ĐỀ THI TUYỂN SINH LỚP 10 MÔN TOÁN CHUYÊN 3 NĂM GẦN NHẤT TỈNH THÁI NGUYÊN (có đáp án năm 20212022)

UBND TiNH THAI NGUYEN
SO GIAO DUC VA DAO TAO

BfeCHnNHIHCTC

K t Tffl TUYEN SINH LClP 10 NAM HOC 2021 - 2022
MON: TOAN
(Danh cho thl sinh thi chuyen Toan)
Thai gian lean bai: 180 phut, khong ke then gian giao de
(Be thi gom c6 01 trang)

f
x+5 ^
Can 1 (1,5 diem). Cho bilu thuc A = -jT-----------— : 1-----------W * + l 2 + yjx ~ x ) \
4-

. Rut gon va tinh

gia trj cua A khi x = yj29 + \2y[3-2y/2.
Cau 2 (1,5 diim ). Giai phuong trinh V lx + 3 + V lx+ T s + Ax -1 = 0.
Cau 3 (1,0 diim ). Cho / (x) la da thuc bac ba thoa man f ( x +1) - f(x ) = -x z, Vx e R
va f (0) = 2021. Xac dinh da thiic / (x), tuc do tinh t6ng T = l2 + 22 + 32 + ...+ 2 0 2 12.
Cau 4 (1,0 diim ). Tim n nguyen duemg d l bilu thurc A = 4«3+ 2n2 - In - 5 co gia tri la
s6 chinh phucmg.
Cau 5 (1,0 diem). Ban Nam du dinh chi dimg cac hinh vuong canh 2cm va 3cm de ghep
lai thanh mot hinh vuong canh 2021cm. Hoi ban Nam co thuc hien duoc ldiong ? Giai thich ?
Cau 6 (2,0 diem). Cho hai duong tron ( O) va (O') cit nhau tai hai diim A, B (O va O'
nlm vh hai phia d6i voi duemg thing AB). Tu diim C thuoc tia doi cua tia AB ve cac
tiep tuyln CD, CE voi duong tron (O), D, E la cac tilp diim va diim E nim trong
duong tron (O'). Duong thing AD, AE cit duong tron (O') lin luot tai cac diim M ,N

(M & A, N & A). Goi K la giao diim cua hai duong thing DE va MN. Chung minha) Tam giac CDA va tarn giac CBD ding dang, tir do suy ra ADBE = BD.AE;
b) K la trung diim cua doan thing MN.
Cau 7 (2,0 diem). Cho diim A c l dinh nim ngoai duong tron (O ). Ke cac tilp tuyln

AE, AF voi duong tron (O) ( E,F la cac tilp (film). Diim D di dong tren cung Ion EF
sao cho tarn giac DEF nhon. Tilp tuyln tai D cua duong tron (O) cit cac tia AE, AF lin
luot tai B, C. Goi M, N lan luot la giao diim cua duemg thing EF voi cac duong thing
OB, OC.
a) Chung minh bln diim B, N, M, C cung thuoc mot duong tron;
b) Goi DK, OI lin luot la duong phan giac cua cac goc EDF va BOC (K thuoc

EF, I thuoc BC). Chung minh duong thing IK luon di qua mot diim c l dinh.
HET
Ho va ten thf sinh:

.So bao danh.


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UBND TiNH THAI NGUYEN
S d GIAO DUC VA DAO TAO

KY THITUYEN SINH LOT 10 NAM HOC 2021- 2022
M 6N THI: TOAN
Danh cho chuyen Toan
HUONG DAN CHAM

( Ban hu&ng ddn chain gom co 06 trang)
L Hiring dan chung
- Giam khao can nlm vung yeu cdu cua hudng dan cham dl danh gia dung bM lam cua
thi sink. Thi sinh lam each khac dap an nlu dung vin cho dilm toi da.
- Khi van dung dap an va thang diem, giam khao can chu d6ng, linh hoat voi tinh than
tran trong bM lMn cua hoc sink
- N lu c6 viec chi tilt hoa dilm cac y cln phM dam bao khong sai lech vdi tong dilm va
durge thing nhlt trong toMi hoi dong cham thi.
- Diem toMi bM la tong diem cua cac cau hoi trong dl thi, chMn dilm le den 0,25 va
khong lMn tron.
H. Dap an va thang diem

