SỞ GIÁO DỤC VÀ ĐÀO TẠO KỲ THI CHỌN HỌC SINH GIỎI CẤP TỈNH GIA LAI TOÁN !"##$%!&'( ĐỀ BÀI: Bài 1: ")&$( (*+, ( ) n n − -!. )/012345678 n 9 :(;<345=78!+; x xy y+ + = 9 Bài 2: ")&$( (> !:$3!* A = ? ? @ − 9 @ + 9 :(AB78!+; C C @ @x x x− + + + − = 9 Bài 3: ")&$( *+,/012!D ) )a b c )!E ( ) ? a b c+ + + ≥ ( ) a b c ab bc ca+ + + + + 9 F!*GB4+#H.I Bài 4: ")&$( .%12!D ) ) )a b x y !JK< ax by+ = ) ax by+ = ) ? ? ax by+ = ) @ @ ax by+ = L%!+M=:$3!* B = ax by+ 9 Bài 5: ")&$( +5N!F!&O Oxy .+:.P " (P y ax= " (a ≠ )&$ Q @ F a ÷ /H&7!F " (d @ y a = − 1.1./0!+R.H9A M PH&$:S!#; !+5+:.P " (P 9T MH /3E/0&7!F " (d !T H 9* +,!P3E MF MH= 9 Bài 6: ")&$( .&7!+U"V(/H6W4XY=&7!+U9O!&7!+U"Z(G /0&.TXY!T/H!-G !+./0"V(!T97!F[!"V( !T\"\#%(9*+, 9PM PN PA+ ≥ 9 "-!( Họ tên thí sinh:………………………………Số báo danh:……… ]^!1JS!_) ĐÁP ÁN VÀ HƯỚNG DẪN CHẤM `Vab Bài 1: ( 1,5 điểm ) a) E ( ) ( ) n n n n− = − 9 ( ) n n + 9 ( ) n n− PH!L12345P5!-5 ( ) n n− M 9 ( ) n n + PH!L12345P5!-5 ( ) n n + M 9 cd4 ( ) n n − M @ eeeeeeeeeeeeeeeeeeeee)& E ( ) n n − = ( ) ( ) n n n− + 9 n 9 -3 ?n k = )!; ( ) n n − M ? 9 -3 ? n k = + )!; n − M ? )5 ( ) n n − M ? 9 -3 ? n k = + )!; n + M ? )5 ( ) n n − M ? 9 cd4 ( ) n n − M ? eeeeeeeeeeeeeeeeeeeee)& .H+ ? /H @ 345!2f39 cd4 ( ) n n − M eeeeeeeeeeeeeeeeeeee999)& b) E x xy y+ + = ⇔ ( ) ( ) x y+ + = eeeeeeeeeeeee )& ) ) ) ) x y x y x y x y + = + = + = − + = − ⇔ + = + = + = − + = − ) ) ? ) ?) x y x y x y x y = = = − = − ⇔ = = = − = − eeeeeeeeeeeee)& cd478!+;E%<345PH ( ) ( ) Q Qx y = ) ( ) ( ) Q Q ?x y = − − ) ( ) ( ) Q Qx y = ) ( ) ( ) Q ?Q x y = − − eeeee9)& Bài 2: ( 1,5 điểm ) a) A = ? ? @ − 9 ( ) ? @ + eeeeeeeee999999999999999999999999999& g ( ) ( ) ? ? @ ? @ − + g ? ? eeeeeeeeeeee)& b) '3#< C Cx− ≤ ≤ 9 N! C C t x x= − + + ≥ eeeeeeeeeeeeeeeeee9)& h34+ @ t x − − = 9 cd478!+;E6T @ t t− − = eeeeeeeeeeee)& AB/HP.T<W)!E< @t = 9 i.&E C C @x x− + + = )4 @ x+ − = )4 x = ± C eee999)& Bài 3: ( 1,5 điểm ) +70-!)!E ( ) a b a b ab− ≥ ⇒ + ≥ 9F!