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DE THI CHON DOl TUYEN HQC SINH GIOI LOP 9. Duong tron nQi ti~p tam giac ABC tiep xuc voi cac canh AB, BC, CA Iftn hrot tai cac diem D, E, F.. a) Tinh dQ dai doan AD.[r]
(1)DE THI CHON DOl TUYEN HQC SINH GIOI LOP 9
. MON THI: ToAN
Ngay thi: Thir Bay 04/0112020
Thoi gian lam bai: 120 phut (khong kJ phdt oJ) NAM HQC: 2019-2020
Biti 1.(1,5 aidm)
Cho a, b,c 1(\ba s5 khac d6i mot Chung minh d~ng thirc :
2
_1_-1- 1 + 1 =[_1_+_1_+_1_]
(a-b)2 '(b-C)2 (c-a)2 a-b b-c c-a
Biti 2. (1,5 aidm)
Giai phirong trinh: )x +1 =x2 - 6x +5
Bili 3. (2diim)
a) Cho x,Y :;c O Chung minh r~ng: .£+< - 3(x +1.]+4 >0
y2 X Y x
b) Cho a, b 1(\hai s5 dirong thoa di~u kien a+b :::;1.Clnrng minh:
3b2 +~+2 > 8b
4a
-Bai (4diim)
Cho tam giac ABC co AB = 5, A C = 7, BC = Duong tron nQi ti~p tam giac ABC tiep xuc voi cac canh AB, BC, CA Iftn hrot tai cac diem D, E, F.
a) TinhdQ dai doan AD.
b) GQiM lit giao diem cua DE va duong thang qua A song song voi BC ;N lit giao diSm cua hai duong thang MF va BC. Chung minh r~ng C Ia trung diem cua EN
c) GQiP litgiao diem cua DE va BF. Tinh ti 86 PE
PM
Bai (1diim)
Haybieu di~n 86 2019 t6ng cua cac 86 nr nhien lien ti~p
(2)DApAN
Bai (1,5 aidm)
Cho a, b, c hi ba s6 khac d6i mot Chung minh d~ng thirc :
1 1 [1 1 1]2( b l' b : kha h d"· ")
-+ 2+ 2= + + a.o,c a aso acn au oi mot
(a-b)2 (b-c) (c-a) a-b b-c c-a
Gilii
[_1_+_1_+_1_]2a-b b-c c-a
= + + +2[ + 1 + 1 ] f:,s-ct
(a-b)2 (b-C)2 (c-a)2 (a-b)(b-c) (b-c)(c-a) (c-a)(a-b)
= + + +2[c - a+a - b+b - c] 01\'
cI-(a-b)2 (b-C)2 (c-a)2 (a-b)(b-c)(c-a)
= + + C I~*
(a-b)2 (b-C)2 (c-a)2
Bili (1,5 iliJm)
Giai phirong trinh : .Jx +1 =x2 - 6x +5 Gilii
.Jx+i = x2 - 6x + <=> x +1+ ~x +1+I_= x2 _ 5x + 25
4 4
t=:': .Jx+i+~=x-~ .Jx+l=x-3 <=> (\I x +1+_)2 =(x )2 {:} 2 {:}
2 2.J x +1+ ~=-x + ~ ~x +1 =-x +2
2
t=:': {x 2>: 7+m
* \Ix -t-1 =X - {:} {:} X =
-x -7 -x +8 =
(O,5d)
(O,5d)
r-:-: {X~2 5-~
* \IX +1= -x +2 {:} {:} X =
-x - -x +3 =°
Bai (2 iliJm)
Cho X,Y:;%: o. Chung minh r~ng: x2 + y2 - 3[X + y] +4 >0
y2 x2 y X
(O,5d)
a)
Gilii
£+i_3[x +1L]+4 =[~+1L]2 _3[x +JL]+2 =[~+JL_l][X +1t.-2] (O,5d)
(3)·
x2 +y2 -xy X2 +y2 -2xy [X2+ +(X-Y?](X-y)2
= . = ? >O,Vx,y 7:° (0,5<1)
~ ~ ~-y
b) Cho a, b la hai s6dirong thoadi€ kien a+b< Chung minh: 3b2 +!!+2 ~ 8b
Giili
3b2 + ~ +2 > 8b ¢:> 3b+ ~ + ~8
4a - 4a
Ap dung b~t ddng thirc Cosi cho 6duong, ta co:
J'_+5a >2~ .5a =5
4a - 4a
~+8b>2J~ 8b =
b - b
-5(a +b) ~ -5
Cong ba b~t ddng thirc lai, ta co di u phai chirng minh
(O,25d)
(O,25d)
(O,25d)
Bai (4tJidm)
Cho tam giac ABC co AB = 5, A
ti~p xuc voi cac canh AB, BC, C
(O,25d)
Giai
=7, BC = Duong trim n<)iti~p tam giac ABC
IAn hrot taicac di~m D, E, F.
al Tinh d<)dai doan AD.
Chiungmm hAD =AF = -+_AB+AC- C
2 (ld)
blChung minh rang C 1fttrung diem cual EN
Ta co : fl.ADM rv fl.BDE ::::}An = AM ::::}AM = AF
b.AMF r-.J b.CNF =} CF =CN =} CEI= CN =}dpcm
(O,75d)
(4)- ,
/ G PI' . d·~ DE 'BF T' h : PE
e 01 a giao rem eua va III ti SO -
. PM
AD = 2=:;.BD = BE = 3 va CF = CE = G9i Q la giao diem cua BF va AM.
tlAQF rv tlCBF =:; AQ = AF =:; AQ = 16 =:; MQ = 26
CB CF 5
Taco: PE = BE =~
PM MQ 26
(0,25d)
(ld)
(0,25d)
Bili 5.(1di~m)
Hay bieu di~n s6 2019 t6ng cua cac s6tvnhien lien tiep Giai
G9i n s6c6t6ngb~ng 2019 lit: k+1,k+2, ,k+n. "n~2)
=:;.(k +1)+(k +2)+ +(k +n) =2019
=:;.nk +1+2+ +n =2019
=:;.nk +n(n +1) =2019
=:;.n(2k +n +1)=2.3.673
(0,25d)
(0,25d)
{n = {n = {n =
=:; 2k +n +1= 3.673 v 2k +n +1= 2.673 v 2k +n +1= 673
{
n =2 {n =3 {n =6 =:; k = 1008 V k = 671 V k = 333
(0,25d)