. Dovay dÚcJng tron (K) tiep xuc trong vdi dtfdng
c)Cho biet O M= 2R vaN la mpl diem bat ki thuoc cung EF chtfa diem Icua
dirdng iron (O; R) (N khac E, F). Goi d la diTcJng thing qua F va vuong goc
dirdng thing EN lai diem P, d c l l dirdng Iron dirdng kinh OM lai diem K (K
khac F). Hai dirdng thing FN va KE c l l nhau lai diem Q. Chufng minh rang: PN.PK + QN.QK <
Bai 5: (1 diem) Giai phiTdng Irinh: x** - x' + x' - x S x"^ - x + 1 = 0
HUCfNG DAN GIAI • i
Bai l : a ) A = n ' + 1 In = n' - n + 12n = n(n + l ) ( n - 1) + 12n
n - 1, n, n + 1 la ba so nguyen lien liep nen c6 mQl so chia het cho 2, mpl so chia hél cho 3. Ma lJCLN(2; 3) = 1 ; (2; 6) = 6 t - m o M 0 J chia hél cho 3. Ma lJCLN(2; 3) = 1 ; (2; 6) = 6 t - m o M 0 J
Nen n(n + l)(n - 1 ) 1 6 . Mat khac 12n ! 6 ... i „ ,
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Luy§n gi^i dS truest KI t t - i ..-an l i ' tíi " ' i ' ^ " ""í. ' ' " ' " J ^'^^^ ToAn _ NguySn BCfc T^n
b) Cdch 1:
, • Neu n = 0 thi B = 1, khong m so nguyen tọ • Neu n = 1 thi B = - 1 , khong la so nguyen tọ • Néu n = 2 thi B = 5, la so nguyen tọ
• Neu n > 3, ta c6:
B = n^ - 3n' + 1 = (n' - 1)^ - n' - (n' - 1 + n)(n^ - 1 - n) Vdi n > 3 ta C O n - 1 + n > 1, n - 1 - n = n(n - 1) - 1 > 1 Do vay B la hcJp sọ
Vay chi c6 n = 2 thi B = n" - 3n^ + 1 = 5 la so nguyen tọ
Cdch 2: B = n^ - 3n^ + 1 = (n^ - 1)^ - n^ = (n' - 1 + n)(n^ - 1 - n) • Neun = OthiB = 1.
• N e u n > 1 t h i n ^ - I >0. D o v a y n ^ - 1 + n > 0
B la so nguyen to nen n^ - 1 - n = 1 o n' - n - 2 = 0 o n = - 1 (loai), n = 2. • n = 2, ta c6: B = 2"* - 3.2^ + 1 = 5 la so nguyen tọ • n = 2, ta c6: B = 2"* - 3.2^ + 1 = 5 la so nguyen tọ
Vay chi c6 n = 2 thi B = n" - 3n^ + 1 = 5 la so nguyen tọ r
Nhan xet:
a) De nhan ra A = n^ + 1 In = (n^ - n) + 12n. b) B = (n' - 1 + n)(n^ - 1 - n) b) B = (n' - 1 + n)(n^ - 1 - n)
B a i 2:a)a = m ' + 2m + 2 = m ' + 2n + 1 + 1 = (m + 1)U 1 > 0 ; c = - l < 0
a va c trai dáu nen phiTdng trinh c6 hai nghiem phan biet X i , X2.
- 2m + 2
^.^j. Theo h? thurc Vi-et, ta c6:
.2 , .,2 X, + X2 = X, + X2 = m^ + 2m + 2 -1 ^^^^ m^ + 2m + 2 Do do: x f + X2 = 2x1X2(2x1X2 - 1) o X? + X j = 4 x ^ X j - 2 x , X 2 O (X, + X2)^ = (2X|X2)^ O m^ - 2m + 2 = -2 m^ - 2m + 2 = 2 ^m^ - 2 m + 2^ m^ + 2m + 2. m^ - 2m + 4 = 0 m^ - 2m = 0 -2 m"^ + 2m + 2 j (m^ - 2m + 1) + 3 = 0 m(m - 2 ) = 0 (m - 1)^ + 3 - 0 m = 0 hoSc m - 2 - 0 o m = 0 hoac m = 2. b) Cdch 1: S = m - 2m + 2
m^ + 2m + 2 . Bieu thtfc S nhan gii tri t khi v^ chi khi phi/cfng
^ , ^ , , •. m^ — 2m + 2
trinh an m sau co nghiem: t = — (*)
m + 2m + 2 . . . -j
Vim^ + 2m + 2 = (m +1)^+1 . ^ ' (adsbygoogle = window.adsbygoogle || []).push({});
Nen (*) <=> tm^ + 2tm + 2t = m^ - 2m + 2
c:>(t- l)m^ + 2 ( t + l ) m + 2 ( t -1 ) = 0(**) • ,
• N e u t = 1 thi(**)<=>4m = 0 o m = 0.
