IMO 1 n, 21n n (x + 2x 1) + (x 2x 1) = A, x, (a) A = 2, (b) A = 1, (c) A = 2?, 3 a, b, c cos x a cos 2 x + b cos x + c = 0 cos 2x a
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- ひでか やぶき
- 5 years ago
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1 1 40 ( ) (IMO) 41 (2000 ) WEB 1
2 IMO 1 n, 21n n (x + 2x 1) + (x 2x 1) = A, x, (a) A = 2, (b) A = 1, (c) A = 2?, 3 a, b, c cos x a cos 2 x + b cos x + c = 0 cos 2x a = 4, b = 2, c = 1, cos x cos 2x 4 c,, 5 AB M AMCD MBEF AB, AM MB AMCD MBEF P, Q, M N AF BC N (a) N N (b) MN, M, S (c) M A B, P Q 6 P Q p A P, C Q,, A B p AB CD ABCD,, B, D P, Q 2
3 IMO 1 N, N 11,, N/11 N N 2 x? 4x 2 ( x) 2 < 2x ABC a BC n n n BC P Q, α = P AQ, A BC h tan α = 4nh (n 2 1)a 4 ABC, A, B h a, h b, A BC m a 5 ABCDA B C D ( ABCD A B C D ) (a) X AB, Y B D, XY (b), XY 1 : 2 Z 6,,, V 1, V 2 (a) V 1 V 2 (b), V 1 = kv 2 k, k, 7 ABCD AB CD, CD = a, AB = c h (a) P, BP C = 90 (b) P (c) P 3
4 IMO 1 x + y + z = a x 2 + y 2 + z 2 = b 2 xy = z 2 a, b, x, y, z, a, b 2 a, b, c, T a 2 + b 2 + c 2 4 3T, 3 x cos n x sin n x = 1 n 4 P 1 P 2 P 3 P P 1 P, P 2 P, P 3 P Q 1, Q 2, Q 3 P 1 P, P 2P, P 3P, 2, 2 P Q 1 P Q 2 P Q 3 5 AC = b, AB = c, AMB = ω ABC, M BC,, b tan ω 2 c < b,,? 6 ε, ε A, B, C, A, B, C ε ε A, B, C L, M, N AA, BB, CC, G LMN (, A, B, C LMN ) A, B, C ε, G 4
5 IMO 1 n (a) n 6 (b) n 6, n 2 x 3 x x + 1 > ABCDA B C D (ABCD, A B C D,, AA, BB, CC, DD ) X ABCD ABCDA, Y X B C CB B C CBB X Y A, B, XY 4 cos 2 x + cos 2 2x + cos 2 3x = 1 5 K A, B, C K D, ABCD, ( ) 6 r, ρ d, d = r(r 2ρ) 7 SABC :,, SA, SB, SC, BC, CA, AB (a) SABC (b), 5
6 IMO 1 x2 p + 2 x 2 1 = x x p 2 A BC,, A, BC 3 n, a 1, a 2,, a n, a 1 a 2 a n, a 1 = a 2 = = a n 4 x 5 + x 2 = yx 1 x 1 + x 3 = yx 2 x 2 + x 4 = yx 3 x 3 + x 5 = yx 4 x 4 + x 1 = yx 5 x 1, x 2, x 3, x 4, x 5 y 5 cos π 7 cos 2π 7 + cos 3π 7 = A, B, C, D, E ABCDE,,,, DAECB,,,,? 6
7 IMO 1 (a) 2 n 1 7 n (b) 2 n n 2 a, b, c a 2 (b + c a) + b 2 (c + a b) + c 2 (a + b c) 3abc 3 a, b, c ABC ABC ABC, a, b, c 4 17 ( ),,,, 5,,,, ( ) 6 ABCD D ABC D 0 A, B, C DD 0, 3 BCD, CAD, ABD A 1, B 1, C 1, 1 ABCD, A 1 B 1 C 1 D 0 3, D 0 ABC,? 7
