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7/23/2019 MO - mediterranean 1998 ~2011
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KAIST
2011 12 8
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2
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I
5
1998 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1999 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2006 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2009 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2011 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1998 , 4 4
. , .
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4
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I
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1998 7
1998
1. ABCD . () AB M , MC M D >3
p3 M A MB . ()
2. n . z2n +zn + 1 z2 +z + 1
n 3 . ()
3. ABC I BC,C A,AB D,E,
F . \BI C BC M , AM EF P
. DP \F DE . ()
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8 1999
1999
1. , ?
2. n , n .
n + 1 .
3. 0 a, b,c x + y+ z = 3 x, y , z
. (-)
3
2r1
a2+
1
b2+
1
c2 x
1 + a2+
y
1 + b2+
z
1 + c2
4. \B = 4\A ABC . BC=a, C A= b, AB= c
. ()
ab2c3 = (b2 a2 ac)((a2 b2)2 a2c2)
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10 2001
2001
1. P Q k . k AC P Q M . ABk P Q kCD , k ABCD . AD BC X k P,
Q .
2. p(x) =x5 nx n 2 n .
3. 2000N 200120012001
N .
4. 1 S
. M2 S ,M BC,C A,AB aM,bM,cM .
f(M) =a3M(bM cM) + b3M(cM aM) + c3M(aM bM)
(a)
fM2 Sj f(M) 0g
.
(b) f(M) , .
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2002 11
2002
1. y j x2 + 1 x2 j y3 + 1 x, y .
2. x,y ,a x + y = x3 + y3 =x5 + y5 =a . a
.
3. ABC M,N AC, B C , K MN .
4ACN,4BC M D . CD 4ABC O K AB .
4. a2 + b2 + c2 = 1 a, b, c .
a
b2 + 1+
b
c2 + 1+
c
a2 + 1 3
4(ap
a + bp
b + cp
c )2
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12 2003
2003
1. x2 + y2 + z2 =x + y+ z+ 1 .
2. BC= C A + 12
AB ABC . AB BP : P A= 1 : 3 P
, \CAP = 2\CP A .
3. a + b + c= 3 a, b, c .
a
b2 + 1+
b
c2 + 1+
c
a2 + 1 3
2
4. ( )
, 6
.
0 1 , .
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2004 13
2004
1. m .
1! 3! 5! (2m 1)! =
m(m + 1)
2
!
2. ABC, A T . O
, OA OT BC Q M .
.SAQC
SCMT= sin B
cos C2
3. ab + bc + ca + 2abc= 1 a,b, c .
2(a + b + c) + 1 32abc
4. z1, z2, z3 jz1j = jz2j = jz3j = 1 1
2 + jz1+ z2j + 1
2 + jz2+ z3j + 1
2 + jz3+ z1j = 1
. A(z1), B (z2), C(z3) ,
.
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14 2005
2005
1. . ,
. .
\ ." . \ .
136." . ?
2. O R R(, R < R) k, k
. O k A, k B (, O A B
). O k E, k F (, E
O F ). 4OAE 4OBF EF ,AB .
3. n 3 , A1; A2; : : : ; An .
1
n
nXi=1
jAij!
+ 1n3
X1i
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2006 15
2006
1. . 1
(a) 1 ?
(b) 2 ?
2. P ABC , A1B2, B1C2, C1A2 P AB, BC,
CA . , A1, A2 BC , B1, B2 CA , C1, C2 AB
. .
s[A1A2B1B2C1C2] 12
s[ABC]
, s[E] E .
3. ABC a,b, c gcd(a;b;c) = 1 . \A
BC D .
(a)4DBA 4ABC c .
(b) c= n2 (n 2), (a) 4ABC .
4. i= 1; 2; : : : ; m j = 1; 2; : : : ; n 0 xi;j 1 , .
nYj=1
1
mYi=1
xi;j
!+
mYi=1
0@1 nY
j=1
(1 xi;j)1A 1
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16 2007
2007
1. x y z xy+ yz + zx = 1 . xz < 12 . , 12 ?
2. ABCD AC BD E . AB = 39,
AE= 45, AD = 60, B C= 56 , C D .
3. = \A a= BC ABC . a =p
rR
(r, R ), AB AC
.
4. x >1 , .x + fxg
[x] [x]
x + fxg
+
x + [x]
fxg fxgx + [x]
>
9
2
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2008 17
2008
1. ABCDEF . AB
BC C D
DE E F
F A = 1 , ,
AD, B E, C F .
2. 1 i < j < k i,j ,k A1; A2; : : : B1; B2; : : : ?
(i) AiAj Bk k= i +j .
(ii) BiBj Ak k= i +j .
3. n .
nXi=1
X1i1
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2010 19
2010
1. a, b, c, d , x,y , z , u
.
x2 yz zu yu = ay2 zu ux xz = bz2 ux xy yu = cu2 xy yz zx = d
2. n >2 a1+ a2+ + an = 1 a1; a2; : : : ; an .
a2 a3 ana1+ n 2 +
a1 a3 ana2+ n 2 + +
a1 a2 an1an+ n 2
1
(n 1)2
3. A,B ,C 4ABC BC,C A,AB . 4ABC R
R
=
1
2rp
2R(2R ha)(2R hb)(2R hc) . ,R r 4ABC ,ha,hb,hc .
4. p >1 . f1; 2; : : : ; pg m n p .
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20 2011
2011
1.
.
P(x) =x10 20x9 + 135x8 + a7x7 + a6x6 + a5x5 + a4x4 + a3x3 + a2x2 + a1x + a0
.
(Gerhard Woeginger, )
2. A .B = f x=y j x; y2 A g ,C= fxy j x; y2 A g .
jAj j
Bj j
Cj2 . (Gerhard Woeginger, )
3. h xh
.
(, ).
x3 + x2 + x= 2 .
4. ABC \A D . ABD
ACD AB, AC M, N . BN
CM AD .
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