MC vdi (O) (C la tiep diem) Ke CH vuong goc vdi AB (He AB), MB cat
fTỉ giac ABOC CO:
A B O + A C O = 90" + 90" = 180" nen noi tiep di/dc trong dúdng iron. nen noi tiep di/dc trong dúdng iron. b) Xet A I C K va A I B C c6 C I K (ehung),
Cty TNHH MTV D W H Khang Vigt
I C K = I B C (He qua goc tao bdi tia tiep tuyen va day cung) Do do A I C K ^ A I B C (g.g) i | = | . V a y C - , K . . B c) A B , A C la cae tiep tuyén eua dúdng trdn (O) (gt)
=> A B = A C (tinh chat eua hai tic'p tuyén eat nhau) => A A B C can tai Ạ Ma B A C = 60" (gt) do d6 A A B C deu => A B C = B C A = 60"
Ta CO : B D C = A B C (He qua goc tao bdi tia tiep tuyén va day cung)
Nen B D C = 60". M a t khae D B C = BCA (so le trong va B D // A C ) Nen D B C = 6 0 " ; A D B C eo B D C = 60", D B C = 60" ^ A D B C deụ
Ta CO A B = AC, OB = OC (= R), DB = DC (ADBC deu)
=> A, O, D cung thuoe dúdng trung tri/c cua BC. Vay A, O, D thang hang. NhSn x e t : Cae cau 1), 2) la rat quen thugc, eau 3) do viee phat hien tarn giac ABC deu, B D C ^ A B C , D B C = B C A giup c6 diTdc Idi giai nhanh chong. Bai 5: Ta c6 : x, y, z e [ - 1 ; 3 ] . Do do - 1 < x < 3 , - 1 < y < 3, - 1 < z < 3
= ^ ( x + l ) ( y + l ) ( z + l ) > O v a (3 - x)(3 - y)(3 - z) > 0 => xyz + xy + yz + zx + x + y + z + l > 0
Va 27 + 3xy + 3yz + 3zx - 9x - 9y - 9z - xyz > 0 => 2xy + 2yz + 2zx > - 2
=> x^ + y^ + 7? + 2xy + 2yz + 2zx > x^ + y^ + z^ - 2 => (x + y + z)^ > x^ + y^ + z^ - 2
= > 3 ^ > x ' + y= + z ^ - 2 j ' ' ' ' => x^ + y^ + z^ < 11
N h a n x e t : B i quyét de giai dang bai toan nay la khai thac dieu k i c n :
X , y, z e [ - 1 ; 3] de tiT do c6 (x + l ) ( y + l)(z + 1) + (3 - x)(3 - y)(3 - z) > 0
giiip CO diTdc Idi giai bai toan.
„ v •, f X , y, z G [ m ; n
B a i t o a n t 6 n g q u a t : Cho x, y, z thoa man : <
[x + y + z = p (vdi m, n, p thoa man 2m + n < p < 2n + m)
ChiJng minh rhng : x^ + y^ + z^ < (m + n - p) H m^ + n^ C6 the giai cae edch khdc nhiT sau :
Cach 1: V a i tro x, y, z nhiT nhau, khong mat tinh tong q u i t , giS sur x > y > z. T a c 6 : 3 x > x + y + z = 3 = > x > l n e n l < x < 3 = > ( x - I ) ( 3 - x ) > 0 => 3 x - x ^ - 3 + x > 0 = > x ^ - 4 x + 3 < 0 , . . ,^ , i;; t K . ; . :
Luygn giai ai truflc kl thi vAo Iflp 10 ba miSn BJc, Trung, Nam mOn ToAn _ Nguygn Ditc Ta"n Do do : + + < + + + 2(y + l)(z + 1) = x ' + (y + z)' + 2(y + z) + 2 = x^ + (3 - x)^ + 2(3 - x) + 2 . , , , , = x^ + 9 - 6x + x^ + 6 - 2x + 2 = 2x^ - 8x + 17 = 2(x^ - 4x + 3) + 11 < 11 Cach 2: Dat x = a + 1 ; y = b + 1 ; z = c + 1 X , y, z e [ - 1 ; 3] => a, b, c e [-2 ; 2] => â < 4
X + y + z = 3 => a + b + c = 0 => a, b, c CO hai so dong thdi khong am hoac dong thcfi khong diTring. Khong mat tinh tong quat gia suf la b, c. Ta c6 be > 0.
Do do : x^ + y^ + = (a + 1)' + (b + 1)' + (c + 1)' = â + b^ + + 2(a + b + c) + 3
= â + b ' + c H 3 < â + b^ + + 3 + 2bc = â + (b + c)^ + 3 ; = á + (-a)' + 3 = 2á + 3 < 2.4 + 3 = 11
Bai nay con c6 cac each giai khae nffa, cac ban hoe sinh hay tiep tue nhẹ
OE THI TUYEN SINH LdP 10 THPT, TJNH HAI Dl/dNG NAM HOC 2011 - 2012
Bai 1: (3 diem)