a Chijfng minh DE la tiep tuyen chung ciia hai du^ng tron ngoai tiep tam gidc DBH vk ECH.. Goi F la giao diem thuT nhi cua hai dudng tron ngoai tigp tam gidc DBH va ECH.. Gia siif 0,1 l
Trang 1H A N G H I A A N H - N G U Y E N T H U Y Mtl - T R A N KY T R A N H
(Tuyen chgn vd gidi thi$u)
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LCFl NOI DAU
Cac em hpc sinh than men!
HQC la qua trinh ren luyen vat va nhat cua ddi ngudi Dieu gl
vat va mdi c6 dupe thi do chi'nh la dieu minh quy nhat
N h i m giup cac em chuan bj thi vao Idp 10, giup cac em c6 them t u lieu va c6 mot cai nhin tong quat ve nhQng van de trong cac de thi tuyen sinh vao Idp 10 trong nhOng nam gan day, Chung toi bien sogn cuon:
Trong qua trinh bien soan kho tranh khoi nhufng thieu sot, rat
mong nhan dUpc S[jl gdp y cua bgn dpc gan xa
Chuc cac em thanh cong trong ki thi sap tdi
CAC T A C GlA
Trang 3DE 1
TRl/dNC PnH C H U Y E N H 6 N G P H O N G - TRXN DAI NGHTA
L6P C H U Y ^ N NGUYiN THl/ONG H I E N - L^P CHUYEN G I A DjNH
De thi tuyen sinh vao I6p 10 PTTH chuyen tqi TP.HCIVI nam hoc 2006 - 2007
Cau 1. Thu gon cdc bilu thiJc sau :
Cau 3. Giai cac phiTcfng trinh va he phuomg trinh :
Cau 5, Cho tam gidc ABC c6 ba goc nhon (AB < AC), c6 dudng cao AH Goi
D, E Ian lucrt 1^ trung diem ciia AB va AC
a) Chijfng minh DE la tiep tuyen chung ciia hai du^ng tron ngoai tiep tam gidc DBH vk ECH
) Goi F la giao diem thuT nhi cua hai dudng tron ngoai tigp tam gidc DBH
va ECH Chilng minh HF di qua trung diem cua DE
' c) Chiifng minh rkng diibng tr6n ngoai tiep tam gidc ADE di qua diem F
B A I G I A I
c a u l A = V16 + 4V(V5 1 ? 1 (Vio A/2) = ( V I 6 T 4 ( X ^ ( A ^
-5
Trang 4A = (V2A/6 + 2V5)(A/10 - V2) = (A^VCVS + 1)^ (VlO - N^)
= A /2(A/5 + - V2) = (VlO + V2)(VrO - V2) =8
V$y h§ da cho c6 nghi$m (x; y) la
c) K h i cdc dieu ki$n xdc d i n h thoa t a c6 :
Tucfng tir n h u t r e n , t a c6 D E tiep xiic difcfng t r o n ngoai tiep t a m gidc
E C H t a i E
Vay D E l a tiep tuyen chung cua hai difdng t r b n ngoai tiep t a m gidc D B H va E C H
Trang 5Tif (1) va (2), suy r a I D = I E hay H F qua trung diem I cua D E
c) V i cac tuf giac B D F H va C E F H npi tiep nen :
D F H + D B H = 180°; E F H + E C H = 180°
Trong t a m giac ABC c6 : BAC + D B H + E C H = 180°
^ ~ ^ b ) 2x2 ^ 2V3^ - 3 = 0 c) 9x^ + 8x2 - 1 = 0
5x + 3y = - 4
L a i CO : D F H + E F H + DFE = 360° Suy ra : BAC + D F E = 180''
Vay dirdng t r o n ngoai tiep t a m gidc A D E qua F
D E 2 THI TUY^N SINH VAO l6P 10 TAI TP HCM NAM HOC 2006 - 2007
C a u 1. Giai cac phifOng t r i n h va he phucfng t r i n h sau :
C a u 3. Cho manh dat h i n h chO nhat c6 dien tich SeOml Neu tang chieu
rpng 2ra va giam chieu dai 6 m t h i dien tich m a n h dat khong doi T i n h
chu v i m a n h dat Iiic ban dau
C a u 4
a) Viet phuong t r i n h di/&ng t h i n g (d) song song vdi difcmg t h i n g y = 3x + 1
va c i t true tung t a i diem c6 tung dp bkng 4
b) Ve do t h i hkm so y = 3x + 4 v^ y = tren cung mot he true tpa dp
2
T i m tpa dp cac giac diem cua hai do t h i ay b i n g phep t i n h
C a u 5 Cho t a m gidc A B C c6 ba g6c nhpn va A B < AC Dudng t r o n t a m O
difdng k i n h B C c i t cdc canh A B , AC theo t h i l t u t a i E va D
C a u 3 Goi chieu rong ciia manh dat luc dau la : x (m), (x > 0)
Chieu dai manh dat luc dau la : 360 (m),
Trang 6C h i l u rong manh dat sau k h i tang 2m la : x + 2 (m)
O C A
C h i l u dai manii dat sau k h i giani 6m la : 6 (m)
X
Tir dieu kien dau bai ta c6 phu'cfng t r i n h : (x + 2) 360 - 6 = 360
Giai phucfng t r i n h nay ta dMc x, = 10 (nhan) va Xv = -12 (loai)
Vay : Chieu rong manh dat liic dAu la : 10 (m)
Chi6u dai manh dat luc dau la : = 36 (m)
10
Chu vi manh dat luc dau la : (36 + 10) x 2 = 92 (m)
C a u 4
a) Phaong t r i n h (d) song song vdi ducyng t h ^ n g y = 3x + 1 nen c6 dang
y = 3x + b, (dj cat true tung tai diem c6 tung dp bang 4 => b = 4
Vdi Xv = -4 =o y, = -8
Vay toa do giao diem ciia (P) va (d) la : (-2; 2) va (-4; 8)
Cau 5 (Hinh 2)
a) Ta CO : ABD = AEC (cung chdn cung DE)
Suy ra hai tam giac vuong ABD va ACE dong dang
Ma H la giao diem ciia BD va CE nen
H la trirc tam ciia tam giac ABC
Suy ra A H 1 BC
1
c) Ta CO : A N M = - M O N (goc tao bdi
tia tiep tuyen va day vdi goc d tam ciing chan MN)
Ma OA la tia phan giac ciia goc M O N (tinh chat hai tiep tuyen cat
1 nhau) nen A O N = - M O N , suy ra : A M N = A O N
d) Ta CO : A N E = A B N (goc tao bdi tia tiep tuyen va day vdi goc npi tiep
ciing chan cung NE) Va NAB la goc chung ciia hai tam giac A N B va
A N E
A N AE Suy ra AANE ^ AABN (g-g)
Trang 7C a u 4 T i m cac so' n g u y e n diTOng c6 2 chuf so', b i e t so' do l a boi ciia t i c h 2
chCf so' ciia c h i n h so do
C a u 5 Cho h i n h b i n h h a n h A B C D c6 goc A n h o n , A B < A D T i a p h a n giac
ciia goc B A D cat B C t a i M v a cat D C t a i N G o i K l a t a m ciia d u d n g
Trang 8V a y b a t dang thufc da cho d i i n g
b ) V d i a > b > 0 t h i Va" - b^ + V2ab - b^ > a
t a ' - b-) + (2ab - b") + 2yl(a^ - b ^ ) ( 2 a b - b ^ ) >
<^ 2b(a - b) + 2V(a^ - b ^ ) ( 2 a b - b - ) > 0 ( d u n g )
V a y b a t d a n g thiJc da cho d u n g
C a u 4 G p i so c a n t i m l a ab (a, b k h a c 0) TCf gia t h i e t c6 ab = m ab
