i i XVI i i

.. , I.. i, .. , I.. , .. i

28 2 2013 . . ii XVI i -i ii .. . i i i i i . i i - 21 . , i i.

i i - 10 , i ii. i 9 i 16 i .

1. . Q(x) n- iii i i i 1 < 2 < : : : < n+1: i

'(x) =n+1Xi=1

aiQ (x+ i) ;

ai =Qj 6=i

(i j)1 ; 1 i n+ 1 ( i 1 j n+ 1; j 6= i).. P (t) n;

P (t) n+1Xi=1

P (i)

Yj 6=i

t ji j

!; t 2 R: ()

i, () n i P (i) t = i; 1 i n+1; n; i n+1, i. ( i i () ii ).

Q(x) = q0xn+q1xn1+ : : :+qn; q0 6= 0: ii () n- i t x

Px(t) = Q(x+ t) = q0tn + p1(x)t

n1 + : : :+ p0(x):

Px(t) =n+1Xi=1

Px(i)

Yj 6=i

t ji j

!=

n+1Xi=1

aiQ(x+ i)

Yj 6=i

(t j)!; t 2 R:

i ii tn i i, i, '(x) = q0 ix 2 R:

ii: '(x) i ii Q(x).

1

2. ii . i i i a; ii

1

x+

1

y>

a

x+ y+

r1

x2+

1

y2

i i x > 0 y > 0:. i i i ii

x+ y

xy(x+ y

px2 + y2) > a;

(1 + t)(1 + tp1 + t2)t

> a;

t = yx> 0: i 1 + t +

p1 + t2:

, (1 + tp1 + t2)(1 + t+p1 + t2) = (1 + t)2 (1 + t2) = 2t; i

f(t) =2(1 + t)

1 + t+p1 + t2

> a; t > 0:

ip1 + t2 1 i t > 0: , i a 1

i. i , i f(t) [0;+1) f(0) = 1; a > 1; t f(t) < a:

ii: a 1:4. i, . N i k i i . N

ii , , i i i i , , i i . i ii i i i, i i i k- i . i i ii, , ii , i i i i.

4.1. N = 2027 k = 2013 i ii i, i i .

4.2. ii S(N; k) ii i, i i .

. k = 1; i i i i . i , k > 1: i , N = 1 i ii ii i, i k + 1: i -, N > 1:

m ii i, i i . xi; 1 i N; ii , i i i , i- i. i, i

x1 = m 1 m1k = (m 1) k1k ;x2 = (x1 1) k1k = (m 1)

k1k

2 k1k;

x3 = (x2 1) k1k = (m 1)k1k

3 k1k

2 k1k;

2

xN =(xN1 1) k1k = (m 1)

k1k

N k1k

N1 k1k

N2 : : : k1k

=

=(m 1) k1k

N+ k

k1k

N (k 1) = (m+ k 1) k1k

N (k 1) :i k1 k i , xN ; i i i xi;

i, m+k1 = l kN l: , xN = l (k 1)N (k 1) : l 1; xN i N; S(N; k) = m = l kN k + 1:

i i :) k = 2 l = N + 1; i S(N; 2) = (N + 1)2N 1:) k 1 i N; l = 1 S(N; k) = kN k + 1:) N , k > 2 l = 1;

i (k 1)N (k 1) i N: S(N; k) = kN k + 1: ii i 4.1: iN = 2027 , S (2027; 2013) = 20132027 2012:

) k > 2 i (k 1; N) = 1: i (k 1)'(N) 1i N; '(N) i , ii , i N N . s , q = s'(N)N + 1 0: i l = (k 1)q i,

xN = l (k 1)N (k 1) = (k 1)(k 1)s'(N) 1

i N; i S(N; k) m = ks'(N)+1 k + 1:) k > 2 (k 1; N) = d > 1: M = N

d: ,

i (k 1;M) > 1: i, p i (k 1;M) : i p k 1 n N i i n. , xN = l(k 1)N (k 1) i pn+1 l; i N .

