1 www.omaths.com 1 1. a b(b 0), q, a = bq a b, b a. a b, b a ( ). b ±1, b a. a b, b a. a t b, a t+1 b, t N, a t b.. (1)b 0, ±1 a, a a(a 0). () b a, a 0, 1 b a. (3) c b, b a, c a. (4) b a, c 0, bc ac. (5) c a, c b, c (ma + nb)(m n Z). (6) k a i = 0, b a 1, a,, a k k 1, b. 1. m, a b m, a b m, a b(modm).. (1)a b(modm) m (b a). ()a b(modm) b = km + a(k Z). (3)a a(modm). (4) a b(modm), b a(modm).
(5) a b(modm), b c(modm), a c(modm). (6) a b(modm), c d(modm), a ± c b ± d(modm), ac bd(modm), a n b n (modm). (7) ac bc(modm), (c, m) = d, a b(mod m ). (c, m) c m. d, (c, m) = 1, ac bc(modm), a b(modm). 3. m, m ( m ). m 0, 1,, m 1 m,, m m : A 0, A 1,, A m 1. A i = {qm + i m, q Z}, i = 0, 1,, m 1. A i (i = 0, 1,, m 1) m 1 4. i=0 A i = Z, m 1 i=0 A i =. m m A 0, A 1,, A m 1, A i a i, a 0, a 1,, a m 1 m ( m ). m 0, 1,, m 1, m. m m. 3 1. 1, 1, ( ); 1,. 1., Z + = {1} { } { }.. 1,. 3. a a. 4.. 5. f(n) = m a i n i, n, f(n). 6. (Wilson) i=0 p (p 1)! 1(modp). 4 1. ( )
3 1, ( ),.. n(n > 1) n = m 3. p αi i. p i, α i, i = 1,,, m. d(n) = 1 1 n, n n = m p αi i, d n 4. d(n) = m (1 + α i ). σ(n) = d 1 n, n n = m p αi i, d n σ(n) = 5. n!, p 5 1. r=1 m p αi+1 i 1 p i 1. (1) c a 1, c a,, c a n, c a 1, a,, a n. [ ] n p r. [x] x. a 1, a,, a n a 1, a,, a n. (a 1, a,, a n ). () a 1, a,, a n a 1 = m p αi i, a = m p βi i,, a n = m p δi i, p i, α i, β i,, δ i, i = 1,,, m, (a 1, a,, a n ) = m p ti i, t i = min{α i, β i,, δ i }. (3) a b, a b b. (4) a b, (a, b) = b. (5) a b 1, d = ax 0 + by 0 ax + by(x y ), d = (a, b). (6) a b. (7) m, (am, bm) = (a, b)m. ( ) a (8) n a b, n, b (a, b) = n n. (9) a b(a > b) a = bq + r, 0 r < b, q r Z. (a, b) = (b, r).
4 a b. a = bq 1 + r 1, 0 r 1 < b. r 1 = 0, (a, b) = b. r 1 0, r 1 b b = r 1 q + r, 0 r < r 1. r = 0, (a, b) = (b, r 1 ) = r 1. r 0, r r 1 r 1 = r q 3 + r 3, 0 r 3 < r., b > r 1 > r > r 3 > r i (i = 1,, ),, n + 1 r n+1 = 0. r n 0, (a, b) = (b, r 1 ) = (r 1, r ) = = (r n 1, r n ) = r n. (5) ax + by d = ax 0 + by 0.. (1) a 1 b, a b,, a n b, b a 1, a,, a n. a 1, a,, a n a 1, a,, a n. [a 1, a,, a n ]. () a 1, a,, a n a 1 = m p αi i, a = m p βi i,, a n = m p δi i, p i, α i, β i,, δ i, i = 1,,, m, [a 1, a,, a n ] = m p ri i, r i = max{α i, β i,, δ i }. (3)a 1, a,, a n. (4)[a, b] = ab (a, b). 6 1. (1) (a 1, a,, a n ) = 1, a 1, a,, a n ( ). n ( ).., 1 ; ; ; p, p a, p a () (a, b) = 1, (a ± b, a) = 1, (a ± b, ab) = 1. (3) (a, b) = 1, a bc, a c. (4) a c, b c, (a, b) = 1, ab c. (5) (a, b) = 1, (b, ac) = (b, c). (6) (a, b) = 1, c a, (c, b) = 1. (7) (a, b) = 1, (a, b k ) = 1. (8) a 1, a,, a m b 1, b,, b n, (a 1 a a m, b 1 b b n ) = 1.
5. : m m (Euler), ϕ(m). ) m = n p αi i, ϕ(m) = m (1 n 1pi. p i, α i (i = 1,,, n). m, ϕ(m) = m 1. : (1)ϕ(m), (a, b) = 1, ϕ(a)ϕ(b) = ϕ(ab). () p, ϕ(p) = p 1, ϕ(p k ) = p k p k 1. ( (3) m = p α1 1 pα pα k k, ϕ(m) = m 1 1 ) ( 1 1 ) ) (1 1pk. p 1 p (4) d 1, d,, d T (m) m, T (m) 3. (1) ϕ(d i ) = m. m, (a, m) = 1, ϕ(m), a ϕ(m) 1(modm). () (Fermat) p, (a, p) = 1, a p 1 1(modp). : m. 4. m 1, m,, m k k. x b 1 (modm 1 ), x b (modm ), x b k (modm k ) x M 1M 1 b 1 + M M b + + M k M kb k (modm). M = m 1 m m k, M i = M m i, i = 1,,, k, M i M i 1(modm i ), i = 1,,, k. :. 7 1., ; 1,.
