Đề thi HSG Toán 9

     ! "#$%&'(')'(( *+,-./,012 -(3'1$456không k thi gian giao đê7 89(-  99ab cd eg+ + M  deg 99abc M   ( 1)( 2)( 3)( 4) 1n n n n+ + + + +    !"#$%&'( n N∀ ∈  89)( 0k > " 1 1 1 ( 1) 1 1k k k k k k = − + + + + *$+, /012  1 1 1 1 2 1 1 2 3 2 2 3 4 3 3 4 2025 2024 2024 2025 + + + + + + + + 89:3-$.$;  1$<=,>5?@,$- 2 1 x ay ax y + =   − =  456  $%&'( 3a =  "7"78/6"9 a :6  $%&";6   0; 0?x y> < 89A<=%>?'?@$A0"=BCD%!E??@"F %>?BG"F%>?@BHI  $/"="9 %>H ∈ ?JI ∈ ?@41K=:"9%LGH'I   · 0 90EMD = M KC$/""9%>?'?@M "KGKHNKIKM 893O<=%>?'?@"FBC'GPQ"7/KCK ∈ ? ∈ ?@41I4R%S:"9CKC 7"?I4?@MB =M '8#"9K: ="=  +K(TM 89BO= 0 0 ( ; )x y   6  "96  $%& ax by c bx cy a cx ay b + =   + =   + =  U 3 3 3 3a b c abc+ + = VVVVVVVVVV-WVVVVVVVVVVV A B C D E M O O' I  CC* GX"; deg 10000. 100. (9999. 99. ) ( )abc ab cd eg ab cd eg ab cd eg= + + = + + + + + X ) (9999. 99. ) 99; ( ) 99ab cd eg ab cd eg+ + + +M M < )E  / 99ab cd eg+ + M  deg 99abc M < "; [ ] [ ] 2 2 ( 1)( 2)( 3)( 4) 1 ( 1)( 4) ( 2)( 3) 1 ( 5 4)( 5 6) 1n n n n n n n n n n n n+ + + + + = + + + + + = + + + + + <+ DY   2 5 4n n t+ + = "; 2 2 ( 2) 1 2 1 ( 1)t t t t t+ + = + + = + <+ / 2 5 4t n n= + + %S" 2 2 ( 5 5)n n+ +  !"#$%&'( .n N ∀ ∈ )E  / ( 1)( 2)( 3)( 4) 1n n n n+ + + + +  !"#$%& .n N ∀ ∈ < GO )(Z["; 1 1 1 ( 1) 1 . 1( 1 ) . 1( 1 )( 1 ) 1 1 1 (2 ) 1( 1 ) 1 k k k k k k k k k k k k k k k k k k d k k k k k k + − = = + + + + + + + + + + − + − = = − + + − + *$+, 1 1 1 1 2 1 1 2 3 2 2 3 4 3 3 4 2025 2024 2024 2025 1 1 1 1 1 1 1 1 1 1 44 1 1 (2 ) 45 45 1 2 2 3 3 4 2024 2025 2025 d + + + + + + + + = − + − + − + + − = − = − = 89:- )( 3a = "; 3 2 3 1 x y x y + =   − =  < 456  $%&%S" ( ; ) (0,5;0,5)x y = < 45$ 2 1 x ay ax y + =   − =  %S" 2 2 2 1 2 1 1 a x a a y a +  =   +  −  =  +  X H= 2 2 0a a+ > ∀ 6 2 2 0 0; 0 1 2 1 0 2 a a x y a a > −  + >   > < ⇔ ⇔   − < <    ) { } 1;0a Z a∈ ⇒ ∈ − X 89A-)\0< %S" · µ 2DOA B= 4;"="9 OBD∆ J · µ ' 2CO A C= 4;"="9 ' )O CE∆ < µ µ · · ' 2 DOA EO A B C + + =  O O' A B M N E G K · · · · 0 0 0 0 ' 360 ( ' ) 360 180 180 (0,5 ) DOA EO A ODE O ED d + = − + = − = 6 µ µ 0 0 180 90 2 B C+ = = )E  / · 0 90BMC = < 41]=:"9CK'HI%S"7"KHCI" 6 · · IDA IAD= < %S" · · ODA OAD= < ^/ · · 0 90IAO ODI= = 6 'MA OO ⊥ )E  /KC$/""9?'?@< "%S" 2 2 . ; . . .MA MB MD MA MC ME MB MD MC ME= = ⇒ = X 893)\0O< %S" ; ' // 'OE MA O G NA OE O G⊥ ⊥ ⇒ < )E  /7"?I4?@'O< "; ; 2 2 2 MA AN MN FA AG EG= = ⇒ = < K 'EG OO≤ _6  `"'%A6 < 2 'MN OO ⇒ ≤  PKaa??@'?I4?@ "`E  'I4N??@KNOI4NO??@ ";  +(T< 89B ) 0 0 ( ; )x y 6  "96  $%&"; 2 2 3 0 0 0 0 2 2 3 0 0 0 0 2 2 3 0 0 0 0 2 2 2 2 2 2 3 3 3 0 0 0 0 0 0 (*)(0,75 ) ac x bc y c ax by c bx cy a a bx a cy a cx ay b b cx ab y b ac x bc y a bx a cy b cx ab y a b c d  + = + =    + = ⇔ + =     + = + =   ⇒ + + + + + = + + KY  Z7" 2 2 0 0 0 0 2 2 0 0 0 0 2 2 0 0 0 0 2 2 2 2 2 2 0 0 0 0 0 0 3 (**)(0,75 ) a bx ab y abc ax by c bx cy a b cx bc y abc cx ay b ac x a cy abc ac x bc y a bx a cy b cx ab y abc d  + = + =    + = ⇔ + =     + = + =   ⇒ + + + + + = bc'cc / 3 3 3 3a b c abc+ + = . 100. (99 99. 99 . ) ( )abc ab cd eg ab cd eg ab cd eg= + + = + + + + + X ) (99 99. 99 . ) 99 ; ( ) 99 ab cd eg ab cd eg+ + + +M M < )E  / 99 ab cd eg+ + M  deg 99 abc M < "; [. đê7  89 (-  99 ab cd eg+ + M  deg 99 abc M   ( 1)( 2)( 3)( 4) 1n n n n+ + + + +    !"#$%&'( n N∀ ∈   89 )( 0k > " 1. <  89 A<=%>?'?@$A0"=BCD%!E??@"F %>?BG"F%>?@BHI  $/"=" ;9  %>H ∈ ?JI ∈ ?@41K=:" ;9 %LGH'I   · 0 90 EMD = M KC$/"" ;9 %>?'?@M "KGKHNKIKM  89 3O<=%>?'?@"FBC'GPQ"7/KCK ∈ ? ∈ ?@41I4R%S:" ;9 CKC 7"?I4?@MB