(k 1;M) = 1 i i ) s , q = s'(M) N + 1 0; l = (k 1)q: il(k 1)N1 1 = (k 1)s'(M) 1 i M; xN i Md = N: i i S(N; k) m = ks'(M)+1 k + 1:

5. .5.1. i P (x) i ii, i x 2 R

ii

Px+ x2

= x+ x2 + : : :+ x2013 + x2014?

5.2. i P (x) i ii, i i a i b i x 2 R ii

Px+ x2 + x3

= x+ x2 + x3 + : : :+ x2010 + ax2011 + bx2012 + x2013?

3

.5.1. x = 1 i P (2) = 2014; x = 2 P (2) = (2 + 4) + (8 + 16) + : : :+ 22013 + 22014 = 2 + 8 + : : :+ 22013 6= 2014;

i. , P (x) i.5.2. i i ii ii ,

i i. ii i i i x: x = 1 x = i i P (1) = a+b1 P (1) = i1ai+ b+ i ii. i ii i , i a + b 1 = b 1 2 a = 0; i. , P (x) i.

6. i . !, A, B,C. BF i CE ABC, M AC. i i BF i EM i A i !.

. O !, H ABC,T BF i EM , G i, BC CO i (. 1). i \CGO = \CGB = 90; B;O;G i i.

\MGO = \MCO = 90 \MOC = 90 \EBC = \ECB = \EGB = \EGO; M;G;E ii.

i i , . 1. \BAC = ; 45 < < 90: \EBT = 90 ; \AEM = ;\ETB = \AEM \EBT = 2 90; i

BT = BE sincos 2

= BH sin2

cos 2:

, T H i B ii sin2

cos 2: ii i

, - i i 0 < < 180; 6= 45; 6= 135: = 45 = 135 i BF EM i i, T .

. 1.

BC ! 180; iii i 45 135:i \BHC = \EHF = 180 \EAF = 180 , H

4

!0, ! i BC. , A i !; A 6= B; A 6= C; H i !0 . ii T i , !0 i B ii sin2

cos 2; .

7. i ii. i i i i - i a; b; c; ii un = an + bn + cn; n 1; .

. u2 > u1 i ii un; n 1:, . i, ii i- n 2

un1un+1 =(an1 + bn1 + cn1)(an+1 + bn+1 + cn+1) (an12 an+12 + bn12 bn+12 + cn12 cn+12 )2 = (an + bn + cn)2 = u2n:

iun+1un

unun1

: : : u2u1

> 1;

un+1 > un; n 1: , ii un; n 1; i i, u2 > u1;

a2 + b2 + c2 > a+ b+ c: ia 1

2

2+b 1

2

2+c 1

2

2>p

32

2;

i i (a; b; c) , i i i R =

p32 i

12; 12; 12

:

8. i i. i i i i ii [1; 2]i f : [0;+1) ! R; i i r 0 i ' 2

6; 4

iif (r cos') + f (r sin') = f (r) :

. r = 0 f(0) + f(0) = f(0); f(0) = 0: ' = 4

i r 0 i

f

rp2

+ f

rp2

= f (r) ; f (r) = 2f

rp2

; f

rp2

= 2f

r2

; f (r) = 4f

r2

:

i , f(r 2k) = 22kf(r); k 2 Z: i f ii [1; 2] ; i ii [2k; 2k+1]; k 2 Z: f , i ii, i f (0;+1) :

i g : [0; +1) ! R , g(x) = f (px) : i g(0) = 0 f(x) = g(x2): i i f (0;+1) , ig (0;+1):

i g (r2 cos2 ')+ gr2 sin2 '

= g (r2) i i r 0

' 2 6; 4

: , g(u) + g (v) = g (u+ v) i u; v > 0 , 1 u

v 3:

5

ii n 1;

g(nx) = ng(x); x > 0: ()

n = 1 . , () i i n k 1; () n = k. i, k = 2m; u = v = m

g (kx) = g (mx) + g (mx) = mg (x) +mg (x) = kg (x) :

k = 2m+ 1; u = (m+ 1)x v = mx 1 < uv= m+1

m< 3;

g (kx) = g ((m+ 1)x) + g (mx) = (m+ 1) g (x) +mg (x) = kg (x) :