6.. ( ), ( ). ( ). ( ).. 3..,. 8 1. a, a a.. 0, 1, 4, 5, 6, 9. 3.. 4. 5,,. 5. 6,. 6. 4 ; 4 1. 7. 8 0 4; 8 1. 8. 3, 3 ; 3, 3 1. 9. 5, 5 ; 5, 5 +1 1. 10.,,,, 0, 1, 4, 7, 9. 11.. 1.,. 13. p, p. 9 1... 4 4. 3. 5 0 5.
7 4. 3 3. 5. 9 9. 6. 11 11. 7. 10n 1(n ), n, 10n 1, A A = 10x + y, y {0, 1,, 9}, (10n 1) A (10n 1) (x + ny). A 9, 19, 9, 39,. 8. 10n + 1(n ), n, 10n + 1. A A = 10x + y, y {0, 1,, 9}, (10n + 1) A (10n + 1) (x ny). A 11, 1, 31, 41,. 10 1. A A = n a i 10 i, a i {0, 1,, 9}, i = 0, 1,, n 1, a n {1,,, 9}.. A n A n, A n a n 0 (mod10). 3. A n 4. 4. A S(A) = n a i 9, A n a i (mod9). i=0 5. A S(A) = n a i S(A + B) S(A) + S(B), S(AB) S(A)S(B). i=0 6. a b, 1 a 5 b. 7. 1 n, n = a 5 b, a b. 8. 1, n 1. n 9. (n, 10) = 1, 1 n r, r 10r 1(modn). 11 k 1. k ( ), A k, : A = d 0 +d 1 k+d k + +d n k n = n d i k i. d i {0, 1,, k 1}, i = 0, 1,, n 1, d n {1,,, k 1}. i=0. A k A = (d n d n 1 d 1 d 0 ) k. 3. B, B k, : B = d 1 k 1 + d k + + d n k n + d i {0, 1,, k 1}, i = 1,,, n, i=0
8 : B, ; B,. 1 1. ax + by = c (1) ax + by = c(a b c ) (a, b) c. () (a, b) = 1, (x 0, y 0 ) ax + by = c, x = x 0 + bt, y = y 0 at(t ).. x + y = z (1) x = a, y = b, z = c(a b c ) x + y = z, (a, b) = 1,. () x = a, y = b, z = c x + y = z, a b, c. (3) x = a, y = b, z = c x + y = z, a, m n, m > n, (m, n) = 1, m n(mod), a = mn, b = m n, c = m + n. (4) a = mn, b = m n, c = m + n, a b c x + y = z ; m > n > 0, (m, n) = 1 m n(mod), a b c. 3. (Pell) (1) x dy = 1(d ),. () d, x = ±1, y = 0,. (3) d > 0, x dy = 1. (4) n > 0, (x 1, y 1 ) x dy = 1, x n y n (x 1 dy 1 ) n = x n + dyn, (x n, y n ) x dy = 1. 13,,.,. 1. ( ), S, N, L, S = N + L 1.. (1),, 1. (),.
9 (3) 4. 3. A(r) x + y r, r, A(r) = 1 + 4[r] + 4 A(r) = 1 + 4[r] + 8 [ [ ] r r s ] 4. 1 s r, [x] x. 1 s r [ r s ], r, x + y r A(r) πr. 4.. 5. n 5, n. 14 [x] 1. x R, [x] x.. [x] (1)y = [x] R, Z. ()x = [x] + r, 0 r < 1. (3)x 1 < [x] x < [x] + 1. (4)y = [x], x 1 x, [x 1 ] [x ]. (5) n Z, [n + x] = n + [x]. [ n ] (6) x i n [x i ]. [ n ] (7) x 1, x,, x n x i n [x i ]., x n [x n ] [x] n, [x] [ n x] n. [ y (8) x y x] [y] [x]. [ [ ] x [x] (9) n, =. n] n (10) x, [ x] = [x]; x, [ x] = [x] 1. [ m ] (11) m n, m n. n
10. (1) {x} x, {x} = x [x]. y = {x} : (i){x} [0, 1). (ii){x} 1. (iii){n + x} = {x}(n ). (13) p N, λ ( p )! λ M = p 1. [ ] [ ] [ ] p p p (11) M = + + 3 + = p 1 + p + + + 1 = p 1. 15 1. (a, m) = 1, λ, a λ 1(modm), a k 1(modm), 0 < k < m, λ a m λ ϕ(m), λ ϕ(m).. λ = ϕ(m), a m ϕ(m),, a m. 3. λ (1) a m λ, a 0, a 1,, a λ 1, m. () λ m, a t 1(modm) t, λ t. 1 1. 1 ABC a b c, A B C, r R r 1 r r 3, p, h a h b h c, m a m b m c, t a t b t c, A t a, BC h, BC α, S. I O G H, I 1 I I 3. 1. 1. 1 a sin A = b sin B = 1. 1. c sin C = R.
