D thing hang.
2) Khai thac dieuki en rang buoc cua bien x ,y phat hien
Ihi X + y > 2 g i i i p diTa ve bai toan bat d^ng thuTc rat quen thuoc.
cauiiị
1) Ta CO M N P = 90" (Goc n o i tiep ch^n nuTa difdng tron) va M N D = 90" (Goc n o i tiep ch^n niJa va M N D = 90" (Goc n o i tiep ch^n niJa
di/dng tron).
D o do M N P = M N Q ( = 90") ^' ^ - ' ^ ^ ^ . f => H a i tia NP, N D trung nhaụ
V a y ba d i e m N , P, D t h I n g hang. — \ 2) Ta CO M Q P = 90" (Goc n o i tiep chKn nuTa
du-dng tron)
D A M = 90" (Goc n o i tiep ch^n nijfa di^dng tron)
D o do M Q P = D A M ( = 9 0 " ) => TuT giac D A P Q n o i tiep => Q A P = Q D P
M
M a Q D P = M A N ( H a i goc n o i tiep cilng ch^n cung M N cua diTdng t r o n ( O ) ) .
N e n Q A P - M A N ( = Q D P ) => A P l i tia phan giac cua goc Q A N (1)
M a t khdc Q M P = Q N P ( H a i g6c n o i tiep ciing ch^n cung QP), D M A = D N A ( H a i goc n o i tiep cilng ch^n cung A D cua ( O ) )
N e n D N A - Q N P ( = Q M P ) => NP la tia phan giac cua goc A N Q (2)
TCf (1) va (2) CO P la tam duTdng tron n o i tiep tam giac A N Q .
N h a n xet: D a y la bai toan hinh hoc de, quen thuoc. Phat h i e n tỉ gidc D A P Q noi tiép, chia khoa cua cau 2 de c6 dúdc I d i g i a i cau 2.
Cfiu I V .
2ab + a + b + c(ab - 1) 2ab + a + b + b + abc - c Q =
(a + l ) ( b + l ) ( c + 1) (a + l ) ( b + l ) ( c + 1)
(abc + ab + ac + a) + (abc + ab + be + b) - (abc + ca + be + c) (a + l ) ( b + l)(c + 1) '.
ăb + l ) ( c + 1) + b(c + l ) ( a + 1) - c(a + l ) ( b + 1) rinftJ.s/itrtá' (a + l ) ( b + l)(c + 1) b c a + 1 b + 1 c + 1 a + 1 2 a - 1 b 2 ^ b + I 3 c b - 2 3 - c + + + c + 1 4 5 2(a + l ) 3(b + l ) 4(c + l ) 12 (3 - c) + (b - 2) + (a - ^ (3 - c) + (b - 2) 2(1 + a) 1 1 3(b + 1) 2(1 + a) +(3 - c) ^ 2 ^ 3 4 • ill 1 1 4(c + 1) 3(b + 1) 12 a + b + c (b + 1 - c)(2a - 3b - 1) (3 - c)(3b - 4c - 1) _5_ ^ _5_ 2(a + l ) 6(b + l)(a + l ) ^ 12(b + l)(c + l ) 12 ~ 12 ( V i a < b < 3 < c; c < b + 1; a + b > c) D a u " = " xay ra k h i a = 1, b = 2, c = 3 5 .... V a y gia t r i nho nhaft ciia Q la
N h a n xet: D a y la m o t bai toan bat dang thij-c k h o , trong thdi gian han che kho m ^ t i m dúdc Idi g i a i cua bai loan.
Luy§n g i i i lii UvKsc ki thi vio \dp 10 ba mign BJc. Trung, Nam mOn ToAn _ N(jny(^!i iiilc Jjn
1) Giai he phiTdng trinh:
D6 S6 4 4
THI TUYÍN SINH VAO L(3P 10 CHUYEN TOAN TRl/flNG THPT CHUYEN KHOA HQC Tl/ NHIEN, DAI HQC QUÓC GIA HA NQI
NAM HQC 2012 - 2013 Cfiu Ị (3 di^m) Cfiu Ị (3 di^m)
xy(x + y) = 2
9xy(3x - y) + 6 = 26x^ - 2y^
2) Giai phifdng trinh (Vx + 4 - 2)(>/4 - x + 2) = 2x
1) Tim hai chỉ so cuoi cilng cua so A - 4l'"^ + 57^"'^ .= 2) Tim gia tri Idn nhat cua ham so: y = 3V2x - 1 + \sJ5-4\^ , vdi ^ < x < ^ .