8 IMO 1 0 x 2π x 2 cos x 1 + sin 2x 1 sin 2x 2 2 x 1, x 2, x 3 a 11 x 1 + a 12 x 2 + a 13 x 3 = 0 a 21 x 1 + a 22 x 2 + a 23 x 3 = 0 a 31 x 1 + a 32 x 2 + a 33 x 3 = 0 (a) a 11, a 22, a 33 (b) (c),, x 1 = x 2 = x 3 = 0 3 ABCD, AB, CD a, b AB CD d, ω AB CD ε, ABCD ε AB, CD k, 4 x 1, x 2, x 3, x 4,,, 2 5 AOB < 90 OAB M O M OA, OB P, Q OP Q H M (a) AB, (b) OAB, H 6 n (n 3) d d,, n 8
9 IMO 1 A, B, C 25 A, B C A, A, A, B? 2 a, b, c, α, β, γ, a + b = tan γ (a tan α + b tan β) 2, 3, 4 n x kπ/2 t (t = 0, 1,, n k ), 1 sin 2x + 1 sin 4x sin 2 n x = cot x cot 2n x 5 a 1 a 2 x 2 + a 1 a 3 x 3 + a 1 a 4 x 4 = 1 a 2 a 1 x 1 + a 2 a 3 x 3 + a 2 a 4 x 4 = 1 a 3 a 1 x 1 + a 3 a 2 x 2 + a 3 a 4 x 4 = 1 a 4 a 1 x 1 + a 4 a 2 x 2 + a 4 a 3 x 3 = 1 a 1, a 2, a 3, a 4 6 ABC BC, CA, AB, K, L, M AML, BKM, CLK, ABC 1/4 9
10 IMO 1 AB = a, AD = 1, BAD = α ABCD ABD, A, B, C, D 1, a cos α + 3 sin α 2, 1, 1/8 3 k, m, n, m + k + 1 n + 1 c s = s(s + 1), (c m+1 c k )(c m+2 c k ) (c m+n c k ) c 1 c 2 c n 4 A 0 B 0 C 0 A 1 B 1 C 1 A 1 B 1 C 1, A 0 B 0 C 0 ( A 0 BC, B 0 CA, C 0 AB ) ABC, 5 {c n }, c 1 = a 1 + a a 8 c 2 = a a a 2 8 c n = a n 1 + a n a n 8, a 1, a 2,, a 8, 0, {c n } 0, c n = 0 n 6, n (n > 1), m,, m 1 1/7,, 1/7, n n,,? 10
11 IMO 1,, 2 x, ( ) x 2 10x 22 x 3 x 1, x 2,, x n ax bx 1 + c = x 2 ax bx 2 + c = x 3 ax 2 n 1 + bx n 1 + c = x n ax 2 n + bx n + c = x 1 a, b, c a 0 = (b 1) 2 4ac (a) < 0, (b) = 0, (c) > 0, 4,, 5 f x,, a, x f(x + a) = f(x) (f(x)) 2 (a) f (, b, x f(x+b) = f(x) ) (b) a = 1, 6 n, [ ] [ ] n + 2 k n k+1 = + 2 k=0 [ n ] + + [ n + 2 k 2 k+1 ] + 11
12 IMO 1 a : z = n 4 + a n 2 a 1, a 2,, a n, x, f(x) = cos(a 1 + x) cos(a 2 + x) cos(a 3 + x) n 1 cos(a n + x) f(x 1 ) = f(x 2 ) = 0, m x 2 x 1 = mπ 3 k a, 6 k 1, a > 0, k = 1, 2, 3, 4, 5, 4 AB γ C γ, A, B, D C AB AB γ 1, γ 2, γ 3 γ 1 ABC γ 2 γ 3, CD γ, CD γ 1, γ 2, γ 3 AB 5 n (n > 4) (, ), n 3, 2 6 x 1 > 0, x 2 > 0, x 1 y 1 z1 2 > 0, x 2 y 2 z2 2 > 0 x 1, x 2, y 1, y 2, z 1, z 2, 8 (x 1 + x 2 )(y 1 + y 2 ) (z 1 + z 2 ) 2 1 x 1 y 1 z x 2 y 2 z 2 2, 12