, ii () i n 1: g(1) = c: i m;n 1 () i ng m

n

= g(m) = mg(1) = mc;

gmn

= m

nc: , g(x) = cx i i x:

x i ii . i ig , c 0 i i 0 < r0 < x < r00 cr0 = g(r0) g(x) g(r00) = cr00; i g(x) = cx: i c < 0: , , g(x) = cx i x 0: ii f(x) = cx2; x 0:

i , i i i i c 2 R:ii: f(x) = cx2; x 0; c 2 R i.9. . PQR, ! -

i QR, RP PQ A, B C ii, AB2+AC2 = 2BC2.i, iii PA, QB i RC, !, i ABC, A iii AC i AB i.

. AB = c; BC = a; AC = b;AA1; BB1; CC1 i ABC; M i, O !(. 2). i \AB1O = \AC1O = 90; iB1O; C1O i AC AB. , A; B1; C1 O i i AO:

G i AA1 ii B1C1: i i i i

AA21 =14(2b2 + 2c2 a2) = 3

4a2;

AG MG = 12AA1 16AA1 = a

2

16= a

4 a4= B1G C1G: . 2.

M , AB1C1: , ii PA, QB RC i , -

.

6

P PD PE i AB AC ii (. 3). iPB = PC; \PBD = C \PCE = B; iPDPE

= PB sinCPC sinB

= cb: i (i

i), i AP i , i i \BAC ii i i AB AC i- c b: L AP BQ; ha; hb; hc ii i L iBC; AC AB ii. i hc : hb = c : b i ha : hc = a : c: i hb : ha = b : a; L CR: , -, ii PA, QB RC ii i L:

i, SABA1 = SACA1 ; ii i A1 i AB AC i b c:i , , A1i i \BAC; i AP: \BAL = \A1AC: i . 3.

SBCL : SACL : SABL = aha : bhb : chc = a2 : b2 : c2;

i b2+c2 = 2a2 i, SBCL = 13SABC = SBCM; LM k BC:i, i AP AB1C1 i L1:

i ^C1L1 =^MB; i L1M k C1B1 k BC; L1 L i, .

11. i i . i i i i p; 37p2 47p+ 4 .

. 37p2 47p + 4 = n2; n 2 N: i p (37p 47) = (n 2) (n+ 2) :, n 2 n+ 2 i p:

n 2 = kp; k 2 N [ f0g; n + 2 = kp + 4; p (37p 47) = kp (kp+ 4) ;(37 k2) p = 4k + 47: i k 6 : p = 3 k = 4 p = 71 k = 6:

n+2 = lp; l 2 N; n 2 = lp 4; (37 l2) p = 47 4l: l 6 p = 23 l = 6; l > 6 i i, (l2 37) p > 6l 37 > 4l 47:

ii: p = 3, p = 23 p = 71:13. ii ii. ii

fan; n 1g : a1 = 0; a2 = ; an = an1 + an2; n 3:

i i i ; ; ; p > 2 ap2 i p?

7

. = 6; = 1; = 2: Ii n 1 ,

an = 2n + 2(1)n; n 1:

i, n = 1 n = 2 ii , n = m n = m+ 1; n = m+ 2

am+2 = am+1 + 2am = 2m+1 + 2(1)m+1 + 2(2m + 2(1)m) = 2m+2 + 2(1)m+2:

n = p2; p . 2p1 1i p;

ap2 = 2p2 2 = 2(2p21 1) = 2(2p1 1)(2p(p1) + : : :+ 22(p1) + 2p1 + 1)

i p:ii: i.16. i . i i k 2 Z;

i i i P (u; v; w) i ii, i x 2 R; y 2 R ii

cos(20x+ 13y) = P (cosx; cos y; cos(x+ ky)):

. 1) , cos(ax+by), a, b i ii , i ii i cos x, cos y cos(x + ky), b i k.

i, k = 0; i x; y 2 R

cos(ax by) = P (cosx; cos(y); cos x) = P (cosx; cos y; cos x) = cos(ax+ by);

x = 2a; y =

2bi 1 = 1; i.

i , k 6= 0: i i x 2 R y = k

cosax b

k

= P (cosx; cos(y); cos(x )) = P (cosx; cos y; cos(x+ )) = cosax+ b

k

:

x = bak

ii i 1 = cos(2bk); b

k i.