11 a = b + c bc cos A, b = c + a ca cos B, c = a + b ab cos C. 1. 1. 3 (1)S = 1 ah a = 1 bh b = 1 ch c; ()S = 1 ab sin C = 1 bc sin A = 1 ca sin B = 1 ah sin α; (3)S = abc 4R = R sin A sin B sin C = R (sin A + sin B + sin C); (4)S = a sin B sin C sin(b + C) = b sin C sin A sin(c + A) = c sin A sin B sin(a + B) ; (5) (Heron) S = p(p a)(p b)(p c); ( (6)S = r cot A + cot B + cot C ) ; (7)S = pr = (p a)r 1 = (p b)r = (p c)r 3. 1. 1. 4, ;, ;,. 1. 1. 5 r = 4R sin A sin B sin C ; r 1 = 4R sin A cos B cos C ; r = 4R cos A sin B cos C ; r 3 = 4R cos A cos B sin C ; r 1 + r + r 3 = r + 4R. 1. 1. 6 BIC = 90 + A, CIA = 90 + B, AIB = 90 + C, BI 1C = 90 A, CI A = 90 B, AI 3B = 90 C. 1. 1. 7 A B C p a p b p c; p; B C A p c p b; C A B p a p c; A B C p b p a. 1. 1. 8 AI ABC D, DI = DB = DC = DI 1, I B C I 1, D; AD I I 1, DI = DB = DC = DI 1, I I 1 ABC A. 1. 1. 9 BOC = A, COA = B, AOB = C. 1. 1. 10 (Archimedes) G( ), G 3.
1 1. 1. 11 (Pappus) ( ) m a = 1 b + c a, m b = 1 c + a b, m c = 1 a + b c. 1. 1. 1 t a = bcp(p a), tb = cap(p b), tc = abp(p c). b + c c + a a + b 1. 1. 13 t a = bc(p b)(p c). b c 1. 1. 14 BHC = 180 A, CHA = 180 B, AHB = 180 C. 1. 1. 15 ; ;. 1. 1. 16 BHC CHA AHB R. 1. 1. 17 (Carnot) BHC CHA AHB A B C. 1. 1. 18 AH BC D, ABC K, HD = DK. 1. 1. 19 AH BH CH BC CA AB D E F, AH HD = BH HE = CH HF. 1. 1. 0 BC L, AH OL, AH = OL = R cos A. 1. 1.. 1 n (n )π. 1.. (1) S = ab(a b ); () S = ah = ab sin θ(a b, θ, h a ); (3) S = 1 (a + b)h(a b, h ); (4) S = 1 mn sin ϕ(m n, ϕ ); (5) (Bretschneider) S = 1 4 4m n (a b + c d ) (m n, a b c d ); (6)
13 S = (p a)(p b)(p c)(p d)(p, a b c d ); (7) S = abcd sin A + C (a b c d ); (8) ( ) S = abcd(a b c d ). 1.. 3 a b c d, m n, m n = a c + b d abcd cos(a + C). 1.. 4 n, n. 1.. 5,. 1.. 6 n, n. 1.. 7,. 1. 3 (1) (Menelaus) ABC BC CA AB X Y Z. AZ ZB BX XC CY Y A = 1. () X Y Z ABC BC CA AB. AZ ZB BX XC CY = 1, X Y Z Y A ( ). 1. (3) (Ceva) P ABC, AP BP CP BC CA AB D E F. AF F B BD DC CE EA = (4) D E F ABC BC CA AB. AF F B BD DC CE EA ( ). = 1, AD BE CF (5) (Ptolemy) ABCD, AB CD + BC DA = AC BD. (6) ABCD AB CD + BC DA = AC BD, ABCD. (7)
14 ABCD, AB CD + BC DA AC BD, ABCD. (8) (Simson) ABC P BC CA AB D E F. D E F ( ). (9) ABC P BC CA AB D E F. D E F, P ABC. (10) (Fermat) ABC F. ABC 10, F BC CA AB 10 ; ABC 10, F. (11) ABC G. (Carnot) G ABC, P ABC, P A + P B + P C = GA + GB + GC + 3P G GA + GB + GC ; (Leibnitz) G ABC, P ABC, P A + P B + P C = 3P G + 1 3 (a + b + c ), a b c ABC. (1) ABC G. 1. 4 1. 4. 1., A B A B, AB = A B, (1) : F d, F, F F, T (v), v,. () : F O α( ) F, F F, R(O, α). α = π,,. (3) ( ): F l F, l ( ), U(l). 1. 4.
15, A B A B, A B = kab(k > 0),. (1) : O, F P, P OP ( ), OP = kop(k 0), O k, H(O, k). () : O, k(k > 0), θ, F P, OP O θ, P, OP = kop, P P O θ k, S(O, θ, k). 1. 4. 3 (1) F F,,, F F. l 1 l,,. () F F,,, F F. l 1 l,,. (3), R(O 1, θ 1 ) R(O, θ ), θ 1 +θ π, R(O, θ 1 + θ ), O O 1 O l m, O 1 O l θ 1, m O 1 O θ. (4), H(O 1, k 1 ) H(O, k ), H(O, k 1 k ), O M M O 1 O O. 1. 5 1. 5. 1 :,. 1. 5. : F A B F, F. 1. 5. 3 : M M. M.. 1. 5. 4 : d: d, d d, d.
