Do do 8 ip . N c n p = 2. K h i do p(p' - 6p' + p) = 2(2' - 6.2' + 2) = -28 ^ 8
Do vay (*) khong xay ra vdi moi p nguyen tó. Dieu gia suT tren saị . Vay n = p ' khong phai la so dieu hoạ
c) n = p.q la so dieu hoạ p va q la cac so nguyen to khac nhaụ /, ,
Do do (pq + 3)^ = + p ' + q^ + (p.q)^ <=> 4(pq + 2) = (p - q)\
Ta CO (p - q)^: 4. Nen p - q : 2. Do vay n + 2 = pq + 2 = ^ p - q ^ ^ la so
chinh phming
Nhan xet: Bai toan nay .khong kho doi vdti hoc sinh gioi loan \dp 9. L d i giai
cau b that dep, sang tiiọ
C a u I I I . a) D K X D : x > 1. Dat y = 7x - 1 (y > 0)
Ta CO (x^ - 5x + 4) + 2Vx - 1 = (x - 1 )(x - 4) + 2Vx - 1 ;
• = y^(y^ - 3) + 2y = y"* - 3y^ + 2y = y(y + 2)(y - 1)^ > 0 , vdi moi y > 0.
Vay cac gia tri x e R can tim la x > 1.
b) Theo cau a), ta co, neu y > 0 thi y"* - 3y^ + 2y > 0.
Do vc)y á - 3a + 2Va > 0; b ' - 3b + 2Vb > 0 ; c^ - 3c + 2Vc > 0 . Nen (â + b^ + c^ + 2ab + 2bc + 2ca) - 3(a + b + c) + 2(>/a + Vb + V c )
> 2(ab + be + ca)
<=> (a + b + c)^ - 3(a + b + c) + 2(%/a + Vb + V c ) > 2(ab + be + ca) Ma a + b + c = 3 (gt). Ta eo 2(Va + Vb + V c ) > 2(ab + be + ea) Do vay Va + Vb + Vc > ab + be + ea
Nhan xet: Bai toan nay rat hay v i tii cau a giup c6 du"dc Icfi giai cua cau b,
cac ban hay giai cau b bKng cdc each khac nffa nhe!
C a u I V . a) Ve O H 1 A B tai H . Ta eo B D 1 A B , CA 1 A B , O H 1 A B
=> B D // CA // O H . O la trung diem CD. Do do O H la difdng trung binh cua
hinh thang A B C D ( B D // AC) = Ma A C + B D < CD. Do do O H <
A C + BD
Ta CO d < R. Nen dudng t h i n g A B va diTcfng tron (O) du^dng kinh C D c^t nhau tai hai diem phan biet M , N .
Ta CO O H la dúcfng trung trifc cua doan
Cty TNHH MTV DWH Khang Vi§t
thang A B => O A = OB.
AC // B D => A C D + BDC = 180" .
Khong mat tinh tong quat AACO c6 A C O > 90" => OA > OC => A nam ngoai dirdng tron (O). Ta CO OB = OA > OC
=> B rikm ngoai dudng tron (O). Do do M , N nam tren canh A B .
C M D = 90" , C N D = 90" (Goc noi tiep chan nuTa diTdng tron).
Ta CO dieu phai chiJug minh.
b) Goi L la giao diem cua A E v^ M C , F la giao diem cua M D va BẸ
Ta CO A E // M D (gt), C M 1 M D ( C M D = 90")
A E 1 M C =^ A L M = M L E = 90" . . V VV v A t i V V /^
TMdng tur CO BFD = E F M = 90".
Xet A B M D va A A C M c6 M B D = C A M (= 90"), B M D = A C M (hai goc cung phu vdi goc A M C )
M D BD Do do A B M D A A C M (g.g)