2) , i i a b, b i k, i cos(ax+by) i ii i cos x, cos y cos(x+ ky):

, a 2 Z i Ta(t); cos ax = Ta(cosx): i, T0(t) = 1 T1(t) = t, i Tn(t) Tn+1(t); ii

cos(n+ 2)x = 2 cos x cos(n+ 1)x cosnx

Tn+2(t) = 2tTn+1(t) Tn(t); n 0: , Ta(t) i i i a 0 a < 0 Ta(t) = Ta(t):

8

, a 2 Z cos(ax+ ky) i cos x, cos y cos(x + ky): i, a = 0 a = 1 cos ky = Tk(cos y) cos(x+ ky) = cos(x+ ky); ii

cos(nx+ ky) = 2 cos x cos(n 1)x+ ky cos(n 2)x+ ky

ii i i a = n 2; a = n 1 a = n n 2 i a = n+ 1; a = n+ 2 a = n n 1:

i, i i a 2 Z , cos(ax+ cky) i cos x, cos y cos(x + ky) i c 2 Z: c = 0 c = 1 , ii

cos(ax+ nky) = 2Tk(cos y) cosax+ (n 1)ky cosax+ (n 2)ky

ii i i c = n 2; c = n 1 c = n n 2 i c = n+ 1; c = n+ 2 c = n n 1:

, , cos(ax+by), a, b i ii , i cos x, cos y cos(x+ ky) i i, b i k.

ii i cos(20x+13y) i cosx; cos y cos(x+ky) , 13 i k, k = 1, 13.

ii: k = 1, 13.17. . i

i i . i, i i i i i i i i i, ii i, i .

. i A1A2A3A4. G1, G2, G3, G4 i A2A3A4, A1A3A4, A1A2A4, A1A2A3 ii. i-, ii A1G1, A2G2, A3G3 A4G4 ( i ) i i G, i i i-i 3 : 1, i . G1G2G3G4 A1A2A3A4 i iG i ii k = 13 . i - A1A2A3A4 i G1G2G3G4. , G1G2G3G4 i i i i, A1A2A3A4 i i.

21. i . n 2. i i pa21 + 1 +

pa22 + 1 + : : :+

pa2n + 1

a1 + a2 + : : :+ an

a1, a2, : : : ; an, i a1a2 : : : an = 1.. , q

a21 + 1 +qa22 + 1 + : : :+

pa2n + 1

p2 (a1 + a2 + : : :+ an) : ()

9

i a1 = : : : = an = 1 ii, i , i i

p2:

i f(x) =px2 + 1 xp2 + lnxp

2 , f(x) 0

i x > 0:

f 0(x) =xp

x2 + 1p2 +

1p2x

=2x2 2p2xpx2 + 1 +p2px2 + 1

2xpx2 + 1

:

:

2x2 2p2xpx2 + 1 +

p2px2 + 1 =

p2x

px2 + 1

2+px2 + 1

p2

px2 + 1

:

0 < x < 1 , x > 1 i, px2 + 1 >

p2xpx2 + 1 > 0

px2 + 1

p2 =

x2 1px2 + 1 +

p2>

x2 1p2x+

px2 + 1

=p2x

px2 + 1 > 0;

px2 + 1

p2px2 + 1 < p2xpx2 + 12:

, i f(x) 0 < x < 1 x > 1; i x > 0 f(x) f(0) = 0: ,

f(a1) + : : :+ f(an) =qa21 + 1 + : : :+

pa2n + 1

p2 (a1 + : : :+ an) 0

( , ln a1 + : : :+ ln an = 0). ii () .

10