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2006亚太数学奥林匹克研习营试题


ù++? =?b?S??=?3?, ? ??
2006 ? 2 ~ 13 n

v???: lúüv (8:30–11:30) ?T??, ???à.HC?H°z? ??.)J^?? (?) ^? .)Uà??l?? ???}

1.

8úi$ ?ABC , AC = BC . IO ??-, I ?q-,

/ O ?(¨ CI ,. J ??? D ? BC i,, /ò( OD⊥BI . ?p ID D AC ?W.

2. I R+ [?F ??bFAí??. t°F í?b f : R+ → R+ U)

úL<í??b x, y, x2 (f (x) + f (y ) = (x + y )f (yf (x)) A .
3. úL?#ì5?cbn, ?p 5n ? 1, 13n ? 1, 34n ? 1 .??·ucbí?j. 4. ??_ n _?'í?ˇ2, ??X?'??wF?'>G??. J úkL<s

_? A, B , /?? t _?U) A, B s??=¤ t _?, t?:
n = 4t + 3.

29

ù++? =?b?S??=?3?, ? ???5j?
2006 ? 2 ~ 13 n

v???: lúüv (8:30–11:30) ?T??, ???à.HC?H°z? ??.)J^?? (?) ^? .)Uà??l?? ???} 8úi$ ?ABC , AC = BC . IO ??-, I ?q-, / O ?(¨ CI ,.

???:

J??? D ? BC i,, /ò( OD⊥BI . ?p ID D AC ?W. j: T iíò CE , ??
EIB = 90? ? 1 ABC, 2 ODB = 90? ? 1 ABC, 2


OIB + ODB = 180? ,

] B, I, O, D ??u?. ?¤ IDB = IOB . ? IOB =
IDB = ACB , FJ ID D AC ?W.

1 2

AOB = ACB , ]

?:“O ?(¨CI ,” í??u?7é?? ?“. ?_‘Kuu?ìí, ??(? ?E?A .

30

??ù: I R+ [?F ??bFAí??. t°F í?b f : R+ → R+ U) úL<í??b x, y, x2 (f (x) + f (y ) = (x + y )f (yf (x)) A . j: I f (1) = p, ? p > 0. ? x = y = 1, ?% ?l??)
f (p) = p. (1)

? x = p, y = 1 ? p2 (f (p) + p) = (p + 1)f (f (p)), ? (1) ?)
2p3 = p(p + 1).

j, ) p íM?: ?1/2, 0, 1. O p > 0, FJ p = 1, ?
f (1) = 1. (2)

I t ?L<??b. ? x = 1, y = t, y? (2) ?)
1 + f (t) = (1 + t)f (t).

j, )
1 f (t) = . t (3)

%?? f (t) = 1/t ?—?<.

31

??ú: úL?#ì5?cbn, ?p 5n ? 1, 13n ? 1, 34n ? 1 .??·ucbí? j. j: ??. q 5n ? 1, 13n ? 1, 34n ? 1 ·u?jb. ?
2 5n ? 1 = k 1 2 13n ? 1 = k2 2 34n ? 1 = k3 ,

w2 k1 , k2 , k3 ·u?cb. ? ú ? k3 ?Jb, q k3 = 2m ? 1 Hp) 17n =
2m2 ? 2m + 1, ] n 6uJb.

ù ? ? )
2 2 8n = k 2 ? k1 .

?…V? ?

ù ?? k1 , k2 uXb. ]?q k2 = 2p, k1 = 2q , Hp, )

2n = p2 ? q 2 = (p + q )(p ? q )

?k p2 ? q 2 = 2n uXb, ?¤ p, q °J°X, ] (p + q ), (p ? q ) ·uXb. FJ
2n u 4 íIb, ] n uXb. ?????|í n uJbpe.

32

???: ??_

n _?'í?ˇ2, ??X?'??wF?'>G??. J úkL

<s_? A, B , /?? t _?U) A, B s??=¤ t _?, t?:
n = 4t + 3.

j: 5?L<?X?' A, /cq A /= r _?'. ? r = 2t + 1. 5??ú (X, Y, A) U) A D Y ?= X , ?—¤‘Kí?ú_b?
rt = r r (r ? 1) = ? r = 2t + 1. 2 2

D5??ú (X, Y, Z ) U) X D Y ?= Z , ?—¤‘Kí?ú_b?
t n 2t + 1 =n ? n = 4t + 3. 2 2

33


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