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2007亚太数学奥林匹克研习营试题


ù++? =?b?S??=?3?, ? ??
2007 ? 2 ~ 10 n

v???: lúüv (08:30 – 11:30) ?T??, ???à.HC?H°z? ??.)J^?? (?) ^? .)Uà??l?? ???}

1. t?ì|üís_?cb n U)?? S = {1, 2, · · · , 3n ? 1,

3n} ?\}’A n

_?¤?.ó>í?? {x, y, z } w2 x + y = 3z .
2. °|F í?Pb abcd, w2 a, b, c, d }??—Pb, ìPb,

Pb, _Pb,

U)¤?Pbb/ k
aa + bb + cc + dd . 3. I a, b, c [?úi$íúi? t? ab bc ca a b c + + ≥ + + . c(c + a) a(a + b) b(b + c) c+a a+b b+c 4. #ìúi$ ?ABC ?w?Q?, D uC BC í2? (?.¨? A ?íC), E u

C AC í2? (?.¨? B ?íC), F uC AB í2? (?.¨? C ?íC),
DE D BC, AC }?>k G, H . DF D BC, AB }?>k I, J . ?I M, N }

?u GH, IJ í2?, AD D EF >k P ?
(i) t? ?DM N íú_qià A, B, C [? (ii) ?p ?DM N í?Q??-r? ?P M N í?Q?,

ù++? =?b?S??=?3?, ? ???5j?
2007 ? 2 ~ 10 n

v???: lúüv (08:30 – 11:30) ?T??, ???à.HC?H°z? ??.)J^?? (?) ^? .)Uà??l?? ???} ???: t?ì|üís_?cb n U)?? S = {1, 2, · · · , 3n ? 1, 3n} ?\}’A
n _.ó>í?? {x, y, z } w2 x + y = 3z .

j: cq¤ n _.ó>í??? {xk , yk , zk }, k = 1, 2, · · · , n. ?
3n n n

i=
i=1

(xk + yk + zk ) = 4
k=1 k=1

zk (? xk + yk = 3zk )

FJ 4 .ìc? 1 3n(3n + 1), C6 3n(3n + 1) u 8 íIb ] 2
n ≡ 0 C n ≡ 5 (mod 8). (i) ? n = 5, ¤ n _?¤?.ó>í???? {1, 11, 4}, {2, 13, 5}, {3, 15, 6}, {9, 12, 7}, {10, 14, 8} {1, 14, 5}, {2, 10, 4}, {3, 15, 6}, {9, 12, 7}, {11, 13, 8} {1, 8, 3}, {2, 13, 5}, {12, 15, 9}, {4, 14, 6}, {10, 11, 7} {1, 11, 4}, {2, 7, 3}, {5, 13, 6}, {10, 14, 8}, {12, 15, 9} {1, 8, 3}, {2, 13, 5}, {4, 14, 6}, {10, 11, 7}, {12, 15, 9} (ii) ? n = 8, ¤ n _.ó>í???? {1, 5, 2}, {3, 9, 4}, {6, 18, 8}, {7, 23, 10}, {14, 19, 11}, {16, 20, 12}, {17, 22, 13}, {21, 24, 15}.

??ù: °|F í?Pb abcd, w2 a, b, c, d }??—Pb, ìPb, b, U)¤?Pbb/ k
aa + bb + cc + dd .

Pb, _P

j: 3435 ???íj íl, ì 00 = 1. I m = abcd, s = aa + bb + cc + dd /cq m = s. é??,
103 ≤ m < 104 . JúF í x ∈ {a, b, c, d} / x ≥ 6, ? s ≥ 66 > 104 ¤?pe F

J a, b, c, d ≤ 5. JúF í x ∈ {a, b, c, d}) / x < 5, ? s ≤ 4 × 44 = 1024 / a = b = c = d u?? í ?U) s ≥ 103 . ?7, ?¤8”-, s = 1024 = 4444 = m. ?¤ x = 5 ú/<
x ∈ {a, b, c, d}. .? s_ x íM k 5 C6 s ≥ 2 × 55 (?? m í/<?b}?