Suy r a : 10a + b = m a b , h a y b = a ( m b - 10), suy r a b c h i a h e t cho a
t h a n g A C )
a ) C h i r n g m i n h h a i t a m giac M B C v a M H K d o n g d a n g v d i n h a u
b ) T i m v i t r i ciia M de do d a i d o a n H K I d n n h a t
15
Trang 9V a y abed = 45- = 2025 ThiJr l a i t a t h a y so c a n t i m l a 2025
C a u 5 (Hiiih 5)
a) Ke t i e p t u y e n Cx v o i d u d n g t r o n
(O), k h i do B A G = B C x (cung e h a n cung B m C ) ;
M a t k h a c B A G = M N C (cung bij
v d i M N B ) Suy r a M N C = B C x , tiT do M N // Cx
Trang 10Cau 6. (Hlnh 6)
a) Bon diem A, H , M , K cung n k m tren
mot ducfng t r o n dudng k i n h A M
Dang thilc xay ra <=> H = B, luc do A B M = 90'' <=> A M la dudng k i n h
ciia dirdng t r o n (O) Do do k h i M la diem dol xufng ciia A qua O t h i do
dai H K Idn nhat
= - 1
= 0
Cau 2 Cho X > 0 thoa : x" + A- = 7 T i n h x"' + —
Cau 3, Giai phiforng t r i n h : 3x = A /3X + 1 - 1
Cau 5 Cho t a m giac ABC c6 3 goc nhon npi tiep trong dudng t r o n t a m 0
(AB < AC) Ve dirdng tron t a m I qua 2 diem A, C cat doan A B , BC Ian
lirot t a i M , N Ve dirdng t r o n t a m J qua 3 diem B, M , N cit dirdng t r o n
(O) t a i dii'm H (khac B)
a) Chdng m i n h : OB vuong goc vdi M N
18
b) ChiJng m i n h : tuT giac l O B J la h h i h binh hanh
c) Chijrng m i n h : B H vuong goc vdi I H
Cau 6. Cho h i n h binh hanh ABCD Qua mot dii'm S trong h i n h binh hanh
A B C D ke dudng t h i n g song song vdi AB Ian lirpt cat A D , BC t a i M , P
va cung qua S ke dirdng thang song song vdi A D M n lirpt cat A B , CD tai N , Q ChiJng m i n h 3 dirdng t h i n g AS, BQ, DP ddng quy -
Trang 11Vay phifoing t r i n h c6 2 nghiem la : x = 0; x = 5
C a u 4
a) Ta CO : P = Sx" + 9y- - 12xy + 24x - 48y + 82
= ( X - - 8x + 16) + (4x- + 9y- - 12xy + 32x - 48y + 64) + 2
= (X - 4}- + (2x - 3y + 8 ) ' + 2 > 2
fx = 4 Dau "=" xay ra <=> X = 4
(1; 1; 1), (4; 4; -5), (4; - 5 ; 4), (-5; 4; 4)
C a u 5 (Hiiili 7)
a) Ve tiep tuyen (d) t a i B ciia diidng tron (O)
Ta CO : B i = Ci (goc tao bcfi
mot day va tiep tuyen ciJng
chan A B ciia (O))
Tif (1) va (2) suy ra OIBJ la h i n h binh hanh
c) T a CO OJ 1 B H t a i trung dig'm K ciia doan H B
Goi P la trung diem ciia OJ t h i P cung la trung diem ciia doan B I Vay P K la dudng trung binh ciia tain giac B H I
Suy ra P K // I H => I H 1 H B (dpcm)
C a u 6 (Hinh 8)
Goi I la giao diem ciia DP vdi NQ
Goi K la giao diem ciia AS vdi BC
BP I Q Suy ra SK, IP, BQ dong quy
a) T i m m de phuong t r i n h c6 hai nghiem phan biet deu am
b) Goi X i , X2 la hai nghiem ciia phaang t r i n h T i m m de c6 Xj + = 1 3
Trang 12C a u 4 ChiJng m i n h rhng neu a + b > 2 t h i it n h a t mot trong h a i p h u a n g
t r i n h sau c6 n g h i e m : + 2ax + b = 0; + 2bx + a = 0
Cau 5 Cho diTcrng tron tarn O du&ng k i n h AB Goi K l a t r u n g di§m ciia
cung A B , M la d i 6 m di dgng t r e n cung nho A K ( M k h a c d i e m A va
K) Lay d i e m N t r e n doan B M sao cho B N = A M
a) Chufng m i n h A M K = B N K
b) ChiJng m i n h t a m giac M N K Ik tarn gidc vuong can
c) H a i dudng thSn g A M v a OK c^t nhau t a i D Chufng m i n h M K la dudng
phan giac ciia goc D M N
d) Chufng m i n h r a n g diTdng t h a n g vuong goc vdi B M t a i N luon luon di
qua mot diem co' dinh
Cau 6 Cho t a m giac ABC c6 BC = a, CA = b, AB = c va c6 R la ban k i n h
dudng tron ngoai tiep thoa man h$ thiJc : R(b + c) = aVbc Hay d i n h
dang t a m giac ABC
Cau a)
3
Dat u = X + y, V = x.y H | da cho c6 dang
Giai h$ nay t i m diTOc (u; v) = (6; 5), (5; 6)
x + y = 6 _^ |x + y = 5
u + V = 11 u.v = 30
<—> J <=> l x - y = 16
xy = -64
y - X _ 1 <=>
xy 4 y^ + 16y + 64 = 0
xy = -64
y - X _ 1 -64 ~ 4
X - y = 16 Vay nghiem (x; y) ciia he phirang t r i n h la (8; -8)
4 PhiTdng t r i n h x'^ + 2ax + b = 0 c6 biet so Aj = a^ - b Phirang t r i n h x^ + 2bx + a = 0 c6 biet so Ag = b^ - a
Ta CO : Al + A 2 = (a^ - b) + (b'^ - a) = (a - 1)' + (b - if + (a + b - 2) > 0
Do do trong hai so A ; , A^ C6 i t nhat mot so khong am nghia la c6 i t
nha't mot trong hai phirgmg t r i n h da cho c6 nghiem
Trang 13c) TCr k e t qua cau b) t a m giac M K N vuong
can t a i K nen N M K = 45° ,
D
Do D M N = 90" D M K = 45"
Suy ra M K la ducfng phan giac ciia D M N
d) Gia siif du'cfng thSng vuong goc vd'i B M
M a t khac BAE = 45° nen t a m giac A B E
vuong can tai B, dfin t d i E co dinh (dpcm)
C a u 6. Tij' moi quan he giuTa do dai dudng k i n h va day cung t a c6 :
2R > BC = a va b + c > 2 V b c , suy ra R(b + c) > aVbc
Dang thiJc xay ra k h i va chi k h i tam giac ABC vuong can t a i A TCr do
R(b + c) = aVbc t h i tam giac ABC vuong can t a i A
D E 7
T R U O N G PTTH C H U Y E N L E H O N G P H O N G , T P H C M
De thi tuyen sinh vdo I6p 10 chuyen Todn nam hoc 2003 - 2004
C a u 1
a) Thu gon bieu thiJc A =
nghiem t h i phuong t r i n h sau luon c6 nghiem ;
^yttk (an ~ mb)x- + 2(ap - mc)x + bp - nc = 0
^ ^ ^ u 5 Cho t a m giac ABC vuong t a i A (AB < AC) c6 dUcfng cao A H va trung
tuyen A M Ve ducfng tron t a m H ban k i n h A H , cat A B d diem D, cat
AC d diem E (D va E khac diem A)
a) ChiJng m i n h ; D, H , E thang hang
b) ChiJng m i n h : M A E = A D E va M A vuong goc vdi D E c) ChiJng m i n h bon diem B, C, D, E cCing thuoc mot dudng tron t a m la O Tuf giac A M O K la hinh gi ?