13 IMO 1 M ABC AB r 1, r 2, r AMC, BMC, ABC, q 1, q 2, q ACB r 1 r2 = r q 1 q 2 q 2 a, b, n 1, a b a A n 1 A n,, b B n 1 B n,, A n = x n x n 1 x 0, A n 1 = x n 1 x n 2 x 0, B n = x n x n 1 x 0, B n 1 = x n 1 x n 2 x 0, A n 1 A n (, x n 0, x n 1 0) < B n 1 B n a > b 3 a 1, a 2,, a n, 1 = a 0 a 1 a 2 a n b 1, b 2,, b n, n ( b n = 1 a ) k 1 1 a k ak k=1 (a) n 0 b n < 2 (b) 0 c < 2 c, a 0, a 1,, n b n > c 4 { n, n + 1, n + 2, n + 3, n + 4, n + 5 },,, n 5 ABCD, BDC, D ABC H, ABC, (AB + BC + CA) 2 6(AD 2 + BD 2 + CD 2 ),? 6 100, 70% 13
14 IMO 1 n = 3 n = 5, n > 2 : a 1, a 2,, a n, (a 1 a 2 )(a 1 a 3 ) (a 1 a n ) + (a 2 a 1 )(a 2 a 3 ) (a 2 a n ) + + (a n a 1 )(a n a 2 ) (a n a n 1 ) A 1, A 2,, A 9 P 1 P i (i = 2, 3,, 9) A 1 A i P 1 P 1, P 2,, P 9, 3 2 k 3 (k = 2, 3, ),, 4 ABCD XY ZT X : X AB A, B, Y, Z, T BC, CD, DA (a) DAB + BCD CDA + ABC, (b) DAB + BCD = CDA + ABC,,, 2AC sin(α/2), α = BAC + CAD + DAB 5 m,, S : S A, S m, A 1 6 A = (a ij ) (i, j = 1, 2,, n), n, a ij = 0, i j (2n 1 ) n, A n 2 /2 14
15 IMO 1 10,, 2 n 4,,, n (2m)!(2n)! 3 m, n, m!n!(m + n)! 4 (x 1, x 2, x 3, x 4 ) (x 2 1 x 3 x 5 )(x 2 2 x 3 x 5 ) 0 (x 2 2 x 4 x 1 )(x 2 3 x 4 x 1 ) 0 (x 2 3 x 5 x 2 )(x 2 4 x 5 x 2 ) 0 (x 2 4 x 1 x 3 )(x 2 5 x 1 x 3 ) 0 (x 2 5 x 2 x 4 )(x 2 1 x 2 x 4 ) 0, x 1, x 2, x 3, x 4, x 5 5 f, g,, x, y, f(x + y) + f(x y) = 2f(x)g(y) f(x) 0,, x f(x) 1, y g(y) 1 6, 15
16 IMO 1 O g, OP 1, OP 2,, OP n, P 1, P 2,, P n g,, g (, P i P j g ) n, OP 1 + OP OP n 1 2 M : M, M A, B, M C, D, AB CD 3 a, b, x 4 + ax 3 + bx 2 + ax + 1 = 0 (a, b) a 2 + b 2 4,, 5 f(x) = ax + b (a 0, b ) G (a) f, g G g f G (b) f G f 1 G f 1 f(x) = ax + b f 1 (x) = (x b)/a (c) f G, x f, f(x f ) = x f, k, f G f(k) = k 6 a 1, a 2,, a n n, q 0 < q < 1 n b 1, b 2,, b n (a) a k < b k (k = 1, 2,, n) (b) q < b k+1 < 1 (k = 1, 2,, n 1) b k q (c) b 1 + b b n < 1 + q 1 q (a 1 + a a n ) 16
17 IMO 1 A, B, C, p, q, r 0 < p < q < r,,,,, A 20, B 10, C 9 B r, q? 2 ABC, CD AD DB D AB, sin A sin B sin 2 C 2 n ( ) 2n n 0, 2 3k 5 2k + 1 k= ,, p ( ) (i) (ii) i a i, a 1 < a 2 < < a p p, p, a 1, a 2,, a p a 5 S = a + b + d + b a + b + c + c b + c + d + d a + c + d, a, b, c, d 6 P, n(p ) (P (k)) 2 = 1 k ( ) n(p ) deg P 2 deg P P 17