16,,. 1. 5. 5 : F n G 1, G,, G n, F G 1, G,, G n ; G 1, G,, G n F n, G 1, G,, G n F. 1. 5. 6 (1)F F. () G 1 G, G G 1, G G. (3) G 1 F, G F, G 1 G F. (4) G F, S(G) S(F ), S(X) X,. 1. 5. 7 (1) n, n ; () O, F O r, F r. (3)A B, α, F P AB, A B AP B α, F AB α G. (4) G F, S(F ) > S(G), G F. (5) d F d G. (6) 1, 1,. (7) 1, π,. (8) n, S 1, S,, S n, S, S 1 + S + + S n > S,. (9) S G G 1, G,, G n, S 1, S,, S n. G G i k, S 1 + S + + S n ks; S 1 + S + + S n > ks, G {G i } k + 1.. 1 : n,,., S
17,,,., 360.. (Euler) : V, F, E, V + F E =. 3 3. 1. (1) : x cos α + y sin α = p, p, α x () : r(a cos θ + b sin θ) c = 0, a = cos α, b = sin α, c = p, p, α x. (3) ( ) : l 1 = 0, l = 0, λl 1 + µl = 0. 3. : (x 1, y 1 ) (x, y ) (x 3, y 3 ), x S = 1 1 y 1 1 x y 1. x 3 y 3 1 3. 3 (1) x + y = R (x 0, y 0 ) x 0 x + y 0 y = R ; (x a) + (y b) = R (x 0, y 0 ) (x 0 a)(x a) + (y 0 b)(y b) = R. () y = ax + bx + c (x 0, y 0 ) y + y 0 = ax 0 x + b(x + x 0) 1 + c; y y 0 = ax 0 + b (x x 0). (3) x a + y b = 1 (x 0, y 0 ) x 0 x a + y 0y b = 1; a x x 0 b y y 0 = a b.
18 (4) (x 0, y 0 ) x 0 x a y 0y b = 1; a x x 0 + b y y 0 = a + b. 3. 4 (1) O(a, b), R (x a) + (y b) = R, P (x 1, y 1 ) O P O R = (x 1 a) + (y 1 b) R. () ( ),. C 1 = 0, C = 0, 1, C 1 C = 0. 1 1. 1, n, m 1, m,, n m n, m 1 + m + + m n., n, m 1, m,, n m n, m 1 m m n. 1. 1.. 1 (1) n k(1 k n), n k, k-. A k n, A k n = n(n 1) (n k + 1) = n! (n k)!. k = n, n A k n = n(n 1) 1 = n!. (). n k. (3) n k(k 1), n k- n k,,. k n 1, n,, n k (n 1 + n + + n k = n), n n!, n 1!n! n k!.
19 (4) n ( ) k(1 k n), n k-. k- k-,. n k- A k n k = n! k (n k)!., k = n, n (n 1)!. (5) {1,,, n} {a 1, a,, a n }, a i i, i = 1,,, n, ( ). [ D n = n! 1 1 1! + 1! + 1 ] ( 1)n. n! 1.. (1) n k(1 k n), n k ( ), k-. C k n, C k n = Ak n k! = n(n 1) (n k + 1) k! = n! k!(n k)!. () n k(k 1), n k-. n k- C k n+k 1. (3) x 1 + x + + x n = m (x 1, x,, x n ) C n 1 m 1. (4) x 1 + x + + x n = m (x 1, x,, x n ) C m m+n 1. 1. 3 (1)C r n = C n r n ; ()C r n + C r+1 n = C r+1 n+1 ; (3)C r n = n r Cr 1 n 1 ; (4)C r nc m r = C m n C r m n m; (5)C 0 n + C 1 n + + C n n = n ; (6)C 0 n C 1 n + C n C 3 n + + ( 1) n C n n = 0; (7)C 0 n + C 1 n+1 + C n+ + + C k n+k = Ck n+k+1 ; (8) C 0 mc k n + C 1 mc k 1 n + C mc k n + + C k mc 0 n = C k m+n.
0 A B, f A B. f,,, f ( ). A B f,, A B, A = B. A B, f A B. f,, A B. A B, f A B. f,, A B. S 1, S,, S n, S 1 S Sn = S i S i1 Si + +( 1) k 1 S i1 Sik + +( 1) n 1 S 1 Sn. 1 i n 1 i 1<i n 1 i 1< <i k n : S 1, S,, S n S, S 1 S Sn = S S i + S i1 Si +( 1) k S i1 Sik + +( 1) n S 1 Sn. 1 i n 1 i 1<i n 1 i 1< <i k n ( 1) r n(n 3),, n n r Cr n r(n r)! = n! n n 1 C1 n 1(n 1)! + n(n 3), [ D n = n! 1 1 1! + 1! + 1 ] ( 1)n. n! n n C n (n )! + ( 1) n. (Catalan) n., n 5, n 10,. 1 n + 1 Cn n,,. n n n 1 n 1 Cn n 4. n, n 1 n + 1 Cn n.
