I Do 66 khong c6 45 bi nao cung m^ụ Bai toan khong dung neu ta chi lay ra
c)Tii cau b) giup ta nghl den diem O, tam dúdng tron ngoai tiep tam giac
A B C va c6 IJ = OẠ
Bai 4: a) Dat 2a + b = x^ 2b + c = y^ 2c + a = z^ (vdi x, y, z 6 N ) Khong mát tinh tdng quat, gia sijr : 3.
Ta co: x^ + y^ + z' = 3(a + b + c) i 3. Ma i 3 nen x^ + y^ i 3. D a t x = 3t + r, y = 3h + m ( t , h e Z ; r , m e { 0 ; 1 ; - 1 } ) + ý = 9t' + 6u- + r^ + 9h' + 6hm + m^ = 3(3t^ + 2lr + 3h' + 2hm) + r^ + m^ : 3 Nen r ' + m^ : 3. Ma 0 < r^ + m^ < 2. Ta co: r^ + m^ = 0 <=> r = m = 0. V l vay X : 3, y : 3. Do do: 2a + b : 3 ; 2b + c : 3 ; 2c + a : 3. Ma 2a + b = 3a - (a - b ) ; 2b + c = 3b - (b - c ) ; 2c + a = 3c - (c - a) Nen a - b : 3 ; b - c : 3 ; c - a : 3 .
Vay tich (a - b)(b - c)(c - a) chia het cho 27. •'' b) a = 0, b = 1, c = 2. Ta co: 2a + b = 1, 2b + c = 4, 2c + a = 4 thoa man (*) va :
(a - b)(b - c)(c - a) = (-1 ) . ( - 1 ).2 = 2 khong chia het cho 27. Vay ton tai a = 0, b = 1, c = 2 thoa man dieu kien (*) sao cho:
Tny^ri giai as trUBc ki trngaóiop iu ea mien gae, Trung, Nam monTo.'.n muysn VM lan
(a - b)(b - c)(c - a) khong chia hét cho 27. Nhan xet: a = 0, b = k', c = 2k' (k e N, k ^ 3)
2a + b = k^ 2b + c = (2k)^ 2c + a = (2k)' thoa man dieu kien (*) v^: (a _ b)(b - c)(c - a) = (-k')(-k')(2k') = 2k'' khong chia het cho 27.
Nhir vay ton tai v6 so c^c bo so nguyen (a; b; c) thoa man dieu ki^n (*) sao cho (a - b)(b - c)(c - a) khong chia het cho 27. j <^
Bai 5: a) Chia hinh chiir nhat ABCD thanh 6 hinh chff nhat nho c6 kich thiTdc 1 x 2.
Vi 7 : 6 = 1 (du-1), do d6 ton tai 2 diem thuoc A 2 2 B mot hfnh chff nhat nho (nguyen t^c Di-rich-le)
Goi hai diem do la A", B'.
D
De MyWB' < + ¥ = S
Ta CO dieu phai chtfng minh.
b) Chia hinb chff nhat ABCD thanh 5 hinh nho nhi/hinh ve ben.
Vi 6 : 5 = l(dir 1), do d6 ton tai 2 diem thuoc mot hinh nho do (nguyen t^c Di-rich-le)
Gpi hai diem do la M, N, de thay MN < N/S .
Ta CO dieu phai chiJng minh. Nhan xet:
a) Con so Vs giup nghi den tam giic vuong c6 do d^i cdc canh g6c vuong la 1
va 2. Nguyen t^c Di-rich-le se giup c6 di/dc Idi giai va hien nhien phai nghi den chia hinh chff nhat ABCD thanh 6 hinh chff nhat nho c6 kich thiTdc 1x2. b) Chia hinh chiyehat ABCD thanh 5 hmh nhọ Phai that sir linh hoat, sdng tao
thi mdi CO diTdc Icfi giai d tren.
DE THI TUYEN SINH VAO LOP 10 CHUYEN,
TRJdNG THPT CHUYEN KHTN, DHKHTN, DHQG HA NOI NAM HQC 2011 - 2012
Bail:(3digm) •
(x - l)y^ + X + y = 3
(y-2)x^-(-y = X + 1
a) Giai he phiTdng trinh:
b) Giai phi/dng trinh: ^ X 2(x + l)l. . x ^ + 7
Cty IMÍM iVi! V i ) V \ " ! K!l,,,;,| VI
Bai 2: (3 diem) (adsbygoogle = window.adsbygoogle || []).push({});
a) Chrfng minh ring khong ton tai cAc bp ba s6' nguyen (x ; y ; z) thoa man ding thiJc: x" + ý = Iz" + 5. ding thiJc: x" + ý = Iz" + 5.
b) Tim tát ca cac cap só nguyen (x ; y) thoa man ding thtfc: ;/ (x + l ) ' - ( x - l ) ' = y^
Bai 3: (3 diem) Cho hinh binh hanh ABCD vdi BAD < 90". DiTdng phan giac
cua gdc BCD cit diTdng Iron ngoai tiep tam gidc BCD tai O khac C. Ke dirdng thing (d) di qua A va vuong goc vdi CỌ DiTdng thing (d) lan lirpt cat cac dirdng thing CB. CD tai E, F.
a) Chii-ng minh rSng: AOBE = AODC.
b) ChiJng minh rang O la tam diTdng tron ngoai tiep tam giac CEF.
c) Gpi giao diem cua OC va BD m I, chufng minh r^ng: IB.BẸEI = ID.DF.FI
Bai 4: (1 diem) Vdi x, y la nhuTng so thuTc difdng, tim gia tri nho nhat ciia bieu iMc:
P =
V x^+8ý ^y-^+(x + y)^" 4y^
HI/(5NG DAN GIAI Bai 1: Bai 1: a) ( x - l) y 2+ x + y = 3 f(x - Oy^ + (x - l) = 2 - y 'i-i \ <=> (y - 2)x2 + y - X + 1 [(y - 2)x^ + (y - 2) = x - 1 .. +:x ỵ 2 - y ( x - l) ( y 2 + l) = 2 - y ( y - 2) ( x 2 + l ) = x - l X - 1 = y^ + 1 -1 = 1 ^ <=> { ( y - 2 ) y^ +1 y + 1 2 - y y^ + 1 o = 0 x - 1 = X - 1 = 0 o:j '6 y - 2 = 0 [y = 2 fx = l
i _ 2' phiTdng trinh c6 nghiem duy nhat (x ; y) = (1 ; 2)
b) Dieu kien: x > 0.
T A I 3 x' + 7
Tacd: J x + - = — ^
X 2(x + l)
f 1 C:>2 1 + - C:>2 1 + - V X 3 7 3 ^ 3 4 2 3 ^ x + — = x + — <=>x+ 2/ X + —+ , ; x + — = 0 X X x V x x x V x X + — ( / 3 . X + - - 2 2 / / 3 ] + — 2 - . X + - IV X > X V X j = 0 o / 1 N / f / 3 2 . x + - - 2 X + - IV X 0 X xj = 0 o x + - - 2 <=> 3 2 X + — = — X X X + — = 4 X 3 4 x + - = — X X^ x' + 3 = 4x x' + 3x = 4 (x - l ) ( x ' + X + ( x - l ) ( x- 3 ) = 0 • - • 4) = 0
PhiTdng trinh c6 hai n g h i e m la X| = 1 , X2 = 3.
Nhan xet:
( x - l ) ( y ^ + l ) - 2 - y
X = 1 (thich hdp)
X = 3 (thich hdp)"
- ,4