2
<=>(V2x + 3 - l ) + (V |2 x + 1 8 -4 ) + (4 x + 4 ) = 0
„

2 ( * + l) , 2 -fX+1) + 4 (* + l ) - 0
-J2x + 3 + 1 V2j:+ 18 + 4
v
'
f
i
1
^
< = >2(x+ l) ■■
+ i--------+2 =0
v
'W 2 x + 3 f l v 2 x + 1 8 + 4
j

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x = -1 (t/m)
<=>

1
V 2x+3+l

— ------- + 2 = 0
V2 x+18 + 4

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Voi x > — thi yjlx- f 3 + 1 > 0 ,V 2 x + 1 8 + 4 > 0 nen

2

1

;

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1

------ + 2 > 0
>/2x+3 +1 V 2x+1 8 + 4
Vay phurcmg trinh co iighiem duy nhit x = -1 .

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Vi / ( x ) la da thuc bac ba nen

fix') = ax* + bx2 + cx + d {a *0; a, b ,c ,d <=R).
Taco: /( x + 1 ) = a (x + 1) + 6 ( x + 1) + c (x +1) + d.

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= a(x + 3x2 + 3x +1) + £>(x2 + 2x +1) + c (x +1) + d
/ ( x + l ) ~ / ( x ) = 3ox + (3a + 2&)x + or+ Z>+ c.

Thee bM ra: 3ax2 + (2 a + 2b)x + a + b + c = - x 2, V x e R .
a— I
°
3

'3a = - l

Nen ■3a + 2b = 0 <=> ■b = Can 3

.

2


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_

1
6
Mat]khac / ( 0 ) = 202 nen J = 2021.
V

Vay / ( x ) = - - x 3++ x2- —x+ 2021.
6
yw
3
2
Tir / ( x ) - / ( x + 1 ) = jc2 ta co:

/ ( 1 ) ~ / ( 2 ) = 12
/ ( 2 ) - / ( 3 ) = 22
/ ( 3 ) - / ( 4 ) = 32

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/ (2021) - / (2022) = 20212
D odo: l 2 + 2 2 + 32 + j .+20212 = / ( l) - /( 2 0 2 2 )
Thay s6 vao ta c6: T ==2 753 594 311

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3
Bat 4n3+2n2-7 n -5 = k2, k e N .
Ta co 4n3+ In2- 7 n -5 = (n + l)(4n2- In - 5) = k2.
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Dat LTCLN(ft+1,4n2- 2n- 5) = d e N*.
Taco ft+l:J=>(ft+l)(4ft-6):<f=>(4n2-2 n -6 ):c /. Talai co (4n2-2 n -5 ):d ,
suy ra (4ft2 - 2n - 5) - (4k2 - 2n - 6): d =^>1: d => d = 1. V&y

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DCLN(ft+l,4ft2-2 ft-5 ) = l .
C au4
Ma (n+l)(4ft2-2 ft-5 } = £2 ndn ft+l;4n2-2 ft-5 la cacsO chinhphuang
=>ft+l = a2 va 4ft2-2 ft-5 = 62;a,6eN *. Do n+1 la s6 chinhphtrong, neN*
nen n > 3.

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Voi « > 3 , taco (2ft-2)2 <4ft2 -2 n -5 <(2ft+l)2.
Do b le nen J2 = (2 n -l)2 0 4«2- 2 n - 5 = 4n2 - 4n+ 1 » 2« = 6 0 ft= 3 .
Voi ft= 3 ,ta c o J = 4«3+2«2 - 7 « - 5 = 102 la mot sd chinh phuong.

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Gia sue ta pM kin dugc hinh vuong canh 2021cm chi bod cac hinh vuong canh 2 cm
va3cm.
Xet mot hinh vuong canh 2021cm. Ta chia hinh vuong nay thanh cac hinh vuong
canh 1cm. Ta Id hieu cac hang t£rtren xu6ng dudi la hang 1,hang 2 ,... , hang 2021.
Ta t6 mau do cho cac hang co sd thu tu la s6 le (hang 1, hang 3 ,..., hang 2021).