1 (Fubini) A = {a 1, a,, a m }, B = {b 1, b,, b n }, S A B, S(a i, ) = {(a i, b) S b B, S(, b j ) = {(a, b j ) S a A}. S = a A S(a, ) = b B S(, b). 3 1 n + 1 n,,. m n,, k,, [x] x. m k = [ n, m n ; m ] + 1, m n. n 3 n,,. Erdös-Szekeres k a 1, a,, a k, k > mn(m n k ). m,, n. 4 1 M, M. M, M. M, M. 5 5. 1,, {a k }, g(x) = + k=0 ( ). g(x) ; ( ),,. a k x k
,. a 0, a 1, a,, g(x) = + a k x k. 5. 1 + (1) 1 x = x n, x < 1; k=0 1 + () 1 + x = ( 1) n x n, x < 1; k=0 (3)(1 + x) α = + Cαx n n, α R, Cα n = 6 k=0 6. 1 k=0 α(α 1) (α n + 1), Cα 0 = 1. n! ( ). G, V, E, G(V, E). G V G. V E,., ( ).,,., v k, deg v = k, k 0 1, v ( ); ( ) ( ). n,,, K n, E = 1 n(n 1). r,, G r (, G ),, deg G = r., G K n, deg G = n 1. G. : (1). () ( ). G, : e 1, e,, e m, e i = (v i 1, v i )(i = 1,,, m), v 0 v m. m, v 0 v m, v 0 v 1 v m. G u v,, d(u, v).,.
3,. 1. G u v, u v,.,. v 0 v 1 v m, v 0 v m,,. G,, ;,. 1,. G n(n 3). u v, d(u) + d(v) n 1, G. 3(Ore) G n(n 3). u v, d(u)+d(v) n, G. 4(Ore) G n m. n 3, m 1 (n 3n + 6), G. 5(Dirac) G n(n 3), u, d(u) n, G. 6(Pósa) G n(n 3). m, 1 m < 1 (n 1), m m, n, 1 (n 1) 1 (n 1), G... 7 = +1, ( 1 )., 1, ; 1,.,. 8, ( ).. G,, G : G, 0, 0, G.,,.,,..
4 9( ) G v e f, v e+f = (, v e + f = 1). 10 G v(v 3) e, e 3v 6. G 1 G,, f : G 1 G, G 1 A B,, f(a) f(b).,,,.,. 6. (Ramsey) :,,. k c 1, c,, c k K n,, K n k K n. 1 K 6,. 3 K 17 3. n, k K n, k K n n r k,. 3 (1) k(k ), r k, r k (r k 1 1) + ; () k, r k 1 + 1 + k + (k 1) + + k!! + k! 1! + k!. K n, p q, n, K n K p, K q. n r(p, q), r(p, q). 4(Erdös) p, q, : r(p, q) r(p 1, q) + r(p, q 1). 5(Erdös-Szekeres) p, q, r(p, q) C p 1 p+q. (Schur) k, n 0, n n 0, {1,,, n} k, x y z {1,,, n}, x + y = z. x y. 6. 3 1(Turán) (Erdös) [ ] n n,. 4 G n(n 5), n + 4, G. 3 n 6, n, 3n 5 G.
5 4 n n, 1 + 1 (n 1)(n ). 5(Mantel) G n, m, G 4m ) (m n. 3n 4 6 1)(n 5). G n(n > 5), G G 1 4 n(n 1 1. 1 1. A B A\B A, A\B = {x x A x / B}. A\B A B. B A, A\B = A B. 1. 3 ( ) A B A B ( ). A B = {(x, y) x A, y B}. 1. 4 (A B) (B A) A B, A B, A B = (A B) (B A) = {x x A x B, x / A B}. A B, 1. xxxxxxxxx xxxxxxxxxxx xxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxx xxxxx xxxxxxxxxxxxxxx xxxxx xxxxxxxxxxxxxx xxxxxx xxxxxxxxxxxxx B xxxxxx xxxxxxxxxxxx xxxxxx xxxxxxxxxxxx xxxxxxx A xxxxxxxxxxxx xxxxxxxx xxxxxxxxxxx xxxxxxxxxxxxxxxxxx xxxxxx xxxxxxxxx xxx 1 1. 5 I, A I, B I. I (A B) = ( I A) ( I B), I (A B) = ( I A) ( I B). 1. 6
6 (1)A (B C) = (A B) (A C); ()A (B C) = (A B) (A C); (3)(A B) C = (A C) (B C); (4)(A B) C = A (B C) = (A B) (A C); (5)A (B C) = (A B) (A C); (6)C (A B) = (C A) (C B); (7)A (A B) = A B. 1. 7 n n. 1. 8 1 M, M. M, M,. 1. 9 A, CardA( A, n(a)). 1. 10 A = A 1 A Am. A = A i A i Aj + A i Aj Ak +( 1) m 1 A 1 A Am. 1 i m 1 i<j m 1 i<j<k m A 1, A,, A m, A 1 A Am = A 1 + A + + A m. 1. 11 1. 11. 1 C Q n, Q = {a 1, a,, a n }. Q C, C Q : A 1, A,, A n. A = {A 1, A,, A n}. Q C A, 1. 1. 11. R Q n, A = {A 1, A,, A k }. r k 1, (1)A i1 Ai Air, 1 i 1 < i < < i r k; ()A i1 Ai Air Air+1 =, 1 i 1 < i < < i r+1 k.