d) • Cho goc A C B = 30° va A H - a T i n h dien tich tam giac HEC
C a u 6. Cho h i n h thang ABCD c6 hai dudng cheo AC va BD ciing bang canh day Idn AB Goi M la trung diem ciia CD Cho biet MBC = CAB T i n h cac goc ciia h i n h thang ABCD
3V2-2V3
V 2 - V 3 \3V2 + 2V3
b) T i m gia t r i nhd nhat ciia : y = V x ^ ^ l ~ - " 2 V x ^ + Vx + 7 - 6Vx - 2
C a u 2 Giai cac phucfng t r i n h va he phuforng t r i n h :
Trang 14Dau "=" xay ra <=> Vx - 2 - 1 > 0
3 - Vx - 2 > 0 1 < Vx - 2 < 3 c:> l < x - 2 < 9 <=> 3 < x < l l Vay gia t r i nho nhat ciia y l a 2 c? 3 < x < l l
(X - y)^ = y- + X- - 2xy = 6 - 12 - 8 V2 = -6 - 8A/2 < 0 (loai)
Vay nghifm ciia he phuang trinh da cho la : X = 2
y = >/2'
X = V2
y = 2 b) + x^ - 4 = 0 (Dieu kien : -2 < x < 2)
0 X > 0 x^ = 4 - x'^
X > 0 2x^ = 4 X = V2 (thoa dk -2 < x < 2) Vay nghiem cua phuomg trinh da cho la x = -\/2
Ap dung : Dieu ki#n - 2 ^ 0 c=> x^±^/2
+4 x^ - 2 = 5x x" + 4 = 5x(x^ - 2) x" - 5x' + lOx + 4 = 0 (X - 2)(x + IXx^ - 4x - 2) = 0
x - 2 = 0
X + 1 = 0 x^ - 4 x - 2 = 0
Trang 15B^ - AC > 0 Vay A' > 0 => (3) c6 nghiem
Trircfng hop phirong t r i n h (2) v6 nghiem, chiing minh tirong tiT nhiT tren
M la trung diem day B C nen O M ± B C
H la trung diem day D E nen O H 1 D E
Ta CO : A H 1 B C (gia thiet), O M 1 B C => A H // O M
va M A 1 D E (cau b), O H 1 D E M A // O H TiJ giac A M O H c6 A H // O M va M A // O H nen tuf giac A M O H la h i n h binh hanh
d) Ve H K vuong goc vdi A C
A H A C vuong t a i H c6 A C H = 30° (gia thiet)
=> A H A C la nufa t a m giac deu => H A E = 60° AC = 2AH = 2a
=D M N la dudng trung binh AADC
=> Tuf giac A N M B noi tiep duoc
Mat khac ABAD can t a i B (AB = BD) c6 B N la trung tuyen
=> B N la duo'ng cao ciia t a m giac BAD => ANB = 9 0 °
=> AB la ducyng kinh ciia ducmg tron ngoai tiep AMAB
=> AMB = 90° Ma M A = M B (tinh chat doi xufng true)
Do do A A M B vuong can t a i M s>
Ta CO : A A B C can tai A ( A B = A C )
M A B = MBA = 45°
29
Trang 16De thi tuyen sinh v a o Idp 10 c h u y e n Todn n a m hoc 2002 - 2003
C a u 1, T i m gia t r i cua m de phifang t r i n h sau c6 nghi^m va t i n h cac
nghiem ay theo m :
X + I - 2x + m I = 0
C a u 2 Phan tich thanh nhan tilf : A = x'" + x'' + 1
C a u 3 Giai cac phucfng trinii va he phirong t r i n h :
C a u 5 Cho tam giac ABC c6 3 goc nhpn noi tiep trong dudng t r o n (0) va
CO A B < AC Lay diem M thuoc cung BC khong chiifa diem A ciia dudng
t r o n (O) Ve M H vuong goc BC, M K vuong goc CA, M I vuong goc A B
(H thuoc BC, K thuoc AC, I thuoc AB)
Chufng m i n h : BC AC A B +
M H M K M I
C a u 6 Cho tam giac ABC Gia sijf cac dacrng phan giac trong va phan gi^c
ngo^i cua goc A cua tam giac ABC Ian liTot dt duemg t h ^ n g BC t a i D,
E va CO A D = AE
ChiJng m i n h : A B " + AC^ = 4 R - vdi R la ban k i n h diTcJng tron ngoai
tiep tam giac ABC
P > 0 ) khong thoa x < 0
+ Xet m = 0 • ( 1 ) CO ngliiem 0 va 1 ; (2) c6 nghiem 0 va 3 N h a n x = 0 + Xet m < 0 : phUdng t r i n h da cho c6 bon nghiem, hai nghiem duang hai nghiem am (do P < 0 ) thoa dieu kien x < 0
1 - Vl - 4 m 3 - V 9 - 4 m Hai nghiem am la : x = ^ ; x = ^ Vay m < 0 t h i phifong t r i n h c6 nghiem :
Trang 17x,( = 2 + 4 = 6; X 4 = 2 - 4 = - 2 PhUong t r i n h c6 bon nghiem la :
32
+ 3 = 0 CO X-' - 2(x'' + 2) + 3(x-' + 2 ) ' = 0
(x=^ + 2f x^ + 2
CO X'' - 2x'' - 4 + 3x" + 12x'' + 12 = 0 3x" + l l x ' ' + 8 = 0 CO (x'' + l)(3x''' + 8) = 0 x'^ + 1 = 0
Trang 18Vay : Gia t r i nho nhat ciia y la 0 v6i x = 0
Gia t r i Ion nhat ciia y la — vdi x = —
M I , M H Ian luot la diro'ng cao dng vdi
canh AB, DC cua tam giac MBA, MDC
tiep Cling chan cung MC)
TRUCfNG PTTH CHUYEN LE HONG PHONG, TPHCM
De thi tuyen sinh Idp 10 (Ban A, B) nam hoc 2001 - 2002
C a u 1 Cho phUcfng t r i n h : (m + l ) x ' - 2(m + 2)x + m - 3 = 0 (c6 x la an so)
a) D i n h m de phucng t r i n h c6 nghiem
b) D i n h m de phUOng t r i n h c6 h a i nghiem X j , x^ thda : (4xi + l)(4xv + 1^ = 18
C a u 2 Chufng m i n h cac bat d^ng thufc sau : a) a' + b ' + c' > ab + be + ca vdi moi a, b, c
, , a^ + b*^ + 1 1 1 , n u n
b) — > - + - + - (a > 0, b > 0, c > 0) a^b^c a b c
C a u 4 Cho tam giac ABC cd ba gdc nhon noi tiep trong difdng t r o n t a m 0
va true t a m la H Lay diem M thuoc eung n h o BC
a) Xae d i n h vi t r i ciia diem M sao cho tuf giac B H C M l a mot h i n h b i n h hanh
b) Vdi M lay bat k i thuoc cung nho B C , goi N , E Ian lugt la cac diem doi
xiing cua M qua AB va AC Chilng m i n h ba diem N , H , E thSng hang
Xac d i n h vi t r i cua M thuoc cung nho BC de cho N E cd do dai Idn nhat
Trang 19C a u 5 Cho dLfdng tron co dinh t a m O, ban kinh bang 1 Tam giac ABC'
thay doi va luon ngoai tiep dudng tron (O) M o t dirdng thang di qua
tam O cat cac doan A B , AC Ian lugt t a i M , N Xac dinh gia t r i nho
nhat ciia dien tich tam giac A M N
6
CaU 2
a)
^ 2a- + 2b' + 2c- - 2ab - 2bc - 2ca > 0
^ (a- + b- - 2ab) + (b" + c" - 2bc) + c" + a" - 2ca) > 0
^ (a - b)- + (b - c)- + (c - a)- > 0 (dung)
Do do a" + b" + c' > ab + be + ca la bat d i n g thufc dung,
b) Ap dung cau a) ta c6 :
a» + b« + c« > a^b^ + b^c^ + c^a^ = ( a V ) " ' + ( b V ) " + ( c V ) ' ' ' >
> (a-b-)(b-c-) + (b-c')(c-a-) + (a-b-Kc'^a") = a-b-c-(a" + b" + c")
> a"b"c' (ab + be + ca)
0 do
+ b** + c*" ^ a2b'c''(ab + be + ca)
a^b^c^ " a^b^c^
+ b ' + c^ + d" + e' > a(b + c + d + e) a" + b" + c~ + d ' + e' - a(b + c + d + e ) > 0 a" + b" + c' + d ' + e" - ab - ac ad - ae) > 0
> 0 (bat dang thuTc dung)
Do do a" + b" + c" + d- + e'- > a(b + e + d + e) la bat dang thufc dung
Trang 20X = 0 khong la nghiem ciia phifang t r i n h vi 0 - 1
* Xet x*<d Chia ve trai ciia phuo'ng trinh cho x, phucfng t r i n h trot thanh
Vay phirang trinh c6 hai nghiem la
« A M la difcfng kinh cua (O) ^
<=> M la diem doi xufng ciia A qua O
b) A M B = A N B (tinh chat doi xiJng true)
== tir giac N A H B noi tiep N H B = N A B
Ma N A B = B A M (tinh chat doi xiirng true) Suy ra N H B = B A M ChiJng minh tuong t u ta c6 : C H E = M A C
Ta CO : A M = A N , A M = A E (tinh chat doi xufng true)
=> A N = A E A A N E can tai A, ma A K la duo'ng cao
iz> A K vCra la phan giac, vi^a la trung tuyen
==> A N E = 2 N A k , N E = 2NK do do B^AC = N ' A K
A K A N CO K = 9 0 " nen N K = A N s i n N A K
Do do N E = 2AM-sin B'AC
Vi A M < 2R; sin B A C khong doi
N E 16-n nhat CD A M idn nhat co A M la dudng kinh cua (O) <=> M la diem doi xiing ciia A qua O
Vay khi M la diem doi xufng ciia A qua O t h i N E 16'n nhat
Trang 21Ta CO : S AMN AMN SAMN ^ 2S AMN
Dau "=" xay ra o I = A <:= M N 1 OA, BAG = 90°