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To mau xanh cho cac hang co sd thu tu la s6 chin (hang 2, hang 4 ,..., hang 2020).
Moi hinh vuong canh 1cm dirge to mau do ta ggi la 6 do.
Mdi hinh vu6ng canh 1cm duac to mau xanh ta goi la 6 xanh.
Vay so 6 do nhidu hon s6 6 xanh la 2021.
Can 5

(*)

V ci each to mau nhu tren ta thiy:
- Vdd mOi hinh vuong canh 2cm se phu kin 4 6 vuong canh 1cm, trong d6 luOn
cd 2 6 dd va 2 6 xanh.
- Voi mdi hinh vuong canh 3cm se_phu kin 9 6 vuong canh 1cm, trong do luon
c6 6 6 dd va 3 d xanh hoac 3 6 do va 6 6 xanh.
Do do hieu sd giua t6ng s6 6 dd vh tdng s6 6 xanh cua hinh vudng canh

2021cm dugc phu kin chi bod cac hinh vudng canh 2 cm, 3cm luon la mot sd chia
hit cho 3. Dieu nay mau thuin voi (*) vi 2021 khong chia h it cho 3.
Vay dilu gia su la sai. Tire la, ban Nam khong thl thuc hien dugc du dinh cua
minh.

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Cau 6

y / \

a) Xet ACDA va ACDB co: DCA chung, CDA = CBD.
Suy ra ACDA dong dang voi ACBD.
1
CD _ AD
|
^ CB~ BD’
j
(~ir* aJp
Tucmg tur, ACEA dong dang voi ACBE => — = — .
j
CB BE
Ah

AF

Ma CD = CE nen =+■ = — <=>AD.BE = BDAE ( l) .
BD
BE
w

i
b) Ta co BDE = BAE = BMN. Suy ra tur giac BDMK noi tiSp.
Do tur-giac BDMK noitiep =^BDM+BKM = 180°, ma BDM+M a = 180° (do
tur giac AEBD noi tiep)
BKM = BEA.

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0,25

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Mat khac, BMK = BAE. Do do, ABKM dong dang voi ABEA => — = ^ (2 ).
[
BK BE
!

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V i BAD = BNK,ADB = NKB => AABD ddngdang vdi ANBK=> — = — (3).
|
BD BK

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! KM KN
Tir (l),(2 ),(3 ) suy ra----- = — =>KM = KN.

BK

BK

Vay K la trung di&pa doan thang MN.

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/


5

a) Ta co OB,OC lan lugt la phan giac cua cac goc B va C cua tam giac
Cau7

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B C
ABC. D o do: NOB = m ~ B O C = —+
2 2

(>)•

M at khac, tam giac AEF can tai A nen AEF = —^180° - 2 j =

B +C
W-

T il ( l) ,(2 ) ta co AEN = NOB hayturgiac BENO noitiep.

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Suy ra ONB = OEB = 90° hay CNB = 90°.
C hiing minh tuong tur ta co BMC - 90°.

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V ay b6n dilm B, N, M, C cimg thuoc mOt ducmg tron.

f

^
b) Ta co EDF = 180°- EDB - FDC = 1 8 0 °- 9 0 ° - 2

r

B+C

9 0 °-—

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2

A
__ ^ _
1
_____ fi 4-C
+ 9 0 ° - — = 9 0 °+ ———

D o d o KDC = KDF+ FDC = —EDF + FDC =
_

2

4

(
va OIC = -+ B O I = - + - B O C = - + - 1 8 0 °-

2

2

2

2

2

B+C

90° +

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—

—

—

Tu do suy ra KDC = QIC nen OI I IDK .
G oi L,H lan lucrt la giao diSm cua AO v d i (O) va EF. Ta co A,L,0
th in g hang va L, D, K thin g hang.
V i OH IDK nen DOI = ODK.M& KLO = ODK nen DOI = KLH.
M at khac, ODI = LHK = 90° nen ta co ALHK ddng dang voi LODI_____

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6
L K LH
0 1 OD w
AOFA vuon gtai F, F H lO A ta i H nen OA.OH = OF2.
p
OF OH
OL _ OH
AL _ LH _ LH
Suy ra — = ----- =>
J
OA OF
OA OL ^ AO OL OD' ^ '
Tu (3 ),(4 ) ta co -£L = H L' K et hop v o i LK! 101 ta co A ,K ,I thing
hang hay IK luon di qua diem A co dinh.
—

h£ t—

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