18 IMO 1 x i, y i (i = 1, 2,, n), x 1 x 2 x n, y 1 y 2 y n z 1, z 2,, z n y 1, y 2,, y n ( ), n (x i y i ) 2 i=1 n (x i z i ) 2 i=1 2 a 1, a 2, a 3, (, i < j a i < a j ) p 1, a m = xa p + ya q (x, y, q q > p ) a m 3 ABC ABR, BCP, CAQ, BCP = CAQ = 45, BCP = ACQ = 30, ABR = BAR = 15, QRP = 90, QR = RP A, A B B ,, 6 P (i) n t, x, y, P (tx, ty) = t n P (x, y) (ii) a, b, c P (b + c, a) + P (c + a, b) + P (a + b, c) = 0 (iii) P (1, 0) = 1 18
19 IMO 1 32, 16 2 P 1 (x) = x 2 2, j = 2, 3, P j (x) = P 1 (P j 1 (x)), n, P n (x) = x, ( ) 3, 1,, 2,,, 40%, , 5 q = 2p x 1, x 2,, x q p a 11 x 1 + a 12 x a 1q x q = 0 a 21 x 1 + a 22 x a 2q x q = 0 a p1 x 1 + a p2 x a pq x q = 0, a ij { 1, 0, 1 }, (x 1, x 2,, x q ) (a) x j (j = 1, 2,, q) (b) j x j 0 (c) x j q (j = 1, 2,, q) 6 { n n } u 0 = 2, u 1 = 5/2, u n+1 = u n (u 2 n 1 2) u 1 (n = 1, 2, ), n, [u n ] = 2 [2n ( 1) n ]/3, [x] x 19
20 IMO 1 ABK, BCL, CDM, DAN ABCD KL, LM, MN, NK, AK, BK, BL, CL, CM, DM, DN, AN, 2, 7, 11 3 n n > 2, V n 1 + kn ( k = 1, 2, ) V n m, pq = m V n p, q, r V n, r V n,,, 4 a, b, A, B, f(θ) = 1 a cos θ b sin θ A cos 2θ B sin 2θ θ f(θ) 0, a 2 + b 2 2, A 2 + B a, b a 2 + b 2 a + b q, r q 2 + r = 1977 (a, b) 6 f(n),, n f(n + 1) > f(f(n)), n f(n) = n 20
21 IMO 1 m, n 1 m < n,, 1978 m, 1978 n m, n m + n 2 P P, U, V, W, Q P U, P V, P W P P, Q 3 { f(1), f(2),, f(n), } { g(1), g(2),, g(n), }, f(1) < f(2) < < f(n) <, g(1) < g(2) < < g(n) <,, n 1 g(n) = f(f(n)) + 1, f(240) 4 AB = AC ABC, ABC,, AB, AC P, Q, P Q ABC 5 {a k } (k = 1, 2, 3,, n, ), n, n k=1 a k k 2 n k=1 1 k , 1, 2,, 1978,,,, 21
22 IMO 1 p, q p q = , p A 1 A 2 A 3 A 4 A 5 B 1 B 2 B 3 B 4 B 5, A i B j (i, j = 1,, 5),,,,, 10 3 A A,,,, A, P,, P 4 π, P, π Q, π R (QP + P R)/QR 5 x 1, x 2, x 3, x 4, x 5, a kx k = a, k 3 x k = a 2, k 5 x k = a 3 k=1 k=1 k=1 6 A, E A E, E, a n n A E, a 2n 1 = 0, a 2n = 1 (x n 1 y n 1 ), (n = 1, 2, 3, ), x = 2 + 2, y =
23 IMO 1 P ABC, D, E, F P BC, CA, AB P, BC P D + CA P E + AB P F P 2 r 1 r n, { 1, 2,, n } r F (n, r),, F (n, r) = n + 1 r m, n m, n { 1, 2,, 1981 } (n 2 mn m 2 ) 2 = 1 m 2 + n 2 4 (a) n,, n 1, n > 2 (b), n > 2 5 O,,,, O 6 f(x, y) x, y, (1) f(0, y) = y + 1 (2) f(x + 1, 0) = f(x, 1) (3) f(x + 1, y + 1) = f(x, f(x + 1, y)) f(4, 1981) 23
