D thing hang.
1) DB, D Cm cdc tiep tuyen ciia dxXdng tron (O)
ciia dxXdng tron (O)
=> DB J- OB, DC 1 OC. Tac6:
/ M — BJ<: I \
DBO = DCO = DHO = 90" =:> B, C, H cOng thuoc difdng =:> B, C, H cOng thuoc difdng
tr6n dirdng kinh OD. Vay
nam diem D, B H, O, C cilng n^m tren dúcJng trbn diTdng kinh OD.
Mat khac D B , DB, DC la cdc tiep tuyen cua diTdng tron (O)
=> DB = DC, DO la tia phan giac cua g6c BDC (Tinh chat cua hai tiep tuyen c^t nhau)
ADBC can tai D, DO la diTcJng phan giac => DO la diTcJng cao T t f g i d c D I H A npi tiep
Ta CO DIA = DHA = 90"
2) Xdt AOIA va AOHD c6: l O A (chung), 6lA = 6H D ( = 90") 01 OA 01 OA
Do do AOIA <^ AOHD (g.g)
OH OD AOBD vuong tai B, B I la diTdng cao AOBD vuong tai B, B I la diTdng cao
=^ O B ' = OỊOD. Do do OẠOH = OỊOD = O B ' = O M '
OỊOD = OẠOH
(in
Xet AOMA va AOHM c6: M O A (chung). OM OA (vl OẠOH = O M ' )
).iu
OH O M
• D O do AOMA ^ AOHM (c-g-c) => OMA = O H M = 90"
• T a c6 A M 1 O M tai M , M e (O)
• f V a y A M la tiep tuyen cua diTdng tron (O)
^) X6t (O) CO M B D = MCB (hp qua g6c tao b5i tia tiep tuyen day cung)
X6l (MBHOC) CO B D H = BCH (hai g6c noi tiep cilng ch^n cung BH) B M H - M B D + B D H ( B M H la goc ngoai cua tam giac B D M ) Do do D M H = MCB + BCH = HCM
Xet AHMB va AHCM c6:
Luygn giii dg tru6c kl thi vao lap 10 ba miSn BJc. Trung, Nam mOn Toan _ NguySn DCfc Ta'n
HB HM
Do do AHMB ^ AAHCM (g.g) HB.HC = HM^ .
> HM khong đị HM HC
VI A va dirfJng tr6n (O) códinh nen M, H co dinh
Vay tich HB.HC khong doi khi d quay quanh A
N h a n x e t : 1), 2) Day la Mi toan quen thuoc. 3) la cau khọ Ldi giai d trcn that hay khi phat hien AHMB ^ AHCM va MH khong đị
Bai 5. Chia hinh tron da cho thanh 4024, hinh quat bing nhau, moi hinh quat cd dien tich la: 2012 : 4024 = 0,5 (cm^). Xay ra hai triTdng hdp
• Neu ton tai mot hinh quat c6 chiJa ba diem. Ba diem nay la ba dinh cua mot tam giac c6 dicn tich khong vUdl qua dicn tich hinh quat, tufc la khong virut
qua 0,5cm^. , • Ncu khong ton tai mot hinh quat cd chiJa ba diem. ,j ^.^
Ta cd 6037-4024 = 2013
Do vay so hinh quat it nhaft cd chtfa hai diem la 2013. Theo nguyen t^c Di-
rich-le ton tai hai hinh quat lien tiép nhau ma moi hinh quat nay deu chufa 2
diem. Bon diem nay la cac dinh cua hai tam giac ma c6 tdng dien tich khong qua 0,5 X 2 = l(cm^). NhiT vay trong hai tam gidc nay cd it nhaft mot tam giac cd dien tich nho hcJn 0,5cm^.
N h a n x e t : Day la bai toan van dung nguyen tdc Di-rich-le, bai toan nay kho nham de phan loai hoc sinh. Vice chia hinh tron thanh 4024 hinh quat bang nhau la hoan toan tiT nhien. 6 trÚdng hdp khong ton tai mot hinh quat c6 chiira ba diem lai ticp tuc van dung nguyen t^c Di-rich-le d trcn that dac s^c.
s D £ S 6 4 9
M( < . D i THI TUYEN SINH VAO LdP 10 THPT CHUYEN TOAN,
TINH Hl/NG YEN. NAM HQC 2012 - 2013
Bai 1.(2 diem) , " '
a) Cho A = ^2012^ + 2012^2013" + 2013^ . ChiJng minh A m mot so tiT nhien. b) Giai he ph^dng trinh 2 1 X - x'' + — + - = 3 y y • 1 X • X -I- — 4-y y — = 3 Bai 2. (2 diem) ;^
a) Cho parabol (P): y = xNa diTdng th^ng (d): y (m + 2)x - m -f- 6. Tim m d*^' dúdng thing (d) c^t Parabol (P) tai hai diem phan biet cd hoanh do difdng
-CtiL^lliMw^^ Viet
t,) Giai phiTdng trinh: 5 + x + 2^(4 - x)(2x - 2) = 4(V4~::7 + ^/2^r^)
3ai 3. (2 diem)
a) Tim tat ca cac sd hCTu ti x sao cho A = x^ + x + 6 la mot s6 chinh phiWng.