*GB4+"j>( ⇔ a b = 9 78!D b c bc+ ≥ 9j> ⇔ b c= 9 c a ca+ ≥ 9j> ⇔ c a= 9 i.&E a b c ab bc ca+ + ≥ + + ) ) )a b c R∀ ∈ 9j> ⇔ a b c= = "(e99)& PTE ( ) a − ≥ ⇒ a a+ ≥ "(9j> ⇔ a = 9 78!D b b+ ≥ "?(9j> ⇔ b = 9 c c+ ≥ "@(9j> ⇔ c = 9 k."()!E ( ) ( ) a b c ab bc ca+ + ≥ + + "(eeeeeeee)& O"()"?()"@()"(!k./-)!&7l ( ) ? a b c+ + + ≥ ( ) a b c ab bc ca+ + + + + eeeee999)& j> ⇔ a b c= = = eeeeeeeeeeeeeeeeeee9)& Bài 4: ( 1,5 điểm ) jm!< ax by+ = "( ax by+ = "( ? ? ax by+ = "?( @ @ ax by+ = "@( W"(/0 x y + )!E ( ) ( ) ? ? ax by xy ax by x y+ + + = + 9 4"(/H"?(/H.&F!*!+5)!E ( ) xy x y+ = + "( W"?(/0 x y + )!E ( ) ( ) @ @ ax by xy ax by x y+ + + = + 4"(/H"@(/H.&F!*!+5)!E ( ) xy x y+ = + "( cd4 )x y !JK< ( ) ( ) xy x y xy x y + = + + = + 9 AB<H4)!&7l x y xy + = = − ⇔ ) ) x y x y = − = = = − eeeeeeeeeee)& c0 ) x y= − = )<&K.E6T @ C a b a b a b a b − + = + = − + = + = ⇒ a b = = 9 +7l ) x y= = − )!78!D!n!3&7l a b= = eeee99)& cd4 B ax by x y= + = + = − eeeeeeeeeee99)& Bài 5: ( 1,5 điểm ) ( Giám kh#o t$ v& h'nh ) jm! a > "/0 a < )!78!D(9 AB1o ( ) QM x y " (P∈ 9-!; y ax= 9 E Q @ H x a − ÷ eeeeeeeeeeeeeeeeeeee99)& i.&E @ MF x y a = + − ÷ @ x ax a = + − ÷ @ x a x a = + + "(eeeeeeeeeeee99)& cH @ MH y a = + ÷ @ ax a = + ÷ @ x a x a = + + "( p"(/H"(134+&'3B*eeeeeeeeeee99)& Bài 6: ( 2 điểm ) E ) )O I N !FH9h34+ · · MNI PNO= 9 .H+)%!% MIN /H PON PqP7l!W!T I /H O 9 i.&E · · NMI NPO= 9 h34+ MI rr PO 4 PO AB⊥ 9 cd4 P PH&$Ls=3 AB "#* N (eeeee9)& h34+ · · PAB PNA= 4 · · PAM PNA= 9 i.&E!% ANP /H MAP &t6Teeeeeeeeee999)& h34+ 9PA PM PN= eeeeeeeeeeeeeeeeeee99)& PM PN+ ≤ "&(eeeeeeeeeeeeeee)& . Qx y = ) ( ) ( ) Q ?Q x y = − − eeeee9)& Bài 2: ( 1,5 điểm ) a) A = ? ? @ − 9 ( ) ? @ + eeeeeeeee 999 999 999 999 999 999 999 999 999 & g ( ) ( ) ? ? @ ? . AB⊥ 9 cd4 P PH&$Ls=3 AB "#* N (eeeee9)& h34+ · · PAB PNA= 4 · · PAM PNA= 9 i.&E!% ANP /H MAP &t6Teeeeeeeeee 999 )& h34+ 9PA PM PN= eeeeeeeeeeeeeeeeeee 99 )& . b = 9 78!D b c bc+ ≥ 9 j> ⇔ b c= 9 c a ca+ ≥ 9 j> ⇔ c a= 9 i.&E a b c ab bc ca+ + ≥ + + ) ) )a b c R∀ ∈ 9 j> ⇔ a