7 A Q r R. R C r 1 k 1. 1. 11. 3 K Q n, A = {A 1, A,, A k }. A A i A j A i A j, A j A i, A K.. n Q K, C [ n ] n.. 1 ( )... 1 D f(x). M, x D, f(x) M( f(x) M), f(x) D ( ), M ( ). f(x) D, f(x) D., M, x D, f(x) M. f(x),, f(x) D, sup ( f(x) ). f(x),, f(x) D, inf ( f(x) ). y = ax + bx + c, a > 0,, 4ac b. 4a y = sin x y = cos x, ±1... D f(x), x 1 x D, x 1 < x. f(x 1 ) < f(x ), f(x) D ( ) ; f(x 1 ) > f(x ), f(x) D ( ). : 4ac b ; a < 0,, 4a (1) y = f(x) D ( ), D E, y = f(x) D y = f 1 (x), E ( ).
8 ). () f(x) g(x) D, (i)f(x) + g(x), f(x) g(x) ; (ii) f(x) g(x) D ( ), f(x)g(x), f(x) g(x) ( (3). u = g(x) D g,, y = f(u) D f, g(x) G, G D f y = f ( g(x) ) ( )., u = g(x) y = f(u) ( ),,,,,,,,. (4),. (5),,,... 3 (1) : f(x) D. x D, f( x) = f(x), f(x) ; f( x) = f(x), f(x). y (x = 0) ; ((0, 0) ). () ( ) : R, a, f(a+x) = f(a x), f(x) ; f(a + x) = f(a x), f(x). x = a ; (a, 0). (3) : (i) R f(x), g(x) h(x), f(x) = g(x) + f(x) f( x) f(x) + f( x) h(x),, g(x) =, h(x) =. (ii)f(a + x) = f(a x) f(x) = f(a x) f( x) = f(a + x); f(a + x) = f(a x) f(x) = f(a x) f( x) = f(a + x). (iii) f(x)(x R) x = a F (x), F (x) F (x) = f(a x). x). f(x)(x R) (a, 0) F (x), F (x) F (x) = f(a.. 4
9 f(x) D, t, f(x) : (1) x D, x + t D; ()f(x + t) = f(x). f(x), t. : (1) D. () t f(x), nt(n Z) f(x). (3)f(x),,. (4) λ 0. f(x) D x, : (i)f(x + λ) = f(x); (ii)f(x + λ) = 1 f(x) ; (iii)f(x + λ) = 1 f(x) ; (iv)f(x + λ) = f(x) + 1 f(x) 1 ; (v)f(x + λ) = 1 f(x) 1 + f(x) ; (vi)f(x + λ) = f(x λ); (vii)f(x), f(λ + x) = f(λ x); (viii)f(x), f(λ + x) = f(λ x). f(x) λ. (5) λ 0, f(x) D x, : (i)f(x + λ) = f(x λ); (ii)f(x), f(λ + x) = f(λ x); (iii)f(x), f(λ + x) = f(λ x); (iv)f(x + λ) = 1 + f(x) 1 f(x) ; (v)f(x + λ) = f(x) 1 f(x) + 1. f(x) 4λ... 5
30 f(x) R. (1) f(a + x) = f(a x) f(b + x) = f(b x), f(x) x = a, x = b, f(x), b a. () f(a + x) = f(a x) f(b + x) = f(b x), f(x) (a, 0) (b, 0), f(x), b a. (3) f(a + x) = f(a x) f(b + x) = f(b x), f(x), f(x), 4 b a... 6 f(x) D. x 1 x D, α [0, 1], f ( αx 1 + (1 α)x ) αf(x1 ) + (1 α)f(x ), f(x) D ; x 1 x D, α [0, 1], f ( αx 1 + (1 α)x ) αf(x1 ) + (1 α)f(x ), f(x) D. α = 1, f ( x1 + x. 3. 3. 1 ) f(x 1) + f(x ) f ( ) x1 + x f(x 1) + f(x ). f(x) D D, f (0) (x) = x, f (1) (x) = f(x), f () (x) = f ( f(x) ),, f (n) (x) = f ( f (n 1) (x) ), f (n) (x) f(x) D n, n. f(x) f 1 (x), f (n) (x) f 1 (x) n, f ( n) (x).. 3. n(n ), f (n+1) (x) = f(x), f(x), n. N +,, n 0, f ( f (n0) (x) ) = f(x), n 0 f(x). n 0 f(x), n f(x) n 0 n.. 3. 3 (1)f(x) = x + c, f (n) (x) = x + nc, f ( n) (x) = x nc. ()f(x) = ax, f (n) (x) = a n x, f ( n) (x) = 1 a n x. (3)f(x) = ax, f (n) (x) = a n 1 x n, f ( n) (x) = a n 1 x n.