Vay gia t r i nho nhat ciia dien tich tani giac A M N la 2
D E 10 TRl/dNG PTTH CHUYEN LE HONG PHONG, TPHCM
De thi tuyen sinh vdo I6p 10 c h u y e n Todn n a m h o c 2001 - 2002
C a u 1 T i m t a t ca cac so' nguyen x thoa : x"* + 8 = VVSx + 1
C a u 2. Cho n la so nguyen duong Chufng m i n h ta luon c6 bat dang thufc
C a u 4 Cho cac so' thufc duong a, b, c thoa a" + b" + c" = 27
T i m gia t r i nho nhat ciia S = a' + b ' + c'
C a u 5. Cho tam giac ABC c6 ba goc nhon D la mot diem tren doan BC
Dat BC = a, CA = b, AB c, AD = d, DB = m, DC = n
1. ChiJng niinh : d"a = b"m + c'n - amn
2. Cho biet AD la phan giac trong goc A Chufng m i n h : A D " < AB.AC
X = 3 x^ + 3x'* + 9x3 + 43x2 + 129x - 5 = 0
Ta thay x = 0 khong la nghiem ciia (*) vi : -5 0
x ? ^ 0 t a c 6 x > 0 v a x € Z nen x > 1
Do do : x"' + 3x'' + 9x' + 43x^ + 129x > 5 => (*) v6 nghiem nguyen khac 0 Vay phifong t r i n h chi c6 mot nghiem nguyen la x = 3
Cdch 2 ; x > 0; X G Z
Vdi X = 0; 1 ; 2; 4 dang thufc khong thoa
V<Ji X = 3 t h i dang thiJc thoa
Trang 22T a c o : (2a + 3)(a - 3 ) " > 0 <=• (2a + 3)(a^ - 6a + 9) > 0
<=:> 2 a ' - 12a- + 18a + 3 a - - 18a + 27 > 0
* X e t D n a m t r e n duo'ng t h a n g H B Chufng m i n h tUo'ng t i f t r e n t a cung c6 : d ' a = b ' m + c ' n - a m n
Trang 23D E 11 TRl/dNG PTTH CHUYEN LE H 6 N G PHONG, T P H C M
De thi tuyen sinh vdo Idp 10 (Ban A, B) nam hoc 2000 - 2001
C a u 1 Cho phuong t r i n h : x ' - 2(m + l)x + m " - 4m + 5 = 0 (c6 an so la x)
a) D i n h m de phiidng t r i n h c6 nghiem
*
b) D i n h m de phuong t r i n h c6 hai nghiem phan biet deu difOng
C a u 2 Giai cac phuong t r i n h va he phiTcfng t r i n h :
C a u 4 Cho diTong tron (O; R) c6 dudng k i n h AB va mot diem C bat k i
thuoc diJorng tron khac A va B Goi M, N Ian lugt la trung diem ciia
cung nho AC va CB
a) Ke N D vuong goc voi AC (D thupc AC) Chufng m i n h N D la tiep tuyen
ciia duofng t r o n (0)
b) Goi E la trung dii'm ciia doan BC Dudng thSng OE cAt diTong tron (0)
tai diem K (khac diem N) Chufng m i n h tiif giac A D E K la mot h i n h
b i n h hanh
c) Chufng m i n h rang k h i C di chuyen t r e n dudng t r o n (O) t h i M N luon
luon tiep xiic voi mot duang tron co dinh
C a u 5 Cho hai tam giac ABC va DEF c6 cac goc deu nhon va c6
ABC = D E F , BXC = E D F , AB = 3DE
Chufng m i n h rang ban k i n h dudng t r o n ngoai tiep t a m giac ABC bang
ba Ian ban k i n h dUcJng t r o n ngoai tiep tam giac DEF
(a + b)(a - b ) ' > 0 (a + b ) [ ( a ' - ab + b") - ab| > 0
<=> a'' + b'' - ab(a + b) > 0 c:> a'' + b ' > ab(a + b)
*
a + b a^_+b^
2ab
Trang 242ab
Cau 4. (Hinh 18)
a) ACB = 90° (goc noi tiep chdn nijfa difong tron)
N la trung diem cung BC =^ ON ± EC
Ma AC 1 BC
Suy ra AC // ON
DN 1 AC (gia thiet)
Do do DN 1 ON
=> ND la tie'p tuye'n cua dudng tron (0)
b) N la trying diem cung BC,
E la trung diem day BC (gia thiet)
Tii giac CDNE c6 CBN = DNE = DCE = 90° nen la hinh chuf nhat
Goi I la giao diem cua DE va CN
T a c o : IE = IN => AlEN can tai I lEN = INE
Hinh thang ACNK (AC // ON) noi tiep dudng tron (0) nen la hinh
thang can =- INE = AKN
Taco: AKN = lEN (= I N E ) ^ AK = DE
> tiJ giac ADEK la hinh binh hanh
OH = (H la hinh chieu ciia O tren MN)
Do do M N luon luon tiep xuc voi dudng tron co dinh tam O, ban kinh s
AOAB can tai O
AIDE can tai I
OA _ AB R ^ 3DE
ID" " DE ^ r ^ DE R = 3r.i
D E 12
T R U O N G PTTH CHUYEN LE HONG PHONG, TPHCM
De thi tuyen sinh v a o Idp 10 chuyen Toan nam hoc 2000 - 2001 Cau 1. ChiJng minh :
A + a + B + b B + b + C + c C + c + A + a
C + c + A + a + b + d
A + a + B + b + c + d B + b + C + c + a + d trong do A, a, B, b, C, c, d la cac so diTcfng
Cau 2 Cho X, y > O v a x + y < l Tim gia tri nho nhat ciia bieu thiJc :
Cau 4. Tim tat ca cac so' nguyen x, y, z thoa phuong trinh :
3x- + 6y' + 2z' + 3y-z- - 18x - 6 = 0
Cau 5 Dien tich tam giac A B C la 1 Goi A , , B,, C, Ian luot la trung diem
BC, C A , A B Tren cac doan A B , , C A , , B C , Ian lugt chon cac digm K , L ,
M Tim gia tri nho nhat ciia dien tich phan chung cua hai A K L M va
A A , B , C ,
47
Trang 251 x^ + y^ 2xy
C a u 3. Goi cac irdc nguyen to cua so N la p, q, r va p < q < r
N nho nhat nen N = 2-.5.13 = 260
CaU 4- D§ thay z" chia het cho 3 => z' chia het cho 9
• Xet z- = 0 Ta c6 3x- + 6y- - ISx - 6 = 0
C 5 3(x - 3)- + 6y' + 2z' + 3y^z' = 33 thi 2z^ + 3y-z^ > 2.9 + 3.1.9 > 33 (loai) thi 3(x - 3)- + 2z' = 33
thi 3(x - 3)- = 15 (loai)
=> z' > 6' = 36 Ta c6 : 3(x - 3)" + 2z- > 33 (loai) Nghiem nguyen ciia phucfng t r i n h la : (x = 6; y = 1; z = 0);
Trang 26Do do S > — Dau "=" xay ra <=>
L = A j ; K = A
K = B i ; M B
M ^ C i ; L EE C Gia t r i iilio nhat ciia dien tich phan chung cua hai tam giac K L M va
A , B | C , l a -
8
D E 1 3
T R L / O N G P T T H C H U V A N A N V A TRUCfNG P T T H HA NOI - A M S T E R D A M
D e thi t u y e n sinh v d o Idp 10 c h u y e n Todn, Tin n a m h o c 2005 - 2006
C a u 1. Cho P = (a + bXb + cKc + a) - abc vdi a, b, c la cac so nguyen
Chu'ng m i n h rSng neu a + b + c chia het cho 4 t h i P chia het cho 4
^ - „ ^ , , , , ( X + y)'' +13 = 6xV^ + m
C a u 2 Cho he phu-olig t r i n h • \ ,
xylx" + y") = m
1. Giai he phifang t r i n h vdi m = -10
2. Chirng minh rSng khong ton tai gia t r i cua m de he c6 nghiem duy nhat
C a u 3. Ba so duo'ng x, y, z thoa man he thiJc - + — + - = 6 Xet bieu thufc 1 2 3
P = X + y" + z'
1. Chdng m i n h rSng P > x + 2y + 3 z - 3
2 T i m gia t r i nho nhat ciia P
C a u 4 Cho tarn giac ABC, lay ba diem D, E, F theo thuf tii tren cac canh
BC, CA, AB sao cho A E D F la tuf giac noi tiep Tren tia A D lay diem P
(D nam giCfa A va P) sao cho DA.DP = DB.DC
1. Chijfng m i n h rang tuf giac ABPC noi tiep va hai tam giac DEF, PCB
dong dang voi nhau
2 Goi S va S' Ian luat la dien tich hai tam giac ABC va DEF Chirng
m i n h : — <
S
EF
2 A D ;
C a u 5 Cho h i n h vuoiig ABCD va 2005 diro-ng thang dong thdi thoa man
hai dieu kien :
1. M 6 i duong thang deu cat hai canh do'i ciia hinh vuong
2 M o i d U o n g thang deu chia hinh vuong thanh hai phan c6 t i so' dien
P dircjc viet l a i : P = (T - a)(T - b)(T - c) - abc
= (ab + be + ca)T - 2abc + T ' - (a + b + c)T'
Vi T chia het cho 4 nen trong ba so a, b, c c6 i t nhat mot so chSn
Tir (1) suy ra P chia het cho 4 (dpcm)
(1)
C a u 2
1
( X + y)"* + 13 = 6 x V ^ + m xy(x^ + y^) = m
Dat u = (x + y)", v = xy
Vdi m = -10 he c6 dang : - 6v^ + 23 = 0
uv - 2v^ = -10
(1) (2) ITi^ (2) C O u = 2 v ' - 10 , thay vao (1) ta diTcfc : 2v'' + ITv" - 100 = 0 (3)
25
