VI x^+ xy =2 giup c6 diTdcb) Nhan ra rkng x +
b) Dp dai BE chinh la dp dai diTdng cao uTng vdi canh CM cua tarn giac CK M-
Bai 5: Gpi O la lam diTdng tron ngoai tiep tam giac M A N .
1
Ta c6: M A N = - M O N (he qua g6c npi tiep) va O M = O N . Do d6: M O N = 2 M A N = 60"
C t y T N H H MTV [ l y V ' l K'-anu
A O M N c a n t a i O => A O M N deu
= ^ O M N = O N M = 60"
Tỉ giac O M C N c6 :
M O N + M C N = 60" + 120" = 180" nen npi tiep
= > O C N = 6 M N = 60". 6 C M = O N M = 60" Do vay 6 C M = O C N
=> O thupc tia phan giac cua g6c BCD. Vay O thupc diTcJng t h i n g co dinh AC.
Nh§n xet: Phat hien O thupc diTdng t h i n g co djnh AC. Dieu n^y chtfng minh
dircJc V I de dang chuTng minh diTcJc tu" giac O M C N npi tiep.
}jki 6: Ta c6: . _ J _ > _ J _ . _ _ L _ > _ J _ . . 1 , 1 |;^/^ + ^/2 ^ / 2+ ^ / 3 ' ^ / 3 + ^/4 V4 + ^/5 V79 + V80 TsO + VsT Do d6: + + - _ L ^ + ... + Vl + V2 V3 + V4 75 + V6 779 + x/80 1 > - 2 I Vi + 72 ^fl + ^/2 73 + 74 73 + 74 ••• 7 7 9 7 ^ ^ ^ TTFTTSO — ! — + + — ! — + — ! — + 1 1 7 l + 7 2 V ^ + 73 73 + 74 74 + 75 "' 779+780 780 + 781 = ^(72 - 71 + 73 - 72 + 74 - 73 + ... + 780 - 779 + 7 8 1 - 78o) = ^(-71 + 781)= ^ ( - 1 + 9 ) = 4. J Nhan xet: >) .r.
Viec phat hipn - = = > - = = ; - = = > - = _ ; . . . ;
7 l + 7 2 7 2 + 7 3 73 + 74 7 4 + 7 5
1 ^ 1 7 7 9 + ^ ^ 780 + 781 7 7 9 + ^ ^ 780 + 781
roi "long gh6p" diTa ve tSng :
LuyOn giki tru6c ki thi vao Iflp 10 ba mign Mc, Trung. Nam mOn Toan _ iMguygii OiTc lan
Dfe S 6 60
Di THI TUYEN SINH VAO LOP 10 CHUYEN TOAN,
TRaCiNG THPT CHUYEN DAI HQC S U PHAM HA NQI NAM HOC 2011 - 2012
^4-
Bai 1: (2 diem) Cho a =
a) ChiJng minh rKng: 4â + N/2 a - N/2 = 0. b) Tinh gia tri cua bieu thuTc: S = a H ^
Bai 2: (2,5 diem)
JO + a + 1.
a) Giai he phiTdng trinh: X + y + y = - y
b) Cho hai so hOfu d a, b thoa man ding thtfc: áb + ab' + 2a V + 2a + 2b + 1 = 0 Chiang minh r^ng 1 - ab la binh phiTdng cua mot so hiTu tị
Bai 3: (1,5 diem) Tim tát ca cac so nguyen to p c6 dang p = â + b^ + c^ vdi a, b, c la cac so nguyen diTdng thoa man a" + b"* + c'* chia hét cho p.
Bai 4: (3 diem) Cho tarn giac ABC co ba goc nhon ngi tiep diTdng tron (O), BE
va CF la cac dirdng caọ Cac tiep tuyen vdi difdng tron (O) tai B va C cat
nhau tai S, cdc diTdng thing BC va OS dl nhau tai M.
. ^ , . • u ^ AB BS a) ChiTng mmh rSng: — = — .
AE ME b) Chu-ng minh ring: AAEM ^AABS.
c) Gpi N la giao diem cua AM va EF, P la giao diem cua AS va BC. ChiJng minh ring NP 1 BC.
Bai 5: (1 diem) Trong hop c6 chiJa 2011 vien bi mau (moi vien bi chi c6 dung
mot mau), trong do c6 655 vien bi mau do, 655 vien bi mau xanh, 656 vien bi mau tim va 45 vien bi con lai la cac vien bi mau vang hoSc mau tring (moi mau CO it nhat mot vien). NgiTdi ta lay ra tir hop 178 vien bi bat kị ChiJng minh ring trong so cac vien bi viTa lay ra, luon c6 it nhat 45 vien bi cung maụ Neu ngirdi ta chi lay ra tijr hop 177 vien bi bát ki thi ket luan cua bai toan con dung khong? HI/(5NG DAN GIAI Bail: r j2 a + .2V 8 j \ á+ ^ . a + ^ = ^72 + ^ ^ 4 a ^ + 72 a - = 0 b) Ta c6: 4â + 72 a - = 0 ^ = ^ll^Ll^ • 4 n SB
Cty TNHH M!V DWH Khang Vi^t
Do do :S = â+7ana + l = a ^ + j l i : - i l i + a + l = â + J^Z^TT^ (a + 3)' 2 ^ |va + j ; 2 a = a + ,/ = a + 8 + 3 ^ ^ ( 1 - a ) ^ (a + 3 )72 272 4 4 , _ 72 - 72a + 72a + 372
Nhan xet: Kheo leo bien đi, chu yeu la "nang len luy thiTa", gidp c6 Ic(i
giai cua bai toan.
Bai 2: a) DK: X + y > 0
Ta cd: x^ + y^ + ^ = i ^ + y)^ - 2xy + ^ = 1
X + y X + y
o (x + y)' - 1 - 2xy + 2xy = 0
o ( x + y + l)(x + y - I ) - = 0 X + y 2xy(x + y- l ) X + y ' o ( x + y - l)[(x + y)(x + y + l ) - 2 x y ] = 0 | o ( x + y - I)(x2 + y2 + x + y) = 0 o x + y - 1= 0 (Do x + y > 0 nen x^ + y^ + X + y > 0) <=> y = - X + 1 Nen tir 7x + y = x^ - y ta c6 1 = x^ - (-x + 1) o x^ + x - 2 = 0 I o X = 1, X = - 2 (vi a + b + c = 0) , I X = 1 thi y = - 1 + 1 = 0 ; X =- 2 thi y = 2 + 1 = 3
Vay he phÚdng trinh da cho c6 hai nghipm li (x ; y) = (1 ; 0) vk (x ; y) = (-2 ; 3). b) âb + ab' + 2a'b^ + 2a + 2b + I = 0 o ab(â + b^ + 2ab) + 2(a + b) + 1 = 0
o ab(a + b)^ + 2(a + b) + 1 = 0 o (ab - l)(a + b)^ + (a + b + 1)^ = 0 (*) Ta + b + l V
TCr (*) CO a + b ^ 0 va 1 - ab =
a + b Vay 1 - ab la binh phiTdng cua mot so hihi tị
Nhan xet:
I a) Hien nhien phai nghl den tijf x^ + y^ + = i de den vdi x + y = 1. Q X + y X + y
Luy§n giii truOc ki t h i v a o 1 0 b a 1T11611 B S c . T r u n g , N a m r r f i n l o a n r v i j u y o n D i r r T3'n