24 IMO 1 f(x) n,, m, n f(m + n) f(m) f(n) = 0 1 f(2) = 0, f(3) > 0, f(9999) = 3333, f(1982) 2 a 1, a 2, a 3 A 1 A 2 A 3 (a i A i ) i = 1, 2, 3, M i a i, T i a i, A i T i S i, M 1 S 1, M 2 S 2, M 3 S 3 3 {x n } x 0 = 1, i 0 x i+1 x i (a), n 1 (b) n x 2 0 x 1 + x2 1 x x2 n 1 x n 3999 x 2 0 x 1 + x2 1 x x2 n 1 x n < 4 4 n x 3 3xy 2 + y 3 = n (x, y),, n = 2891, 5 ABCDEF AC, CE M, N AM AC = CN CE = r, B, M, N, r 6 S 100, L S, L, A 0 A 1, A 1 A 2,, A n 1 A n (A 0 A n ), S P, L P 1/2,, L X, Y, X Y 1,, X Y L 198, 24
25 IMO 1 f (i) f(xf(y)) = yf(x) x, y (ii) x f(x) 0 2 C 1, C 2 O 1, O 2, A C 1 C 2 C 1 C 2, C 1 P 1 C 2 P 2, C 1 Q 1 C 2 Q 2 M 1 P 1 Q 1, M 2 P 2 Q 2, O 1 AO 2 = M 1 AM 2 3 a, b, c, 2abc ab bc ca xbc + yca + zab ( x, y, z ) 4 ABC AB, BC, CA (A, B, C ) E E,,? , 3? 6 a, b, c, a 2 b(a b) + b 2 c(b c) + c 2 a(c a) 0? 25
26 IMO ( ) 1 x, y, z x + y + z = 1, 0 yz + zx + xy 2xyz (i), (ii) a, b (i) ab(a + b) 7 (ii) (a + b) 7 a 7 b O, A O X, OA OX a(x) (0 a(x) < 2π) C(X), O OX + a(x) OX,, Y, a(y ) > 0, C(Y ) Y 4 ABCD, CD AB, AB CD, BC AD 5 d n (n > 3), p n 3 < 2d p < [ n 2 ] [ ] n , [x] x 6 a, b, c, d 0 < a < b < c < d ad = bc, k, m, a+d = 2 k b + c = 2 m, a = 1 26
27 IMO ( ) 1 ABCD AB ( ),, AD + BC = AB 2 n, k k < n M = { 1, 2,, n 1 },, (i) i M, i n i (ii) i k i M, i i k, M 3 P (x) = a 0 + a 1 x + + a k x k, w(p ), i = 0, 1, 2,, Q i (x) = (1 + x) i i 1, i 2,, i n 0 i 1 < i 2 < < i n, w(q i1 + Q i2 + + Q in ) w(q i1 ) M, M, 5 ABC A, C, O, AB, BC (A, C ) K, N, ABC, KBN B, M, OMB ( 6 x 1, x 1, x 2,, n 1, x n+1 = x n x n + 1 ) n, x 1, n 0 < x n < x n+1 < 1 27
28 IMO 1 d 2, 5, 13, { 2, 5, 13, d } a, b ab 1 2 A 1 A 2 A 3, P 0 s 4, A s = A s 3, P 1, P 2, : P k+1 P k A k+1, 120 (k = 0, 1, ), P 1996 = P 0, A 1 A 2 A 3 3 x, y, z y < 0 : x, y, z x + y, y, z + y 4 A, B O n (n 5) OAB XY Z OAB XY Z : X n, Y, Z n, X 5 f, (1), (2), (3) (1) f(xf(y))f(y) = f(x + y) (x, y 0) (2) f(2) = 0 (3) 0 x < 2 f(x) 0 f 6, A A,A, ( )? ( ) (x y ) l, l ( ) 28