i G i a i (3) ta diTOc : v" = 4 hoac V " = - — (loai) V = ±2
i T i r (2) ta thay v < 0 nen chon v = - 2 IVdi v = - 2 t h i u = 1 x + y = 1 hoac x + y = - 1
fx + y = l [x + y = - l cGiai cac he : < _^ va <^ ^ ,
• [xy = - 2 [xy = -2 ' ta diroc nghiem (x; y) la : ( - 1 ; 2), (2; - 1 ) , ( 1 ; - 2 ) , (-2; 1)
I 2 Neu (x; y) la nghiem ciia he t h i (-x; - y ) cung la nghiem cua he do
Do do he c6 nghiem duy nhat t h i x = y = 0
IfThay x = y = 0 vao he thay kh6ng thoa
Vay khong c6 gia t r i nao ciia m de he c6 nghiem duy nhat
^ a u 4 (Hinh 22)
DP DC
DB DA
Trang 27Ta CO BDP = ADC (2)
Tir (1) (2) suy ra ABDP <^ AADC
=> D P B = D C A
=> tiir giac A B P C noi tiep
Vi cac tuf gidc A E D F va A B P C npi tiep nen
B A P = D E F = B ^ ; D F E = D A E = C B P
=^ A D E F to A P C B (dpcm)
2. Do ADEF c/3 APCB nen S' =
DP Mat khac
4DA.DP D A
C a u 5 (Hlnh 23)
Dat A B = A D = a Goi E F va H K la
hai true doi xiJng ciia h i n h vuong
ABCD (EF // A D , H K // AB)
Giai suf mot doan t h i n g c^t cac doan
AD, BC, E F \in \mi t a i G, J , M sao
cho ScDcij = 2SABJG t h i
cho E M = M N = N F = H P = PQ = Q K
-3
Taang tir nha t r e n ta thay cac du6ng thang thoa man dieu k i e n ciia d§
bai phai di qua mot trong bon diem co dinh M , N , P, Q
Theonguyen l i Dirichlet tii 2005 duomg t h i n g thoa man d i l u k i e n cua
p i thi tuyen sinh v a o I6p 10 c h u y e n Khoa hoc TiJ nhien n a m 2005 - 2006
x V x - 1 x V x + 1 X + 1
rail 1. Cho bieu thufc P = 7=- + —r=r-
C a u 2. Cho bat phuomg t r i n h : 3(m - l ) x + 1 > 2m + x (m l a t h a m so)
1 Giai bat phiTOng t r i n h vdri m = 1 - 2A/2
2, T i m m di bat phuang t r i n h nhan moi gia t r i x > 1 la nghiem
C a u 3 Trong mat phang toa dp Oxy cho dubng t h i n g (d) : 2x - y - a" = 0
va parabol (P) : y = ax^ (a la t h a m so' difcfng)
1. T i m a de (d) cat (P) t a i hai diem phan biet A, B Chiing m i n h r i n g k h i
do A, B n i m ve ben phai true tung
2 Gpi u, V theo thuf tif la hoanh dp ciia A, B T i m gia t r i nho nhat ciia
bieu thufc T = + —
U + V U V
C a u 4. DucJng t r o n t a m 0 co day cung A B co dinh va I la diem chinh giOfa
cung I6n AB Lay diem M ba't k i t r e n cung Idn A B , difng tia A x vuong
goc vdi ducmg t h i n g M I t a i H va c i t B M t a i C
1. ChiJng m i n h cac tam giac A I B va A C M l a cac t a m giac can
2. K h i diem M d i dpng t r e n cung \6n AB Chufng m i n h rang diem C di
chuyen t r e n mot cung tr6n co dinh
3 Xac dinh v i t r i ciia dig'm M de chu v i tam giac AMC dat gia t r i 16n nhat
C a u 5. Cho t a m giac vuong a A co A B < AC va trung tuyen A M , ACB = a,
A M B = p Chijfng m i n h r i n g : (sina + cosa)^ = 1 + sinp
Trang 28Xet ham so' f{x) = (3m - 4)x + 1 - 2m
Do t h i ciia ham so f(x) la mot ducfng thSng nen de bat phiicng t r i n h (1)
f 3 m - 4 > 0 dung voi moi x > 1 t h i : <=> m > 3
|f(l) = m - 3 > 0 Vay vdi moi m > 3 t h i phu'cfng t r i n h da cho nhan moi gia t r i x > 1 la
nghiem
C a u 3
1 Phucfng t r i n h hoanh do giao diem cua dudng thfing (d) va parabol (P)
CO dang : ax" - 2x + a^ = 0 (1)
Ducfng thang (d) cat (P) tai 2 diem phan biet A, B c=>
Goi u, V Ian liiort la hoanh do ciia A va B
Theo dinh l i Viet cho (1) ta c6 :
a > O c : > 0 < a <
A' = l - a 3 > 0
u + V = - > 0
a u.v = a > 0 Suy ra A, B nam ve ben phai true tung (dpcm)
2 TO ket qua cau 1) ta c6 : T = 2a + - > 2j2a - = 2V2
a V a Vay gia t r i nho nhat ciia T la 2V2 , dat difdc k h i a = —
Neu M thuoc cung I B (khong chiJa A) (nhu hinh 24b)
Chufng minh tUcfng tif
ta cung CO /\AMC can tai M
Theo ket qua cau 1) thi l A = IC, ma l A khong doi nen C nSm tren duong tron tam I
K h i M = A t h i C = A, con k h i M = B t h i C = C, (trong do C, la giao diem ciia dUo'ng tron tam I ban kinh l A voi duo'ng thang qua A vuong goc IB)
Do do k h i M chuyen dong tren cung 16'n AB t h i diem C chuyen dong
tren cung A C j (dpcm)
Gia siif 0,1 la giao diem ciia A I vdi dudng tron tam I ban kinh lA, con
Mil la diem doi xufng ciia A qua O
Nhan xet rang khi tia Ax triing vcJi tia A I t h i M triing vd'i M„, liic do
B, Ml), C» thang hang TiT moi lien he giOfa day cung va dUo'ng kinh ta
CO A M < AM,,, AC < ACo
Goi p la chu vi Ta c6 : p ( A M C ) < p (AM„C|,)
Vay gia t r i Idn nhat cua p ( A M C ) la p (AMiiC,), dat dUcJc k h i M la diem
xuyen tam doi ciia A doi vai dUo'ng tron (0)
Trang 29D E 15 TRl/dNG PTTH CHU VAN AN VA TRl/CfNG PTTH HA NOI - AMSTERDAM
De thi tuyen sinh vdo I6p 10 chuyen Todn, Tin nam hoc 2004 - 2005
Cau 1 ChiJng m i n h rang so t u nhien
A = 1.2.3 2003.2004 1 + — + — + + 1 1 ^
2 3 2003 2004 chia het cho 2005
Cau 2 Cho phuong t r i n h : x + 3(m - 3x'-)- = m
1. Giai phirang t r i n h vdi m = 2
2 T i m m de phirong t r i n h c6 nghiem
Cau 3 Giai bat phuang t r i n h : ^25x(2x^ + 9) > 4x + -
X
Cau 4 Cho t a m giac ABC c6 ba goc nhon, ke difdng cao BE, CF
1. Cho biet so do goc BAG hKng 60", t i n h do dai EF theo BG = a
2 Tren nilfa dUofng t r o n dudng k i n h BC khong chila E, F lay mot d i e m ' M
bat k i Goi H , I , K \An lugt la h i n h chieu vuong goc ciia M t r e n BC,
CE, EB T i m gia t r i nho nhat cila tong S = — + — + —
M H M I M K •
Cau 5 Cho mot da giac C.Q chu v i bang 1, churng m i n h rang c6 inot h i n h
t r o n ban k i n h r = - chufa toan bo da giac do
4
BAI GIAI Cau 1 Bien ddi : C = 1 + 1
Vi A = 2005k.B (vd'i B = 1.2.3 2004) ma k.B la so nguyen nen A chia
Tij' (1) va (2) suy r a de bat phifolig t r i n h c6 nghiem, dau dSng thiic
phai xay ra d bat dang thilc (2), liic do : 5x" = 2x2 + 9 => x = ^
Trang 30C a u 5 (Hinh 27)
Lay diem A tren bien ciia da giac
Lay diem B tren bien ciia da giac sao cho AB
cilia chu vi da giac thanh hai phan c6 dp dai
moi phan bang —
2
Goi O la trung diem ciia AB Gia suf M la mot
diem tiiy y thuoc bien hoac mien trong da giac,
lay M' doi xurng M qua O
Tii giac AMBM' la hinh binh hanh va AM + MB < -i
Hinh 27
Ma MM' < AM + MB MM' < - => OM < - nen M nam trong
dUong tron tam O ban kinh r = - (dpcm)
4
D E 1 6
TRl/dNG P T T H C H U V A N A N V A T R I / O N G P T T H H A NOI - A M S T E R D A M
De thi tuyen sinh vdo Idp 10 chuyen Khoa hoc TiJ nhien nam 2004 - 2005
Cau 1 Cho bieu thiJc P =
1. Rut gon P
Vx - 1 _ 4x +1
+1 - 1 2V^
2. Tim X de > 2
Cau 2 Cho phuong trinh : x" - (m - 2)x - m" + 3m - 4 ="0 (m la tham so)
1 Chufng minh phuong trinh c6 2 nghiem phan biet vd'i moi gia tri ciia m
2. Tim m de ti so giiJa hai nghiem ciia phaang trinh c6 gia tri tuy^t doi
bang 2
Cau 3 Tren mat phSng toa do cho dudng thSng (d) c6 phiTcfng trinh :
2kx + (k - l)y = 2 (k la tham so)