( x ' + y ' ) - ( x ^ + v ^ ) > 8 Cho X > 1 va y > 1. ChiJng minh r^ng:
(x - l)(y -1) ,
Bai 4. (3 diem) Cho tam giac ABC nhon noi tiep diTdng trbn tam O, duTdng cao BE va CF. Tiep tuyen tai B va C c^t nhau tai S, goi BC va OS c^t nhau tai M. a) ChiJng minh AB.MB = AẸBS
b) Hai tam giac AEM va ABS dong dang
c) Goi AM cat EF tai N, AS cit BC tai P. ChtjTng minh r^ng NP vuong gdc vdi BC. Bai 5. (1 diem) Trong mot giai bong da cd 12 doi tham dy, thi dau vong tron
mot lúdt (hai doi bat ki thi dau vdi nhau dung mot tran).
a) ChiJng minh rang sau 4 v6ng dáu (moi doi thi dau dung 4 tran) luon tim dÚdc ba dpi bong doi mot chúa thi dau vdi nhaụ
b) Khing dinh tren con dung khong neu cac doi da thi dau 5 tran?
Hxidng dSn g i a i
Bai 1. a) A = ^2012^ + 2012^2013^ + 2013^
= 7(2013 - 2012)^ + 2.2012.2013 + 2012^2013^ = Vl^ + 2.2012.2013 + (2012.2013)^ = Vl^ + 2.2012.2013 + (2012.2013)^
= 7(1 + 2012.2013)^ = 1 + 2012.2013
Vay A la mot so tif nhien. 1 7 % -4 i: T b) x 2 + . 1
x + l + i i . 3 . Dieu kien y^Ọ Datx + -!- = a . - = b y y
Ta CO X + —
y^ y
^ 0 vay, he phiTdng trinh da cho trd thanh a + a = 6 a + b = 3 |âb = 3 - a + 3 ^ - 6 = 0 a ^ - b = 3 a + b = 3 [(a-2)(a + 3) = 0 b = 3 - a
LuyOn giii 66 truflc kl thi vAo Iflp 10 ba mign BJc, Trung, Nam mOn Toan _ NpuySn Dure Tfln "a = 2, b = 1 (1) a = -3, b = 6 (2) (1) cho ta y ^ = 1 xy + 1 = 2y X = y [y = X x^ + l - 2 x , y = x y = X y = l (2) cho ta 1 X + - = -3 y xy + 1 = -3y X = 6y 6y^ + 1 = -3y X =^ 6y 6y^ + 3y + 1 = 0 y e 0 X = 6y [x = 6y
Vay he phiTdng trinh c6 nghỉm (x; y) (1; 1).
Nhan xet: Day la bai toan quen thuoc doi vdi cdc ban hoc sinh gioi toan Idp 9. Bai 2. a) Phu"c(ng trinh hoanh dp giao diem cua (?) va (d) 1^
x^ = (m + 2)x - m + 6 <=> m^ - (m + 2)x + m - 6 = 0 (*) Ta c6 (d) c^t (P) tai hai diem phan biet c6 hônh dp diTdng
fA > 0
<=> (*) c6 hai nghiem dUdng phan biet <=>
(m + 2f - 4(m - 6) > 0 m + 2 > 0 o m - 6 > 0 ' • • S > 0 P > 0 m^ + 4m + 4 - 4m + 24 > 0 m > -2 m > 6 1 m^ + 28 > 0 (; m > -2 o m > 6 ,v; tn > 6 ,. , ! : V b) Dieukien 1 < x <4 ,f Dat y = V 4 - X + V 2 X - 2 . Ta c6 y^ = 4 - x + 2x - 2+2^/(4 - x)(2x - 2) o X + 2 + 2^(4 - x)(2x - 2) = y^ o X + 2^(4 - x)(2x - 2) - y^ - 2 PhiTdng trinh trd thanh 5 + y^ - 2 = 4y ^
<=>y^-4y + 3 = 0<=> y = 1 Ly = 3 wnmrmv um Rhano Visr (vi a + b + c = 0) . y = 1. Ta c6 V4 - X + >/2x - 2 = 1 o y/l^T^ = i _ V ^ T I ^ N/4 - X < 1 ,s 2x - 2 = 1 - 2V4 - X + 4 - X 1 - V4 - X > 0 2x - 2 = ( l - V 4 ^ ) ' ' 0 < 4 - x < l [ 3 < x < 4 v a 7 - 3 x > 0 <=> 4(4 - X ) = (7 - 3x)^ o X e 0 2 V 4 - X = 7 - 3 x x e 0 4(4 - x) - (7 - 3xf ' ' • . y = 3. Ta c6 V4 - x + V2x - 2 = 3 o V2x - 2 = 3 - N/4^ r I 7 4 - X < 3 ;, jr..', 3 - V4 - X > 0 2 <=> ^ 2x - 2 = (3 - V 4 - x ) [2x - 2 = 9 - 6V4 - x + 4 - x 7 4- X < 3 [ 0 < 4 - x < 9 v a 5 - x > 0 <z> 2V4 - X = 5 - X [4(4 - X ) = (5 - x)^ -5 < X < 4 [-5 < X < 4 x^ - 6x + 9 = 0 [ ( X - 3)2 = 0 o X = 3
Vay phiTdng trinh c6 nghiem duy nhat x = 3 , " ~ NhSn xet:
a) (d) c^t (P) tai hai diem phan bi$t c6 hônh dp diTdng
|| O PhiTdng trinh hônh dp giao diem cua (d) va (P) c6 hai nghiem diTdng phan biet. Dieu neu tren giiip chiing ta c6 Idi gidi b^i todn