• Neu 17i 1 thi (**) c6 nghiem o Á = (t + 1)^ - 2(t - 1)
[(t + 1) + 72 (t - l)][(t + 1) - ^ ( t - 1)] > 0
o [(1 + V2)t + 1 - V 2] [ ( l - V2)t + 1 + V2] > 0 "
J V2 + 1
^-f= ^ t < ^ = 0 3 - 2 N y 2 < t< 3 + 272 i<..
V2 + 1 V2 - 1
• Vdi t = 3 - 2^y2 thi (**) CO nghiem m m = ^1^= 4l
t - 1
• Vdi t = 3 + 2 V2 thi (**) CO nghiem la m = = f >.
Vay: Gid tri nho nhat cua S la 3 - 2 %/2 0 m = >y2 • 1 v, ' (. i - Gia trj Idn nhát cua S la 3 + 2 o m = - V ^
Cdch 2:
5 ^ m^ - 2 m + 2^ (3 - 2V2)(m^ + 2m + 2) + {l^J - 2)W - 2V2n-
m^ + 2m + 2 m^ + 2m + 2 (m + i r + i
Dau " = " xay ra o m - >^ = 0 o m =
Vay gia tri nho nhat cua S la 3 - 2 72 <=> m = 72
- 2 m + 2 _ (3 + 2 ^ ) ( m ^ + 2m + 2) - (2 + 2 7 2) ( m 2 + 2v^m + 2)
S =
m + 2m + 2 m + 2m + 2 (m + 0+ 1
Dau " = " x a y r a o m + N / ^ = 0 o m = - 7 2
Vay gia tri Idn nhát cuaSla 3 + 2 7 2 o m = - 7 2
Nhan x e t : *
a) Taco: xj^ + \\ 2xiX2(2x,X2 - l ) o xf + X2 + 2x1X2 = 4 x f x 2
<=> ( X i + X2)' = (2x1X2)^
Luy^n 'ii Trung, Nam mfln Join _ Nguyln Dtfc Jin
V a n dung he thuTc V i - e t , giiip lim di/cJc m = 0 hoftc m = 2.
, . , „ - 2m + 2
b) B i e u thuTc S = — -. Do vay vi^c t i m gid t r i nho nhat, gia tri Idn m^ + 2 m + 2 (adsbygoogle = window.adsbygoogle || []).push({});
nhat la quen thuoc.
B a i 3: a) A p dung bat dang thiJc Co-si cho hai so difdng, ta c6: (ấ"" + 9) ấ"" + 10 - H l > 2 V ( a ' ' " " + 9 ) . l : ^ a ^ " ' " + 1 0 > 2 V a ^ " ' " + 9 ' " - \: iiiO >2 (\ «^ <-. ... i\ Vấ"" + 9
Dau " = " xay ra <=> á"'" + 9 = 1 o a = 0 . Do vay, dau " = " khong xay rạ Vay , > 2 • 7 a ^ ' " " + 9 b) y ' - x(x - 2)(x' - 2x + 2) = 0 y ' - ( x ' - 2x)(x' - 2x + 2) = 0 y ' - ( x ' - 2x + 1 - l ) ( x ' - 2x + 1 + 1) = 0 y ' - (x^ - 2x + 1)^ ' ( x - 0' 0 ( X - 1) - y - = 1 <::^ ( x - 1 ) - | y = I V l (X - 1)^ + |y > 0, 1 > 0 ncn (X - 1)^ - |yl > 0 , , D o vay chi xay ra: (x - 1 ) ' + y = 1
o • (x - 1 ) ^ = 1 ( x - 1 ) ' - y = 1 . y = 0 ( x - 1 ) ' - y = 1 . y = 0 f x - l = = 1 hoSc x - 1 = - 1 X - 2,y = 0
y - 0 [ x = 0,y = 0 '
Cdc so nguyen x, y thoa man phiTdng trinh y^ - x(x - 2)(x^ - 2x + 2) = 0 la:
X = 2 ; y = 0 hoac X = 0 ; y = 0. N h a n x e t :
a) V I ấ"" + 2010 = (ấ"" + 2009) + 1 giup nghl den van dung bat dang thiJc ^. Co-si cho hai so dUdng. . , \ ^. Co-si cho hai so dUdng. . , \