1 Vdi gia t r i nao ciia k thi ducyng thing (d) song song vdi du&ng thSng
y = xVs ? Khi do hay tinh goc tao hdi (d) v6i tia Ox
2. Tim kde khoang each tiT goc toa dp den diTong thang (d) la Idn nhat
Cau 4 Cho goc vuong xOy va hai di4'm A, B tren canh Ox (A nam giOra O
va B), diem M bat ki tren canh Oy Ducfng tron (T) dudng kinh AB cat
tia MA, MB Ian lucft tai diem thuT hai la C, E Tia OE cat dUo'ng tron
58
(T) tai diem thOr hai la F
1 Chufng minh 4 diem O, A, E, M nlim tren mot du&ng tron, xac dinh
tam ciia ducmg tron do
2. TiJ giac OCFM la hinh gi ? Tai sao ?
3. ChiJng minh he thuTc OE.OF + BE.BM = OB'
4. Xac dinh vi tri ciia diem M de tiJ giac OCFM la hinh binh hanh Tim moi quan he giCfa OA va AB de tuf giac la hinh thoi
=> a.c < 0 => phuong trinh da cho luon c6 2 nghiem phan biet trai da'u
2 Goi X i , X2 la hai nghiem ciia phuong trinh bac hai da cho
Theo dieu kien de bai ta c6 : Xt = -2x2 hoac x-, - -2x|
c:> ( X , + 2xv)(x2 + 2xi) = 0 CO x i X v + 2(x, + X2)" = 0 (*)
Theo dinh li Viet : X| + Xv = m - 2, x^Xv = - m " + 3m - 4
Thay vao ta dupe : m" - 5m + 4 = 0 <=> ni] = 1; m2 = 4
au 3
1 Vdi k = 1 thi phuong trinh dudng thSng (d) la : x = 1, (d) khong sonj
song vdi dudng thang y - 4Sx
Vdi k ^ 1 thi phuong trinh dUdng thang (d) cd dang :
Trang 31Dieu kien can va du de (d) song song vdi di/dng t h i n g y = VSx la :
- 2 k
Khi do goc nhon a tao bdi (d) vdi tia Ox c6 tga = A/3 => a = 60"
2.- Vdi k = 1 Khoang each tiT 0 den (d) la 1
- V6i k = 0 PhiToiig t r i n h difdng thSng (d) c6
dang y = - 2 Khoang each tii 0 den (d) la 2
- V6i k ^ 0 va k ?^ 1 Goi A B Ian ludt la giao
diem ciia (d) vdi Ox, Oy
Trong tam giae vuong A O B ta c6 :
Nen tu" giac O A E M ngi tiep
dudng tron dirdng k i n h M A
2 TiJ giac O A E M noi tiep
==> O M A = O E A Bon diem A , E , F , C
cung nam tren dirdng tron (T) nen O E A = A'CF
C
Hinh 28
Do do OMA = ACF => FC // C M Vay OCFM la hinh thang
3 AOFA c / ' A O B E (g.g) ^ = ^ ^ OE.OF = OA.OB ( l )
U B UxL
' A O B M CO AEBA (g.g) ^ ^ = ^ =^ BE.BM = BA.BO (2)
EB BA Tir (1) va (2) ta C O : OE.OF + BE.BM = OB'
4 Gpi giao diem ciia CM vdi OF la I va OF vdi OB la H
H i n h thang O C F M la hinh binh hanh <=> I la trung diem O F c:> A la
t r o n g t a m A C O F <=> OA = 2 A H T^ir do ta cd each dirng sau :
- Dung di§m H tren tia OA thoa man OA = 2AH
- Qua H diTng dudng thang vuong goc vdi OB cdt (T) t a i F va C
- Noi CA eat Ox t a i M t h i M la diem can dirng
Khi O C F M la hinh binh hanh, muon no la hinh thoi t h i CI 1 O F Suy ra A O C F deu, A la t a m cua t a m giac deu O C F nen CAT = 60"
AB
=^ OA = AC = AT = —
D E 17 TRUONG PTTH CHU VAN AN VA TRUCJNG PTTH HA NOI - AMSTERDAM
De thi tuyen sinh vdo Idp 10 chuyen Khoa hoc Tii nhien nam 2003 - 2004
3 T i m X de bieu thufc Q = nhan gia t r i la so nguyen
a u 2 Trong mat phang toa do Oxy, cho parabol (P) : y = - x ' v a dudng
thang (d) d i qua di§m 1(0; -1) cd he so' goc k
1. Viet phuong t r i n h dudng thang (d) ChUng minh vdi moi gia t r i k, (d)
luon cdt (P) t a i hai diem phan biet A v a B
2 Goi hoanh do ciia A v a B la x, v a x^., chUng minh | x, - x^; | > 2
3 ChUng minh tam giac OAB vuong
a u 3 Cho doan thang AB = 2a cd trung diem la O Tren cung nUa mat
phang bd A B dung nilfa dudng tron (0) dudng k i n h A B v a nUa dudng
61
Trang 32t r o n (O') diTdng k i n h AO T r e n (O') lay mot diem M (khac A va O), tia
OM cat (O) t a i C, goi D la giao d i e m thuf h a i ciia CA vdi (O)
1. ChiJng m i n h tam giac A D M can
2 Tiep tuyen t a i C ciia (O) cat tia OD t a i E, xac dinh vi t r i tirong doi ciia
dudng thang EA do'i voi (O) va (O')
3 Dircfng t h i n g A M cat OD tai H , difdng tron ngoai tiep tam giac COH cSt
(O) tai diem thuf hai la N Chufng minh ba diem A, M va N t h i n g hang
4 Tai vi tri ciia M sao cho ME // AB, hay tinh do dai doan t h i n g OM theo a
N h a n xet rang vdi x > 0, x ^ 1, ta c6 :
- 1 > 1 (Bat d i n g thilc Cosi)
Phuong.trinh hoanh do giao diem ciia (P) va (d) : x" + kx - 1 = 0 (*)
Vi ('•••) CO A = k - + 4 > 0 Vk nen (d) luon cat {P) t a i hai diem phan biet
Vi ( - x i ) ( - X 2 ) = - 1 , suy ra tam giac OAB vuong t a i O
Ta CO AOAC can t a i O c6 OD 1 AC, suy ra MOD = DOA
=> D M = AD hay A D A M can tai D
D l thay AAOE = A C O E (c.g.c)
Suy ra EXO = ECO = 90° hay EA 1 AB
Hay EA la tiep tuyen chung ciia difdng
t r o n (O) va du'dng t r o n (O')
Gia sif A M c i t difdng t r o n (O) t a i N', de y r i n g AOC = 2 A N ' C nen C^OH = CN^H dan den tuf giac CHON' noi tiep difoc trong mot du'dng
t r o n , tif do ta thay N ' = N , suy ra A, M , N t h i n g hang
De thi tuyen sinh v a o Idp 1 0 chuyen Toan, Tin n a m hoc 2 0 0 3 - 2 0 0 4
C a u 1. Cho hai so tif nhien a va b, Chufng m i n h r i n g neu a' + b ' chia het cho 3 t h i a va b cung chia het cho 3
(V' I 1
C a u 2. Cho phuong t r i n h : = m
1 Giai phuong t r i n h vdi m = 15
2. T i m m de phuong t r i n h c6 4 nghiem phan biet
C a u 3 Cho X , y la cac so nguyen difong thda man : x + y = 2003
Tim giii t r i nho nhat, Idn nhat cua bieu thufc P = x ( x ' + y) + y(y' +
Trang 33C a u 4 Cho duorng tron (O) vdi day BC co dinh (BC < 2R) va diem A t r e n
cung I(Jn BC (A khong trung v6i B, C va diem chinh giiJa ciia cung)
Goi H la h i n h chieu ciia A t r e n BC, E va F Ian lirgt la h i n h chieu ciia
B va C t r e n du'cfng k i n h AA'
1. Chifng m i n h rang H E vuong goc vdi AC
2. Chijfng m i n h tam giac H E F dong dang vdi tam giac ABC
3. K h i A di chuyg'n, chiJng minh t a m dudng tron ngoai tiep t a m giac H E F
co' dinh
C a u 5. Lay 4 diim a mien trong mot t i l giac de cung vdi 4 dinh ta dufOc 8
diem, trong do khong co 3 diem nao t h i n g hang Biet dien tich ciia tiJ
giac la 1, chiJfng m i n h rang ton t a i mot tam giac co 3 dinh lay tic 8
diem da cho co dien tich khong vugt qua ^ Tong quat hoa bai toan
cho n giac loi vdi n diem nam d mien trong ciia da giac do
