/L\ (H) #~ ' maJj D , E, F-&:(£~IIIL 0 .~'frJ $~maJj:!lIJT (III.=.) : m ~.=.:frj ~ ABC ~ ~ AH :fa jf BC ~~~s:rz:.$t~ OA's:rz:fi ' lEI!i:t LHAG = LOA'G
~~f&a';Jm~:
:$'c.::?!;.I.I. B' , C', L , M ~ ~ (!liE) 0 (continuedon page 2)