31 (4)f(x) = ax + b, f (n) (x) = a n ( x (5)f(x) = x 3, f (n) (x) = x 3n, f ( n) (x) = x 3 n.. 3. 4 f(x) f(x) = x f(x). b ) + b 1 a 1 a, f ( n) (x) = 1 ( a n x (1) x 0 f(x),, f (n) (x 0 ) = x 0, x 0 f (n) (x). () f (n) (x 0 ) = x 0, x 0 f(x) n-. n N +, f (n) (x 0 ) = x 0, x 0 f(x) n-. (3) k n, x 0 f(x) k-, x 0 f(x) n-. b ) + b 1 a 1 a. (4) x 0 f(x) k-, f(x) n-, x 0 f(x) (k, n)-, (k, n) k n.. 4... 3 3. 1 (1) f(x) = n a i x i = a n x n + a n 1 x n 1 + + a 1 x + a 0 (a n 0) x n i=0., n f(x), deg f. a i (i = 0, 1,, n), f(x). () f(x) = deg f + deg h. n i=0 a i x i, h(x) = k i=0 b i x i, deg(f ± h) max{deg f, deg h}, deg(fh) (3) n f(x 1, x,, x n ), i j(1 i < j n), f(x 1,, x i,, x j,, x n ) = f(x 1,, x j,, x i,, x n ),. (4)σ 1 = n j=1 x j, σ = n. 3. x j1 x j,, σ i = x j1 x j x ji,, σ n = x 1 x x n 1 j 1<j n 1 j 1<j < <j i n
3 3.. 1 1 f(x) = n a k x k, h(x) = m b k x k n = m a k = b k (k = 0, 1,, n). k=0 k=0 3.. f(x) g(x),, g(x) 0, q(x) r(x), f(x) = g(x)q(x) + r(x),, r(x) deg r(x) < deg g(x). r(x) = 0, g(x) f(x), g(x) f(x), g(x) f(x)., g(x) f(x). 3.. 3 3 f(x) x a f(a). 3.. 4 4 f(a) = 0 x a f(x). 3.. 5 5 q p ((p, q) = 1, p q Z, p 0), p a n, q a 0. 3.. 6 (Eisenstein ) f(x), p p a n, P a k (k = 0, 1,, n 1), p a 0,, f(x). 3.. 7 6 n(n 1). n(n 1) n (k k ). 3.. 8 7, f(x) f(x) = A(x α 1 ) m1 (x α ) m (x α t ) mt,, α 1, α,, α t f(x), m 1, m,, m t. 3.. 9 8 a 0 x n + a 1 x n 1 + + a n 1 x + a n = 0(a 0 0) n x 1, x,, x n, σ j = ( 1) j a j a 0 (j = 1,,, n)., σ j, 3.. 3. 3.. 10 σ j (j = 1,,, n), σ 0 = 1, S k = n x k i (k = 0, 1, ),
33 (1) k > n, n ( 1) i σ i S k i = 0; i=0 () 1 k n, k ( 1) i σ i S k i = 0. i=0 3.. 11 n f(x) = (x x 1)(x x ) (x x n ) (x 0 x 1 )(x 0 x ) (x 0 x n ) f(x 0) + (x x 0)(x x ) (x x n ) (x 1 x 0 )(x 1 x ) (x 1 x n ) f(x 1) + + (x x 0)(x x 1 ) (x x n 1 ) (x n x 0 )(x n x 1 ) (x n x n 1 ) f(x n),, x 1, x,, x n. f(x) = n x x j f(x i ). i=0 0 j n j i x i x j 4. 1 S 1, n = 1; S n = a 1 + a + + a n. a n = S n S n 1, n. 4. d, a n = a 1 + (n 1)d, S n = na 1 + 4. 3 4. 3. 1 n(n 1) d S n = (a 1 + a n )n. {a n },, {b n },, b n = a n+1 a n (n = 1,, ), {a n }. {b n }, {c n },, c n = b n+1 b n (n = 1,, ), {a n }.. p, p.,.,. 4. 3. 1 3 4 {a n } p, p 1. S (k) n = n p=1 p k (k = 1,,, n). S n (k) n k + 1. {a n } p : a n n p. {a n } p, S n n p + 1. 4. 4
34 a 1 (1 q n ), q 1; q 0, a 1 0, a n = a 1 q n 1, S n = 1 q na 1, q = 1. 4. 5 (1) a n = a 1 + (a a 1 ) + (a 3 a ) + + (a n a n 1 ), a n = a 1 + n 1 (a k+1 a k )(n ). () a n = a 1 a a 1 a3 a a n a n 1, a n = a 1 4. 6 4. 6. 1 n 1 k=1 a k+1 a k (a k 0, n ). (1) {a n } n a n k a n 1, a n,, a n k, (k < n, k n N + ) k=1 a n = f(a n 1, a n,, a n k ), k. a n a n 1, a n,, a n k k,. () k k a 1, a,, a k ( ) k. (3) a n = p 1 a n 1 + p a n + + p k a n k (k < n), k. k x k = p 1 x k 1 + p x k + + p k 1 x + p k (p k 0), {a n },. 4. 6. (1) a n+1 = a n + f(n). a n = a 1 + n (a k a k 1 ) = a 1 + n k= k= f(k 1) = a 1 + n 1 () a n+1 = pa n +q. {a n q 1 p }, {a n q 1 p } a 1 q 1 p p. a n = q ( 1 p + a 1 q ) p n 1. 1 p (3) a n+1 = qa p n. k=1 f(k).., lg a n+1 = lg q + p lg a n. b n = lg a n, b n+1 = pb n + lg q, () (4) a n+1 = pa n + qa n 1. x = px + q. α β, (i) α β, a n = λ 1 α n 1 + λ β n 1,, λ 1 λ a 1 a.