B A I G I A I
C a u 1. Dat a = 3k + r vdi r = 0, 1, 2 t h i ta thay a^ chia het cho 3 da 0 hoac
du 1 Tirong tir b" cung chia het cho 3 dif 0 hoac dir 1
TCr do (a" + b') chia het cho 3 =:> a^ va b" deu chia het cho 3
Suy ra a va b cung chia het cho 3
x ^ x + i r
1 x(x + 1) x(x + 1) ' " x(x +- 15 = 0 (*) Dat y = l)
phan biet thoa man
= m CO 4 nghiem <=> y" + 2y - m = 0 co 2 nghiem
Thii l a i vdi m > 8 dung Vay m > 8 la gia t r i cdn t i m
C a u 3 Ta CO : P = (x + y ) ' - 3xy(x + y) + 2xy = 2003'' - 6007xy
Vi (x - y)'^ = (x + y)^ - 4xy = 2003^ - 4xy nen xy tang (giam) k h i I x - y | giam (tang)
Theo gia thiet x, y la so nguyen duong, ta nhan tha'y :
- Gia t r i nho nhat ciia P la 2003'' - 6007.1001.1002 dat duqrc k h i (x; y) bang (1001; 1002) hoac (1002; 1001)
- Gia t r i Idn nhat cua P la 2003'* - 6007.1.2002 dat difoc k h i (x; y) bang (1; 2002) hoac (2002; 1) ^
2. T i l giac A H F C noi tiep suy ra E F H = ACB (1)
T i l giac A B H E noi tiep suy ra H E F = ABC (2) TCr (1) va (2) suy ra AHEF ^ AABC
3, Goi M , N , P Ian lugt la trung diem ciia BC, A B , CA
Vi M N // AC, H E ± AC nen M N 1 H E , ma N each deu bon diem A, B
H , E suy ra M N la ducfng trung trifc ciia doan H E Chilng m i n h tifong
t u M P la dudng trung true ciia H F Vay M la t a m dUofng t r o n ngoai tiep A H E F => dpcm
C a u 5. N o i cac diem tao t h a n h cac tam giac doi mot chi chung nhieu nhat mot canh, phii vCra k i n t i l giac Tdng cac goc trong ciia t a m giac bang tong cac goc trong ciia t i l giac cong vdi 4 Ian 360*^ nen b^ng
360" + 4.360" = 10.180"
Vay CO 10 tam giac ma tong dien tich la 1 nen ton t a i t a m giac co dien
tich khong vifot qua ^
65
Trang 34Tong quat : CJ mien trong da giac loi n - canh (n > 3) c6 dien tich bang
1 lay n diem, trong do khong c6 3 diem nao than^ hang Klii do ton
tai mot tam giac c6 3 dinh lay tCr 2n dinh da cho c6 dien tich khong
vifOt qua — - —
• 3 n - 2
D E 19 DAI HOC OUOC CIA HA N O ' - TRUCfNG DAI HOC KHOA HOC TlJ NHIEN
De thi tuyen sinh vdo I6p 10 chuyen Todn, Tin nam hoc 2005 - 2006
C a u 1 Giai phUo'ng t r i n h : V2 - x + ^J2 + x + yji - x^ = 2
2. T i m gia t r i Idn nhat va nho nhat ciia bieu thufc P = V l + 2x + ^ 1 + 2y
C a u 4 Cho h i n h vuong ABCD va diem P nam trong tam giac ABC
1 Giai svi BPC = 135° Chufng minh rang 2PB' + PC^ = PA'
2. Cac durdng thang AP va CP cSt cac canh BC va BA tUcfng iJng tai cac
diem M va N Goi Q la diem doi xufng vdi B qua trung diem ciia doan
M N ChiJng minh rang k h i P thay doi trong tam giac ABC, dirdng
thang PQ luon di qua D
C a u 5
1 Cho da giac deu (H) c6 14 dinh ChiJng minh rang trong 6 dinh bat ki
ciia (H) luon c6 4 dinh la cac dinh ciia mot hinh thang
2. Co bao nhieu phan so' to'i gian — Idn hon 1 (m, n la cac so' nguyen
n duong) thoa man mn = 13860
(2) CO dang : 4x'' + y"* = (4x + y)(x'' + y'' - xy") (yi x'' + y'' - xy" = 1)
M a t khac (x + y ) ' = x' + y" + 2xy = 1 + 2xy > 1 (vi x~ + y ' = 1)
Suy ra x + y > 1 (2) Dau "=" xay ra <=> x = 0 hoac y = 0
T i ^ ( l ) v a ( 2 ) => 1 < x + y < V2 (dpcm)
2. P = V l + 2x + V l + 2y :=> P' = 2 + 2(x + y) + 2V1 + 2(x + y) + 4xy
Do X + y < >/2 va 4xy < 2(x- + y") = 2
Suy ra P < -^^ + 27^ + 273 + 2 ^ Vay gia t r i Idn nhat cua P la V2 + 2V2 + 2V3 + 2V2 , dat diicfc khi
1
X = y =
67
Trang 35M a t kh&c, do X + y > 1 va 4xy > 0, n e n > 2 + 2 + 2Vl + 2 + 0
=> P > V4T2V3
Vay gia tri nho nhat cua P la ^4 + 2A/3 , dat duac khi x = 0 hoac y = 0
Cau 4. (Hin/i 31a, 31b, 31c)
1. Lay diem P' khac phia v<Ji diem P
doi vdi dudng thang A B sao cho
ABPP' vuong can tai B
2. Tmdc het ta chiifng minh nhan xet sau :
Nhdn xet : Gia siif I la diem nam trong
hinh chut nhat ABCD Qua I ve cac M| > k |N
dirdng thing MN, PQ tiTong ling song
song v(Ji AB, AD Goi dien tich hinh
cho nhat IPBN la Sj, dien tich hinh
chO nhat IQDM la S2 Khi do S, = S2
khi va chi khi I thuoc difdng cheo AC
That vay : Gia sii I thuoc ducfng cheo AC Vi diTcfng cheo ciia hinh chu
nhat chia hinh chOr nhat thanh hai phin c6 dien tich bang nhau nen
I
Hinh 31b
xet cac hinh ABCD, APIM, INCQ ta c6 Si = S^
Ngucjc lai, gia siJr Si = S,, suy ra : IN.IP = IM.IQ IN IQ NC " MA
=^ AMAI CO ANIC ^ MIA = NIC
Do M, I , N thing hang nen A, I , C
thing hang
Trd lai bai toan
D i thay tuf giac NBMQ la hinh chOf
nhat.'Qua P va Q ve cac ducmg thing
song song vdi cac canh ciia hinh vuong
Do P thuoc dudng cheo AM ciia hinh
chuf nhat ABMR nen SBLJ.R = ^mm
Lay 6 dinh ciia (H) thi so day noi hai dinh trong 6 dinh la = 15
P thuoc duTorng cheo CN ciia h i n h chO n h a t NBCH nen SBLI'K = Sivim Vay SpiKs = Spi'HK => SFQKS = S(JITII
Theo chufng minh tren, suy ra Q thuoc dudng cheo PD ciia hinh ch& nhat SPTD Vay PQ di qua diem D
iu 5
1. Cac dinh ciia (H) chia ducrng tron ngoai tiep no thanh 14 cung bing nhau, moi cung c6 so' do la a = ^ Cac day no'i hai dinh ciia (H) chin
cac cung nho c6 so' do la a, 2a, 3a, 7a Do vay dp dai cac day do chi
nhan 7 gia t r i khac nhau
^6x5^
2 ,
Vi 15 day nay c6 dp dai nhan khong qua 7 gia t r i khac nhau nen phai
CO ba day ciing dp dai Trong ba day do luon c6 hai day khong chung
dau mut (vi neu hai day bat ki trong ba day do diu chung deu chung dau miit thi ba day bao nhieu do tao thanh mot tam giac deu, do do so dinh ciia (H) chia het cho 3, trai vdi gia thiet)
Dk thay hai day bing nhau ciia mot dudng tron khong chung diu miit thirc hien 4 diu miit ciia chiing la 4 diim ciia mot hinh thang (can) Tir
do suy ra trong 6 dinh bat ki ciia (H) luon c6 4 dinh la cac dinh ciia mot hinh thang
2. Ta CO : 13860 = 2.2.3.3.5.6.11
Vi m.n = 13680 nen m phai la Udc so ciia 13680 tiirc 1^ tich so ciia mot so' nhan tu trong 7 nhan tiJr tren, con n la tich ciia cac nhan tiJf con lai Neu m CO chiJa nhan tiJf 2 (hoac 3) thi no phai chufa 2' (hoac 3") vi nguac lai thi — khong to'i gian
n
Do do neu ki hieu ai = 2', = 3", a,) = 5, a^ = 7, aj = 11 thi m la tich