29 IMO 1 { 1, 2,, n } (n 1), k P n (k), n kp n (k) = n! k=0 2 ABC, A BC L, A ABC N, L AB, AC K, M, ABC AKNM 3 x x x 2 n = 1 x 1, x 2,, x n, k ( 2), (1), (2), (3) a 1, a 2,, a n (1) (a 1, a 2,, a n ) (0, 0,, 0) (2) a i k 1 (i = 1, 2,, n) (3) a a x 1 + a 2 x a n x n (k 1) n k n 1 4 Z 0 f: Z 0 Z 0, n Z 0 f(f(n)) = n n 3, n,, n 6 n 2 0 k 3 k, k2 + k + n, 0 k n 2 k, k 2 + k + n 29
30 IMO 1 R, r (R > r) C 1, C 2 C 2 P, C 1 B BP C 1 B C, P BP l C 2 A, BP C 2 A = P (1) BC 2 + CA 2 + AB 2 (2) AB 2 n B B A 1, A 2,, A 2n+1 (a), (b), (c) (a) A i 2n (b) A i A j (1 i < j 2n + 1) (c) B A i, B 0 1, A i, n 0 n 3 f: N N f(1) = 1, f(3) = 3 f(2n) = f(n) f(4n + 1) = 2f(2n + 1) f(n) f(4n + 3) = 3f(2n + 1) 2f(n) f(n) = n n 1988 n 70 k 4 x k 5 4 x, k= ABC A = 90 A BC D ABD ACD O 1, O 2 AB, AC K, L ABD AKL S, T, S 2T 6 a, b a 2 + b 2 ab + 1, a2 + b 2 ab
31 IMO 1 { 1, 2,, 1989 } (i), (ii) 117 A i (i = 1, 2,, 117) (i) A i 117 (ii) A i ( ) 2 ABC A ABC A 1 B 1, C 1 B, C AA 1 A 0 B 0, C 0, (i) A 0 B 0 C 0 AC 1 BA 1 CB 1 (ii) A 0 B 0 C 0 ABC 3 n, k (i), (ii) n S (i) S (ii) S P P S k, k < n 4 ABCD, C D AB, AB, AD, BC AB = AD + BC CD h, AP = h + AD, BP = h + BC P, h AD BD 5 n k + 1, k + 2,, k + n k 6 n { 1, 2,, 2n } (x 1, x 2,, x 2n ) P, i { 1, 2,, 2n 1 }, x i x i+1 = n n, P P 31
32 IMO 1 AB, CD E M EB E, B D, E, M E DEM BC, AC AM F, G AB = t, EG t EF 2 n 3 2n 1 E E k, :,, n E E k, k 3 2n + 1 n 2 n 4 Q +, f: Q + Q + : : x, y Q +, f(xf(y)) = f(x) y 5 n 0 > 1 A B n 1, n 2, n 3, : A n 2k, n 2k n 2k+1 n 2 2k n 2k+1 B n 2k+1, n 2k+1 n 2k+2 n 2k+2 A 1990 A, B 1 B, (a), (b), (c) n 0 (a) B, A (b) A, B (c) ( ) 6 (a), (b) 1990 (a) (b) , 2 2, 3 2,, ,
33 IMO ( ) 1 ABC ABC I, A, B, C A, B, C, 1 AI BI CI < 4 AA BB CC n n > 6 n, n a 1, a 2,, a k,, a 2 a 1 = a 3 a 2 = = a k a k 1 > 0, n 2 (2 ) 3 S = { 1, 2, 3,, 280 } n, : S n (n S ), n, ( ) 4 G k, G 1 k, :, ( ) ( ) 5 ABC, P, P AB, P BC, P CA 30 6 x 0, x 1, x 2,, C, i 0, x i C a (> 1), : x 0, x 1, x 2, i, j x i x j i j a 1 33
34 IMO ( ) 1 (a 1)(b 1)(c 1) abc 1 a, b, c, 1 < a < b < c, 2 R x, y R, f(x 2 + f(y)) = y + (f(x)) 2 f: R R ( ) ( ) n : n 4 C l l M P l Q, R, M QR C P QR 5 S S x, S y, S z S yz-, zx-, xy- S 2 S x S y S z A A 6 n S(n) : k S(n) n 2 k a) n 4, S(n) n 2 14 b) S(n) = n 2 14 n c) S(n) = n 2 14 n 34