35 (ii) α = β, a n = (λ 1 n + λ )α n 1,, λ 1 λ a 1 a. (5) a n+1 = ba n 1. ca n + d a n+1 d b t n + c b (). = ca n + d ba n = (6). 4. 7 d b a n + c b. t n = 1 a n, t n+1 = {a n } : M T, n(n M), a n+t = a n, {a n } M T. a n+t a n (modm), {a n } m. 5 5. 1 (1). n H n = 1 + 1 + + 1, G n = n a 1 + a + + a n a a1 a a n, A n =, Q n = 1 + a + + a n. n n a 1 a a n a i > 0(i = 1,,, n), H n G n A n Q n a 1, a,, a n. H n G n A n Q n, a 1 = a = = a n,. ( a r () M r = 1 + a r + + a r ) 1 r n,, ai > 0(i = 1,,, n, r 0), M r a 1, a,, a n r n. ( a α α > β, M α M β, 1 + a α + + a α ) 1 ( ) α n a β 1 1 + aβ + + aβ β n, a 1 = a = n n = a n,. 5. ( n ) ( n a i b i R(i = 1,,, n). a i b i λa i (i = 1,,, n),. 5. 3 a i ) ( n b i ), a i b i 0, b i = a 1, a,, a n ; b 1, b,, b n, a 1 a a n, b 1 b b n. a 1 b n + a b n 1 + + a n b 1 a 1 b j1 + a b j + + a n b jn a 1 b 1 + a b + a n b n.. 5. 4 (Jensen)
36 p i R + (i = 1,,, n), f(x) D. x 1, x,, x n D, ( ) p1 x 1 + p x + + p n x n f p 1f(x 1 ) + p f(x ) + + p n f(x n ). p 1 + p + + p n p 1 + p + + p n f(x) D, ( ) p1 x 1 + p x + + p n x n f p 1f(x 1 ) + p f(x ) + + p n f(x n ). p 1 + p + + p n p 1 + p + + p n, p i = 1 (i = 1,,, n), n ( ) x1 + x + + x n f f(x 1) + f(x ) + + f(x n ) n n ( ) x1 + x + + x n f f(x 1) + f(x ) + + f(x n ). n n 6 6. 1,,,,,,. 6.,,, ( ). 6. 3 : (1) ( ). () (i)tan A + tan B + tan C = tan A tan B tan C( ). (ii)sin A + sin B + sin C = 4 cos A cos B cos C. (iii)cos A + cos B + cos C = 4 sin A sin B sin C + 1. (iv)sin A + sin B + sin C = 1 + 4 sin π A sin π B 4 4 ( = 1 + 4 sin B + C sin C + A sin A + B ). 4 4 4 (v)cos A + cos B + cos C = 4 cos A + B 4 cos B + C 4 (vi)sin A + sin B + sin C = (1 + cos A cos B cos C). (vii)cos A + cos B + cos C = 1 cos A cos B cos C. (3) (i)sin A + sin B + sin C 3 3. (ii)sin A sin B sin C 3 3 8. (iii)cos A + cos B + cos C 3. sin π C 4 cos C + A. 4
37 (iv)cos A + cos B + cos C 3 4. (v)cos A cos B cos C 3 3 8. (vi)cos A cos B cos C 1 8. 7 7. 1 (1) : z = a + bi(a b R). () : z = r(cos θ + i sin θ)(r 0, θ R). (3) : z = re iθ (r 0, θ R)., a z, a = Re(z), b z, b = Im(z), r z, r = z = a + b, θ z, θ = Argz, θ [0, π), z, θ = arg z., Argz = kπ + arg z(k Z). 7. z = a bi z. (1)z 1 ± z = z 1 ± z. ()z 1 z = z 1 z. ( ) z1 (3) = z 1 (z 0). z z (4)z = z z R. (5)Re(z) = 1 (z + z), Im(z) = 1 (z z). i (6)z z = z = z. (7) z 1 z = z 1 z. (8) z 1 + z + z 1 z = z 1 + z. (9) z 1 z z 1 ± z z 1 + z. (10) z max{re(z), Im(z)}. (11) z Re(z) + Im(z) z. 7. 3 z n = [r(cos θ + i sin θ)] n = r n (cos nθ + i sin nθ)(n Z).
38, z = 1, cos nθ = Re(z n ) = 1 (zn + z n ), sin nθ = Im(z n ) = 1 i (zn z n ), 1 + z = cos θ ei θ, 1 z = i sin θ ei θ. 7. 4 x n 1 = 0 n : 1, ε, ε,, ε n 1 n., ε = e i π n n. n : (1)ε nq+r = ε r (n q r Z). ()1 + x + x + + x n 1 = (x ε)(x ε ) (x ε n 1 ). (3) n 1 n, n m; ε km = k=0 0, n m. ax + bx + c = 0(a 0) = b 4ac 0, x 1, = b ± ; a = b 4ac < 0, x 1, = b ± i. a 7. 5 z M OM, z 1 z A B. (1) z 1 z A B. () z z 1 = z z M AB. (3) z z 1 = r M A r. (4) z z 1 + z z = a(a > z 1 z ) M A B a. (5) z z1 z z = a(a < z1 z ) M A B a. (6) z 1 z z z = λ M AB λ. (7) z 1 z z 3 A B C. z = z 1 + z + z 3 ABC. 3 (8) AMB = arg z z z 1 z. (9) z 1 z z 3 z 4 A B C D. z z 1 z 4 z 3 = k AB CD, z z 1 z 4 z 3 = ki AB CD, z z 1 z z 3 R A B C.