Loveable

Hai o to khoi hanh cung luc tu hai dia diem A va B, cach nhau 20km, chuyen dong thang deu cung chieu, theo huong AB, voi van toc lan luot la 40km/h va 30km/h.

1. Chon A lam goc toa do, chieu duong tu A den B, goc thoi gian la luc hai xe xuat phat; lap phuong trinh chuyen dong cua hai oto.

2. Xac dinh thoi diem va vi tri hai xe duoi kip nhau.

3. Ve do thi toa do theo thoi gian cua hai xe tren cung he truc toa do x0t

4. Tim khoang cach giua hai xe sau khoang thoi gian 1h30min, ke tu luc gap nhau

E o Lam Dong, sau khj tot nghjep THCS va thj tuyen sjnh, e da duoc tuyen vao tru0ng gjoj nhat huyen, nhung vj hoan canh, em phai xuong Thanh pho HCM de hoc. Nhung e k pjt lam ho so nhu the nao de duoc nhap hoc o Thah pho. M0g anh chj cho y kjen a

Cho hai duong tron (O1) va (O2) co ban kinh bang nhau va cat nhau tai 2 diem A va B. Ve cat tuyen qua B cat hai duong tron tai E va F ( E thuoc (O1), F thuoc (O2))

a) chung minh AE=AF

b) ve cat tuyen CBD vuong goc AB ( C thuoc (O1) va D thuoc (O2)). Goi P la giao diem cua CE va FD. Chung minh:

AEPF, ACPD la cac tu giac noi tiep. Cm tam giac EPF can va ba diem I, A, P thang hang voi I la trung diem EF

c) khi EF quay quanh B thi diem I di chuyen tren duong nao

Cho (O) co AB la duong kinh (AB=2R) va mot diem C thay doi tren duong tron (O) sao cho tam giac ABC khong can tai C. Goi H la chan duong cao cua tam giac ABC ha tu C. Ve HE, HF lan luot vuong goc voi AC va BC. Cac duong thang EF va AB cat nhau tai K. Tinh theo R dien tich tam giac CEF va do dai cac doan thang KA, KB trong truong hop goc BAC bang 60do

Cho duong tron tam O co AB la duong kinh va mot diem C thay doi tren duong tron (O) sao cho tam giac ABC khong can tai C. Goi H la chan duong cao cua tam giac ABC ha tu C. Ve HE va HF lan luot vuong goc voi AC va BC. Cac duong thang EF va AB cat nhau tai K.

a) tinh theo R dien tich tam giac CEF va do dai cac doan thang KA, KB trong truong hop goc BAC bang 60do

b) Ha EP, FQ vuong goc AB. Chung minh rang duong tron duong kinh PQ tiep xuc duong thang EF

c) goi D la giao diem cua (O) va duong tron duong kinh CH (D khac C). Chung minh rang: KA.KB= KH^2 va giao diem M cua cac duong thang CD va EF luon thuoc 1 duong thang co dinh

Cho (O) duong kinh AB. Tren tiep tuyen tai A cua (O) lay 1diem M. Doan MB cat (O) tai C. Goi E la trung diem cua BC. Tia EO cat MA tai F

a) CM tu giac AEBF noi tiep

b) tren tia AM, lay 1diem D sao cho M la trung diem cua AD. CM DB vuong goc voi FB

c) cm FC la tiep tuyen cua duong tron ngoai tiep tam giac ACD

d) trong tam giac MBF, cm 2< sin^2 M + sin^2 B + sin^2 F < 3

Bai 1: Cho a>b>c>0. Cm rang:

a^3b^2+b^3c^2+c^3a^2 > a^2b^3+b^2c^3+c^2a^3

Bai 2: Tim tat ca cac so nguyen duong x,y,z sao cho:

{(xy+1) chia het z

{(xz+1) chia het y

{(yz+1) chia het x

Bai 3: Chung minh rang neu ABCD la tu giac loi va khong phai la tu giac noi tiep thi: AB.CD+AD.BC>AC.BD

Bai 1: cho a,b, la cac so duong thoa man (1/a^2)+(1/b^2)=1/2. Tim gia tri nho nhat cua bieu thuc K=a+b

Bai 2: gia su cac phuong trinh: ax^2+bx+c=0 va cy^2+dy+a=0 (a va c khac 0) co cac nghiem tuong ung la x1,x2 va y1,y2. Chung minh rang : x1^2+x2^2+y1^2+y2^2 >= 4

Bai 3: xac dinh m de he phuong trinh sau day co nghiem duy nhat

{căn(1-x)+căn y=m

{căn(1-y)+căn x=m

Bai 4: giai va bien luan theo tham so m phuong trinh sau:

(x+2006)/(x+2006-m)=(x-2006)/(x-2006+m)

Bai 5: giai he phuong trinh:

{2x^3= 2y^2+y

{2y^3= 2x^2+x

Bai 6: tim nghiem nguyen cua phuong trinh sau : xy+6x+2006y+12033= 0

Bai 7: chung minh rang luon ton tai mot so tu nhien N co khong qua 2007 chu so sao cho cac chu so cua N chi la 9 hoac 0 va N chia het cho 10030

Cho hinh vuong ABCD. Tren doan AC lay diem M. Goi E va F lan luot la hinh chieu vuong goc cua M len BA va BC

a) so sanh dien tich tam giac DEF va dien tich tu giac AEFC

b) Xac dinh vi tri M de dien tich tam giac DEF la nho nhat

Nguyen li Dirichlet la gi? Giai thjx ro rang jum mjh nha, kam on rat rat nhjeu

Cho tam giac ABC noi tiep duong tron (O). Goi (O') la duong tron bat ki tiep xuc ngoai voi (O) tai D tren cung BC khong chua A. Cac duong thang AD,BD,CD cat duong tron (O') lan luot tai A',B',C'

a) chung minh: AA'/AD=BB'/BD=CC'/CD

b) chung minh: AD.BC=AC.BD+AB.CD

c) goi A1,B1,C1 la cac tiep tuyen cua (O') ve tu A,B,C. CM rang AA1.BC=BB1.AC+CC1.AB

Bai 1: voi a>b>c>0. Cm rang:

a^3b^2+b^3c^2+c^3a^2 > a^2b^3+b^2c^3+c^2a^3

Bai 2: tim tat ca cac so nguyen duong x,y,z sao cho

{(xy+1)chja het cho z

{(xz+1)chja het cho y

{(yz+1)chja het cho x

Bai 3: chug mjnh rang neu ABCD la tu gjac loj va khong phaj tu gjac noi tjep thi : AB.CD+AD.BC>AC.BD

Cho A o ngoai duog tron (O). Ve 2 tjep tuyen AB,AC voi (O) va cat tuyen ADE

a)cm AB^2=AD.AE

b)duong thang AO cat BC tai H. Cm D,H,O,E cung thuoc 1duong tron

c) tu D ve day DK//BC. Cm K,H,E thang hang

d)tu D ve duong thang song song voj BE cat AB taj F va BC tai G. Cm D la trung djem cua doan FG

quan he tduc toj nay, nhug trua maj xuat hjen kjh nguyet thj co the mag thaj k?

Cau 1:

{x+y+(1/x)+(1/y)=5

{xbjnh+ybjnh+(1/xbjnh)+(1/ybjnh)=9

cau 2:

{(x+y)(1+1/xy)=5

{(xbjnh+ybjnh)(1+1/xbjnhybjnh)=49

Nam ngoaj e pj penh nen k the thj tuyen sjnh 10 dc, vj the nam nay e mun dag ky thj laj, vay thj e faj dag ky nhu the nao, thu tuc ra sao?

Cau1: gjaj he pt

{x/4=y/3=z/9

{7x-3y+2z=37

Cau2:xac djnh m de he sau co nghjem

{căn cua(x-4)+căn cua(y-1)=4

{x+y=3m

Cau3: xac djnh m de he sau co nghjem duy nhat

{ybjnh-(x+y)=2m

{xbjnh-(x+y)=2m

Cho (O;R) va day cung BC khog di qua tam O. Lay diem A thuoc cung lon BC sao cho AB<AC. Duog cao BE va CF cat nhau tai H

a)cm AFHE,BFEC noj tjep

b)BE va CF lan luot cat (O) tai B' va C'. Cm OA_|_B'C'

c)goi I la trung diem AH. Cm DFIE noi tjep

d)duog thang qua H cat AB taj djem C1, cat AC taj djem B1. Cm 2duong thang C'C1 cat B'B1 tai 1djem thuoc (O)

CHU Y:hjnh nhu cau c saj de, co pan nao nhan ra duoc loj saj va sua laj k, neu k sua dc thj van mog kac pan jup mjh cau d nhe,cau d k saj de dau,kam on tat ca cac pan

Cho tam gjac ABC (goc B>goc C) co 3 goc nhon n0j tjep (O;R). Lay djem M tuy y tren cung nho AC(AM<MC). Goj H,K va I lan luot la hjh chjeu cua M len AB,BC,CA

a)cm AHMI, MIKC noj tjep

b)cm H,I,K thag hag

c)Goj D,E lan luot la trug djem AB va IK. Cm ME vuog goc vs DE

d) cm (AB/MH)+(BC/MK)=AC/MI