35 IMO ( ) 1 f(x) = x n + 5x n 1 + 3, n 1 f(x) 2 D ABC ADB = ACB + 90, AC BD = AD BC AB CD (a) AC BD (b) ACD, BCD C 3 n 2 n n, n 4 P, Q, R PQR m(pqr) ( P, Q, R m(pqr) = 0 ) A, B, C X m(abc) m(abx) + m(axc) + m(xbc) 5 N = {1, 2, 3, } f: N N f(1) = 2 f(f(n)) = f(n) + n n N f(n) < f(n + 1) n N 6 n 1 n L 0, L 1,, L n 1 ON OFF S 0, S 1,, S i, S j L j ( ) L j 1 ON S j L j ON, OFF L j 1 OFF S j L j modulo n L 1 = L n 1, L 0 = L n, L 1 = L n+1, ON (a) M(n), M(n) ON (b) n 2 k, n 2 1 ON (c) n 2 k + 1, n 2 n + 1 ON 35
36 IMO 1 m, n a 1, a 2,, a m {1, 2,, n} a i +a j n (1 i j m) a i + a j = a k k (1 k m) a 1 + a a m m n AB = AC ABC (i) M BC O AM OB AB (ii) Q BC B, C (iii) E AB F AC E, Q, F OQ EF QE = QF 3 k f(k) {k + 1,k + 2,, 2k} 1 3 (a) m f(k) = m k (b) f(k) = m k m 4 n mn 1 (m, n) 5 S 1 f: S S (i) S x, y f(x + f(y) + xf(y)) = y + f(x) + yf(x) (ii) f(x) x 1 < x < 0 0 < x 6 A S k 2 m A n A m, n S k 36
37 IMO ( ) 1 A, B, C, D, AC BD X Y XY BC Z P XY, Z AC CP C M, BD BP B N AM, DN, XY 1 2 a, b, c, abc = 1, a 3 (b + c) + 1 b 3 (c + a) + 1 c 3 (a + b) n A 1, A 2,, A n r 1, r 2,, r n, (i), (ii) n > 3 (i) A 1, A 2,, A n (ii) i, j, k (1 i < j < k n), A i A j A k r i + r j + r k 4 { x 0, x 1,, x 1995 }, x 0 (1) x 0 = x 1995 (ii) i = 1, 2,, 1995 x i = 2x i + 1 x i 1 x i 5 ABCDEF, AB = BC = CD, DE = EF = F A, BCD = EF A = 60 G H ABCDEF, AGB = DHE = 120 AG+GB+GH+DH+HE CF 6 p { 1, 2,, 2p } A (i) A p, (ii) A p 37
38 IMO ( ) 1 ABCD AB = 20, BC = , 1 r,, r,,, A, B (a) r 2 3, (b) r = 73, (c) r = 97,? 2 P ABC, AP B ACB = AP C ABC, D, E AP B, AP C, AP, BD, CE 3 S = {0, 1, 2, 3, } S S f f(m + f(n)) = f(f(m)) + f(n) S m, n 4 a b, 15a + 16b 16a 15b 5 ABCDEF, AB ED, BC F E, CD AF R A, R C, R E, F AB, BCD, DEF, p, R A + R C + R E p 2 6 n, p, q n > p + q x 0, x 1,, x n (a) x 0 = x n = 0; (b) 1 i n i, x i x i 1 = p x i x i 1 = q, i < j (i, j) (0, n) (i, j), x i = x j 38
39 IMO ( ) 1,,, m, n,,, m, n S 1,, S 2, f(m, n) = S 1 S 2 (a) m, n, f(m, n) (b) m, n, f(m, n) 1 max{m, n} 2 (c) C, m, n f(m, n) < C 2 A, ABC B, C,, U BC A AB, AC AU V, W BV CW T AU = T B + T C 3 x 1, x 2,, x n, x 1 + x x n = 1 i = 1, 2,, n x i n + 1 2, (x 1,, x n ) (y 1,, y n ) y 1 + 2y ny n n S = { 1, 2,, 2n 1 } n, i = 1,, n, i i S, (a) n = 1997 (b) n 5 a 1, b 1 (a, b), a b2 = b a 6 n, f(n), n 2,, f(4) = 4,, 4, 4, 2 + 2, , n 3 2 n2 /4 < f(2 n ) < 2 n2 /2 39