Cfiu IIỊ (3 diem) Cho tarn gi^c nhon ABC (AB > AC), noi tiep dtfcJng tron (O). Gia suT M, N la hai diem thuoc cung nho BC sao cho MN song song vcfi BC va tia AN n^m giffa hai tia AM, AB. Goi P la hinh chiéu vuong goc cua diem C tren AN va Q la hinh chiéu vuong g6c cua diem M tren AB,
1) Gia suf CP c^t QM tai diem T. ChuTng minh r^ng T nam tren di/dng tron (O). 2) Goi giao diem cua NQ va (O) la R khdc N. Gia sur AM c^t PQ tai S. ChuTng
minh r^ng bon diem A, R, Q, S cung thupc mot diTdng tr6n.
Cfiu IV. (1 diem) Vdi moi so n nguyen Idn hdn hoSc bSng 2 co djnh, x6t cic tap n so thifc doi mot khac nhau X = {x,,X2,...,x„}. Ki hieu C(X) la so cac gia tri khdc nhau cua tdng Xj + Xj (1 < i < j < n). Tim gia tri Idn nhat va nho nhat cuaC(X). u .„
HU(5NG DAN GIAI
xy(x + y) = 2
9xy(3x-y) + 6 = 2 6 x 3 - 2 y ' '
xy(x + y) = 2
x^ + x^y + 3xy2 + ý = 27x^ - 27x^y + 9xy^ - y^
xy(x + y) = 2 [x + y = 3x - y x = l y = 1 Cfiu Ị 1. jxy(x + y) = 2 [xy(x + y) = 2 l(x + y)-'= (3x - y)^ X = y J2x^ = 2 X = y <=> < x 3 = l <=> \ X = y
V$y nghi^m (x, y) cua h$ phiTdng trinh 1^ (1; 1)
Cty TMHH M I V UVVH Khaiig Vi^t
2) Dieu kien -4 < x < 4 Ta C O (V;r74 - 2)(V4r7 + 2) = 2x o - ^ = = ^ ( 7 ^ + 2) = 2x <=> X ( V 4- X + 2) - 2(Vx + 4 +2) = 0 o x ( V 4 - x - Is/TTl - 2) = 0 x = 0 X - 0 • 25x^ + 96x = 0 X - 0 74 - X - 2Vx + 4 - 2 = 0 25x' + 96x = 0 x = — X = 0 (thich hdp) X = - — (thich hdp) „ — .
Vay phu-dng trinh c6 hai nghiem la: x, = 0 v^ X2 = - —
23
96
. 2 . . . 0. . . . 2
25
Nhan xet:
1) TH xy(x + y) = 2 cho ta 3x^y + 3xy^ = 6.
Do vay 9xy(3x - y) + 6 = 26x^ - 2y^ o (x + y)' = (3x - y)^ giup c6 Idi giai b^i tôn.
2) Day la bai toan ve phifdng trinh chtfa c5n thtfc quen thupc.
Cfiu IỊ
1) Ta c6 A = 41'"^ + 57^"'^ = 4l(4l'"^ - l) + (s?^"'^ - l) + 42 ''
Ma 41^ = 115856201 ^ 41^ - i;iOO
Nen 41'"^ - 1 = (41^)^' - 1^'\4[^ - 1. Ta c6 41'"^ - lilOO
Matkhdc 572<"2 -1 = ( 5 7 f " - p 3 ; 5 7 ^ - l . T a c 6 573'"2 -ilioo
Nen Achiacho 100dir42
Vay hai chỉ so cuoi cCing cua so A = 4l'"^ + 57^"'^ \h 42.
2) Digu kien i < X < ^ • '
2 2
Ap dung baft ding thiirc C6-si cho hai so khong am, ta c6
y = 3V2x - 1 + xV5-4x^ = 37(2x-l).l + xVs - 4x^
^ ^ 2x - 1 + 1 ^ x^ + 5 - 4x^ _ -3x^ + 6x + 5
Luy$n giJi dí ! a miSn BSC, Trung, Nam nị NguyJn Dilc TS'n
Dafu "=" xay ra <=> x = 1. Vay gia tri Idn nhat cua A la 4 khi x = 1
Nhan xet:
1) Day la bai toan quen thupc doi vdi hoc sinh gi6i toan, phat hien
41^ - 1:100, 57"* - 1:100 mau chot cua IcJi giai, lH do c6 diTdc hai chii" so
cuói cung ciia A la 42.
2) Day la bai toan ciTc tri dai so quen thupc. 0 x
Cfiu IIỊ . r ^ !
1) CT 1 AN (gt) ^ fPA = 90" va M Q 1 AB (gt) => TQA = 90".
Ta CO TPA = TQA = 90'
=> Tỉ giac TAPQ npi tiep => QTP = QAP. Mat khac, xet (O) c6 BC // NM => BN = CM => BAN = MAC
tniDo d6 MTC = MAC =:> TiJ giac TACM npi tiep
Vay T thupc diTdng Iron (O)
2) Ta CO QTP = QAP fcM = B N] Tỉ gidc TQPA npi tiep AQP = ATP Ma ATP = ABC (He qua goc npi tiep)