? 6), ?¤? ?_ x íM? 5. ? s > 55 = 3125 / s ≤ 55 + 3 × 44 = 3893 < 4000, .
a = 3. ?¤

s = 55 + 33 + xx + y y = 3152 + xx + y y , ¤T x, y ∈ {a, b, c, d}.

.?w?O4, cq 0 ≤ y ≤ x ≤ 4. J x = 0 ? y = 0 / s = 3154 = 33 + 11 + 55 + 44 = 3409 J x = 1 ? y = 0, 1 / s = 3154 = 33 + 11 + 55 + 44 = 3409 Jx=2?
y = 0 (.?) s = 3156 + y = 3157, y = 1 (.?) ? ? 3160, y = 2 (.?)
y

? ? ? 3157,

Jx=3?

y 3180, y s = 3179 + y y ? 3183, y ? ? ? 3206, y
? ? 3409, ? ? ? ? ? ? 3409, ? ? ? ? ? ? ?

? 3180, ? ? ? ?

=0 =1 =2 =3 =0 =1 =2 =3 =4

(.?) (.?) (.?) (.?) (.?) (.?) (.?) (?) (.?)

Jx=4?
s = 3408 + y y

y y 3412, y 3435, y 3664, y

??,H, m = 3435 u??íj

??ú: I a, b, c [?úi$íúi? t?
ab bc ca a b c + + ≥ + + . c(c + a) a(a + b) b(b + c) c+a a+b b+c

j: I
a b c x = , y = ,z = . b c a

? xyz = 1 /?.

?Z?A
x?1 y?1 z?1 + + ≥ 0, y+1 z+1 x+1

,

gk
x2 + y 2 + z 2 ? x ? y ? z + xy 2 + yz 2 + zx2 ? 3 ≥ 0 (1)

?à? .

?)
(2)

1 √ x2 + y 2 + z 2 ≥ (x + y + z )2 ≥ 3 xyz (x + y + z ) = x + y + z 3

/
xy 2 + yz 2 + zx2 ≥ 3 3 x3 y 3 z 3 = 3. (3)

? (2)

D (3)

ó??) (1)

/? x = y = z v UA

]?.

A /?

a = b = c v UA

???: #ìúi$ ?ABC ?w?Q?, D uC BC í2? (?.¨? A ?íC),
E uC AC í2? (?.¨? B ?íC), F uC AB í2? (?.¨? C ?íC), DE D BC, AC }?>k G, H . DF D BC, AB }?>k I, J . ?I M, N }?u GH, IJ í2?, AD D EF >k P ? (i) t? ?DM N íú_qià A, B, C [? (ii) ?p ?DM N í?Q??-r? ?P M N í?Q?,

j: (i) ?: D =

B+ C , 2

M=

A+ B , 2

N=

A+ C . 2

? BF D = A/2, F BC = B + F BA = B + C/2, FJ
BIJ = 180? ? A/2 ? ( B + C/2) = A/2 + C/2.

°?, BJI = A/2 + C/2. FJ ?BIJ u 8úi$ ] BN u B íi?}( °?,CM u C íi?}( ?¤…bD AD ( A íi?}() ó>k ?ABC íq - Q ? ?? N I = N J, M C = M Q, ?¤ DN Q = DM Q = 90? , FJ DM N Q ??u?, / DQ u ?DM N í?Q?íò ?¤

DN M = DQM = 90? ? QDM = 90? ? B/2 = A/2 + C/2.

°?, DM N = A/2 + B/2. ] M DN = B/2 + C/2.
(ii) I O u(¨ DQ í2? FJ O u ?DM N í?Q??- Bbb?p O r? ?P M N í?Q?,

? ?ü??? Q u ?DEF í - ??5? ?DEF ?
( ??ì?):

- Q. ???!?

- Q ?ú?? D, E, F í?((¨2?, }r? —úi$ ?P M N

(orthic triangle) í?Q?, (¤?? ??) D O u(¨ DQ í2?, ])?


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