, eX dQc I~p tuyen tlnh trong khong gian cac ham s6 dUdng.. B khl1ng dinh ding co thEl lam cho rna tr~n nMn dUQc kha nghich va khong co diElm biit dQng tUc la t6n t1?oi vector v sao cho
Trang 1HOI ToAN HQC VI~T NAM
HE THI OLYMPIC ToAN SINH VIEN LAN THU XIV (2011)
Mon: D.A-I s6
Call 1 Chung minh rang h$ eX, eX ,ex, , eX dQc I~p tuyen tlnh trong khong gian cac ham s6 dUdng
Call 2 Cho day s6 (xn), (Yn), (zn) thOa man: Xo = Yo = Zo va:
Tfnh X2011
Xn+l = 4xn - Yn - 5zn Yn+l = 2xn - 2zn Zn+l = Xn - 2zn
Call 3 Cho hai rna tr~n A va B cling dip n va rna tr~n e = AB - BA giao hoan voi ca hai rna tr~n A va B Chung minh ding t6n t1?oi s6 nguyen dUdng m sao cho em = o
Call 4 Cho da thuc P(x) co b~c n va co n nghi$m th\)'c (co thEl phan bi$t hoi;Lc
bQi) TIm di§u ki$n c§,n va du cua u va v dEl da thuc sau cling co n nghi$m th\)'c:
Call 5 Co hai b1?on A va B chdi mQt tro chdi nhu sau:
Tren mQt bang 0 vuong n x n, A di§n VaG 0 d vi trf (i, j) mQt s6 nguyen dUdng nao do B1?on B co thEl giu nguyen s6 do hoi;Lc tang, giam s6 do 1 ddn vi B khl1ng dinh ding co thEl lam cho rna tr~n nMn dUQc kha nghich va khong co diElm biit dQng (tUc la t6n t1?oi vector v sao cho Av = v) Hoi B nMn dinh dung hay sai? VI sao?
Call 6
a) TIm di§u ki$n dEl M sau co nghi$m duy nhiit:
(1 + a)xl + (1 + a2)x2 + (1 + a3)x3 + (1 + a4)x4 = 0 (1 + b)Xl + (1 + b2)X2 + (1 + b3)X3 + (1 + b4)X4 = 0 (1 + C)Xl + (1 + C 2 )X2 + (1 + C 3 )X3 + (1 + C 4 )X4 = 0 (1 + d)Xl + (1 + d )X2 + (1 + d 3 )X3 + (1 + d )X4 = 0
[ 11 -11]
b) Cho rna tr~n A =
Tfnh A2012
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Trang 2HOI ToAN HQC VI~T NAM
HE THI OLYMPIC ToAN SINH VIEN LAN THU XIV (2011)
Mon: GIAI TICH
Call 1 Cho ham s6 f(x) = (x~xl)2
(i) Chung minh PT f(x) = x co nghi$m duy nhiit tren [~, I] va ham f'(x) d6ng
bi§n
(ii) Chung minh day (un) voi Ul = I,Un+l = f(u n) co cac pMn tu d§u thuQc
do"n [~, I]
Call 2 Tfnh tfch phan:
1
1 + x + x 2 + vi x4 + 3x2 + 1 -1
Call 3 Cho hai day s6 (xn), (Yn) tMa man: Xn+l ;:;> xnt yn , Yn+l ;:;> Vx~ty~, mQi s6 tv nhien n
(i) Chung minh ding cac day Xn + Yn, Xn-Yn tang
(ii) N§u cho truoc hai day (xn), (Yn) bi cMn Chung minh hai day nay cling hQi t1J v§ mQt di§m
Call 4 Cho et, (3 tMa man biit d11ng thuc: (1 + t)n+a < e < (1 + t)n+ i3 , mQi n
nguyen dUdng TIm min cua Jet - (3J
Call 5 Do"n [m, n] la do"n t6t n§u Ung voi a, b, c la cac s6 thvc tMa man
2a + 3b + 6c = 0 thl PT ax2 + bx + c = 0 co nghi$m thuQc [m, n] TIm do"n t6t
co dQ dai nM nhiit
Call 6
(i) TIm tiit ca cac ham s6 f(x) tMa man: (x - y)f(x + y) - (x + y)f(x - y) =
4xy(x2 - y2), mQi x, y
(ii) Cho ham s6 f(x) kha vi trong do"n [-1, I] va: xf(x) + ~f(~) <:: 2, \Ix E [~, 2] Chung minh ding: JE f(x) <:: 2.1n2
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