40 IMO 1 ABCD, AC, BD, AB, DC AB, DC P ABCD ABCD, ABP CDP 2 a b b 3, k, 2, 2 k k a b 1 2b 3 n, d(n), n (1 n ), d(n 2 ) d(n) = k n k 4 ab 2 + b + 7 a 2 b + a + b (a, b) 5 I ABC ABC BC, CA, AB K, L, M B MK, LM, LK R, S RIS 6 N, f, s, t N f ( t 2 f(s) ) = s ( f(t) ) 2, f(1998) 40
41 IMO 1, S, S A, B, AB S 2 n 2 n (a) ( ) 4 x i x j (x 2 i + x 2 j) C x i 1 i<j n 1 i n x 1,, x n 0 C (b) C, 3 n n n, n 2 N : ( ) N 4 (n, p) : p, n 2p, (p 1) n + 1 n p 1 5 Γ 1, Γ 2 Γ, Γ 1 Γ M, Γ 2 Γ N, M, N, Γ 2 Γ 1 Γ 1 Γ 2, Γ A, B, MA, MB Γ 1 C, D, CD Γ 2 6 f: R R : f(x f(y)) = f(f(y)) + xf(y) + f(x) 1 x, y ( ) 41
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高等学校学習指導要領解説 数学編
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取扱説明書 -詳細版- 液晶プロジェクター CP-AW3019WNJ
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名古屋工業大の数学 2000 年 ~2015 年 大学入試数学動画解説サイト
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数論入門
数学のかたち 共線問題と共点問題 Masashi Sanae 1 テーマ メネラウスの定理 チェバの定理から 共線問題と共点問題について考える 共線 点が同一直線上に存在 共点 直線が 1 点で交わる 2 内容 I. メネラウスの定理 1. メネラウスの定理とその証明 2. メネラウスの定理の応用 II. 3. チェバの定理とその証明 メネラウスの定理 チェバの定理の逆 1. メネラウスの定理の逆
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HITACHI 液晶プロジェクター CP-AX3505J/CP-AW3005J 取扱説明書 -詳細版- 【技術情報編】
B A C E D 1 3 5 7 9 11 13 15 17 19 2 4 6 8 10 12 14 16 18 H G I F J M N L K Y CB/PB CR/PR COMPONENT VIDEO OUT RS-232C LAN RS-232C LAN LAN BE EF 03 06 00 2A D3 01 00 00 60 00 00 BE EF 03 06 00 BA D2 01
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2000年度『数学展望 I』講義録
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学習の手順
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LINEAR ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University
LINEAR ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University 2002 2 2 2 2 22 2 3 3 3 3 3 4 4 5 5 6 6 7 7 8 8 9 Cramer 9 0 0 E-mail:hsuzuki@icuacjp 0 3x + y + 2z 4 x + y
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