ciia mot so nhan tiJf trong cac so' ai, a-,, as, a^, as con n la tich cac nhan
tii con lai Vay ta c6 cac trUdng hop sau :
1 Co 1 phan so c6 tiif so la 1 (mlu so la 13680)
2 Co 5 phan so c6 tuf so' la mot trong nSm so' ai, a^, f ar, (miu la tich
ciia 4 nhan tuf con lai)
3 Co 10 phan so c6 tuf so la tich cua hai nhan tuf trong nam so a,, av,
a.r„ (miu so la tich ciia 3 n h a n tuf con lai)
4 Co 10 phan so c6 tuf so la tich ciia ba nhan tuf trong nam so tren (miu la tich ciia hai nhan tuf con lai)
69
Trang 365 Co 5 phan so c6 tiir so' la tich ciia bon nhan tiif trong nam so tren
con mku la so' con lai
6 Co 1 phan so co tuf so' la tich ciia ca nam so' tren ( m l u so' la 1)
Vay so' phan so' to'i gian — thoa man m.n = 13680 la :
n
1 + 5 + 10 + 10 + 5 + 1 = 32
Cac phan so tren dugc chia thanh ttfng cap nghich dao ciia nhau va
32 khac 1 nen so' phan so Ion hon 1 la — = 16
2
D E 20 DAI HOC OUdc GIA HA NOI - TRlTCfNC DAI HOC KHOA HOC TlJ NHIEN
De thi tuyen sinh vao I6p 10 chuyen Khoa hpc TL/ nhien nam 2 0 0 5 - 2 0 0 6
C a u 4 Cho hai dufcfng tron (O), (0') nam ngoai nhau c6 tam tuong iJng la O
va O' Mot tiep tuyen chung ngoai ciia hai duo'ng tron tiep xiic vd'i (O) tai
A va (0') tai B Mot tiep tuyen chung trong ciia hai duo'ng tron cat AB
tai I , tiep xuc vdi (0) tai C va (0') tai D Biet rang C nam giiia I va D
1 H a i duorng thang OC, O'B cat nhau tai M ChiJ-ng minh rang D M > O'M
2 K i hieu (S) la difdng tron di qua A, C, B va (S') la duo'ng tron di qua A,
D, B Ducfng thang CD cat (S) t a i E khac C va cat (S') tai F khac D
ChiJng minh rSng A F vuong goc BE
C a u 5 Gia sii x, y, z la cac so' duong thay doi va thoa man didu kien :
Vdi
S = 2
P = 1 (S = -4
ci> x = 1 (thoa dieu kien)
Vay phucfng t r i n h co nghiem la x = 1
C a u 3 Phuo'ng t r i n h da cho co dang ; x" + 17[y- + 2xy + 3(x + y)| = 1740
Nhan xet rang vdi so x nguyen, x cd the cd dang sau :
Trang 37De thi t u y e n sinh I6p 10 c h u y e n Toan, Tin n a m h o c 2004 - 2005
1 T i m ta't ca cac v i t r i ciia d i e m M sao cho M A B = M B C = M O D = M D A
2 X e t d i e m M n a m t r e n d u o n g cheo A C G o i N l a c h a n diJdng v u o n g goc
C a u 5 V d i so thU c a, t a d i n h n g h i a p h a n n g u y e n cua so' a l a so n g u y e n
I d n n h a t k h o n g v i r g t qua a v a k i h i e u l a [a| D a y cac so x,,, x , , X v ,
n + 1
V2 H o i t r o n g 200
x „ , duo'c xac d i n h b d i c o n g thufc x,, =
so' |xo, X ) , Xm;,| CO bao n h i e u so' k h a c 0 ? (cho b i e t 1,41 < V2 < 1,42)
Trang 38C a u 2 H e da cho tUong diTong v d i he :
T r t r (1) cho (2) ve theo ve t a dirac : (x + y ) ' ' = 2 7
De thi tuyen sinh v d o I6p 10 c h u y e n K h o a h o c \U nhien n a m 2004 - 2005
C a u 1
1 G i a i phuo'ng t r i n h : i x + 11
2 T i m n g h i e m nguyen ciia he :
| x - l l = 1 + | x ' - l | 2y^ - x^ - x y + 2 y - 2 x = 7
C a u 4 Cho tuf giac A B C D n o i tiep t r o n g dUong t r o n c6 h a i d u d n g cheo A C
va B D v u o n g goc vd'i lahau t a i H ( H k h o n g t r u n g vo'i t a m ciia dUong
t r o n ) G o i M v a N I a n lucft l a c h a n cac d u o n g v u o n g goc h a tij' H x u o n g cac d u o n g t h S n g A B v a B C , P v a Q I a n luoft l a giao d i e m ciia d u o n g
Trang 39b) x - y = : l r:=>x + 2y + 2 = -7=:>x + 2y = -9=>y khong nguyen
c) x - y = -7 =>x + 2y + 2 = l Giai he nay dagc nghiem (x; y) = (-5; 2)
khong thoa man (2)
d) x - y = 7 =>x + 2y + 2 = - l ^ x + 2y = -3=>y khong nguyen
Vay he da cho c6 duy nhat mot nghiem nguyen (x; y) = (1; 2)
Cau 2 T a c o : a"'^ + b'"^ = (a"" + b"" )(a + b) - ab(a"'% b'"")
Tuf gia thiet bai ra va dfing thufc tren suy ra :
1 = a + b - ab hay ( a - l ) ( b - l ) = 0 =^ (a; b) = (1; 1)
Do do P = 2
Cau 3 (Hinh 34)
Goi H, D, P Ian luot la chan cac dirdng cao, phan giac, trung tuyen ha
txi B xuo'ng canli AC
• Vi A A B C vuong nen de thay
SAUC = 6 (cm-), SAHP = Sen, = 3 (cm')
TCr do Sni)p = ScHi) - Scisi' = - (cnr)
Mat khac A A B H ^ A A C B ^ABH
SACB Suy ra SAHU = ^ (cm")
A D B = A C B = B H N = QHD Nen AHQD can tai Q => HQ = QD (2) TCr (1) va (2) suy ra Q la trung diem ciia A D TUong tir ta cung thay P la trung diem ciia CD
2
DE 23 DAI HOC OUOC GIA HA NOI - TRUCfNG DAI HOC KHOA HOC TLT NHIEN
De thi tuyen sinh vdo Idp 10 chuyen Khoa h o c Tii nhien nam 2003 - 2004
C a u l Giai phircfng trinh : (Vx + 5 - Vx + 2)(1 + A/X^ + 7X + 10) = 3
^ ^ , , [2x^+3x^y=5
<^au 2 Giai he phiTOng trinh : <^
[y^ + 6xy^ = 7
Cau 3 Tim cac so' nguyen x, y thoa man dflng thufc :
2y^x + x + y + l = x" + 2y^ + xy
77
Trang 40C a u 4 Cho niJfa dudng tron (O) dieu kien AB = 2R ( R la dp dai cho trirdc)
M , N la hai diem tren nua diTdng tron (O) sao cho M thuoc cung A N va
tong cac khoang each tU A, B den di^dng thang M N bang R V S
1. T i n h do dai doan M N theo R
2 Goi giao diem ciia hai day A N va B M la I , giao diem ciia cac dudng
thang A M va BN la K Chiing minh rang bon diem M , N , I , K c i m i r
nam tren mot dudng tron Tinh ban k i n h ciia dUong tron do theo R
3. T i m gia t r i Ion nhat ciia dien tich tarn giac KAB theo R k h i M , N thay
doi nhiftig van thoa man gia thiet ciia bai toan
Cau 5 Gia suf x, y, z la cac so' thirc thoa man dieu kien :
X + y + z + xy + yz + zx = 6 Chufng minh rang x' + y ' + z" > 3
B A I G I A I
Cau 1. Dieu kien :
X + 5 > 0
X + 2 > 0 x^ + 7x + 1 > 0
X > -2
De y rang x^ + 7x + 10 = (x + 5)(x + 2) Nhan hai ve cua phirang trinh
da cho v6i (Vx + 5 + Vx + 2) > 0 ta duo'c :
1 + Vx^ + 7x + 10 = Vx + 5 + Vx + 2
Vx + 5 = 1
Vx + 2 = 1 C5
(2x + yr = 27 y^ + 6xy^ = 7 ' (2)
(2): y = 1, y = • ^^^^ The vao (1) ta du'ofc he c6 3"'nghiem :
1. Dung AA' va BB' vuong goc vdi M N
Goi H la trung diem ciia M N , luc
do thi OH 1 M N Trong hinh thang AA'BB' CO ;
Do do ban k i n h dudng tron qua M , N , I , K la
3. Diem K nam tren cung chila goc 60" difng tren doan AB = 2R nen SKAH 16-n nhat dudug cao KP Idn nhat o AKAB deu, liic do