êæ#( ekt;9 ÐK ) c#( G ügÿák ekt Ç g 2017 c FÚ 21 F, þ êæ#( G Þ1ügÿÁ ( ). zgÿáok, žm ü Œž. g0ùügÿáák). ò^k 1.x L«1 1 g

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1 êæ#( ekt;9 ÐK ) c#( G ügÿák ekt Ç g 2017 c FÚ 21 F, þ êæ#( G Þ1ügÿÁ ( ). zgÿáok, žm ü Œž. g0ùügÿáák). ò^k 1.x L«1 1 gÿá1 x K, K 2.y Âaq. Š Ñ, Ù ){dg ÆoR, nþ Æ';, är Æ㈠ÓÆJø, 3dL «a. I. Á K K 1.1 AB O», CD O ^R u AB u, E CO :, AE O u F, ã BC AF, DF Ou M, L, DLM O u K. y²: KM = MB. K 1.2 n 2 ½ê, é? v a k a a k 1, k = 2,, n ê a 1, a 2,, a n, ŒŠ. S = a 1 a 2 + a 2 a a n 1 a n K 1.3 {x i } i=1 ês, éz éê m, n k x mn x m(n+1). y²: 3 ê i x i K 1.4 A Ú B ü k 8, ve5ÿn f: A A ê: 3ü N g: A B Ú h: B A g(h(x)) = x, x B h(g(x)) = f(x), x A. K 2.1 n ê, n 2, y²: ny ÂvFÏ: ( k ) < 2. 1 êæ#(

2 K 2.2 I š ABC S%, Sƒ I BC, CA ƒ: O D, E, H ABI R%, K = AI BH, L = BI AH. y²: I DKH 9 ELH n:. K 2.3 X k 8. X = A 1 A 10, X = B 1 B 10 vxe5ÿü y: e A i B j =, 1 i, j 10, K A i B j 10. X Š. K 2.4 m ½ê, d Ïf. {a i } i=0 Ú {b i } i=0 ü dê ê, v: 3ê i, j, k, l (a i, b j ) = 1, (a k, b l ) = m. y²: 3ê t, s (a t, b s ) = d. /. ) K 1.1 AB O», CD O ^R u AB u, E CO :, AE O u F, ã BC AF, DF Ou M, L, DLM O u K. y²: KM = MB. e ){áuhžg ÆoRÓÆ. ÑK ^ CD AB õ{. y² ò KM O u: T, ë AK, KD, MO, T M, T O, T B, BK. Ï D, K, M, L o:, ± CMK = KDF, l KF ø = KC ø + T ø B, Ïd CF ø = BT ø, CT ø = BF ø. y MK = MB, Iy MO KB, =dy² MO AK OMK = AKM OMK = ABT O, M, T, B o: OMB = OT B OMB = OBT OMB = 1 2ø AT = 1 2 ( ø AC + ø CT ) 2

3 OMB = 1 2 ( ø AC + ø BF ) OMB = AMC CME BMO (Ï OBM = ECM). ± CM é OBC Ú AEM ^ Menelaus ½nŒ CM MB BA AO OE EC = 1. = 1 = CE, qï OBM = ECM, ± CME BMO. BM 2 BO µ5 ù KJÝ Ñ ýï, k 32% ÓÆ édk. dkã / E,, nú ê {éjs, éaûuå ép. à KövkuyK ^ CD AB õ{. þ, A, B, C, D ùo : ²(½ž, : E, F, M (½, ±: T (½, : K ( ½ (Œ±uy, : K : D Ã'). K 1.2 n 2 ½ê, é? v a k a a k 1, k = 2,, n ê a 1, a 2,, a n, ŒŠ. S = a 1 a 2 + a 2 a a n 1 a n ){ v ŒŠ n 2. y²ù, P A k = a a k, k = 0,, n 1, ½ A 0 = a 0 = 0. dk Œ, A k a k+1, k = 0,, n 1, S = n 1 X n 2 X X n 1 a k = a k+1 X n 2 A k A k 1 = a k+1 1 a k+1 ( 1 ) + 1 = a k+1 a k+2 = n 1 + a 1 a 2 n n 1 X n 1 X a k a k+1 a k a k+1 = n S. n 2 X 1 A k ( 1 ) + A n 1 a k+1 a k+2 (1 a k+1 ) + 1 a k+2 Ïd, S n 2, a k = 2 k 2 a 1, k = 2, 3,, n, Ù a 1 R + žò á. S ŒŠ n 2. e 0 ŸnÞ Æ';ÓÆy² S n 2 ){. a n ){ y S n 2. ^8B{y² r :(Ø: T n = a a n 2 a n 1 + n a 2 a n 1 a a n 1 2, n 2. 3 êæ#(

4 n = 2 ž, T n 1 w, á. n = 3 ž, T 3 = a 1+a 2 a 2 + a 2 a 1 +a 2 1, - a 1+a 2 a 2 = x, K x (1, 2]. ùž T 3 = x + 1 x = 3 2. b(øé n 1 (n 4) á, =k T n 1 n 1. eä n œ¹. 2 ùžeuy²e ت a n 2 a n 1 + Kd8BbÚ ( ) a n 1 a a n 1 a n 2 a a n , ( ) ù`²(øé n á. ey ( ). 5 ( ) du T n = T n 2 + a n 2 a n 1 + T n 2 + a n 1 a a n 1 a n a a n 2 2 = T n n 1 2 a n 2 (a a n 1 ) + a 2 n 1 a n 1 (a a n 1 ) P A = a a n 2, Kþªdu = n 2, 2a n 2 + a a n 2. 2(a a n 2 ) 2a n 2 A 2 + 2a n 2 a n 1 A + 2Aa 2 n 1 (2a n 2 + A)a 2 n 1 + (2a n 1 + A)Aa n 1, = (2a n 2 A)a 2 n 1 + A 2 a n 1 2a n 1 A 2. eyþª á: d^ a n 1 A, 2a n 2 A 0. (2a n 2 A)a 2 n 1 + A 2 a n 1 (2a n 2 A)A 2 + A 2 A = 2a n 2 A 2. ( ) y. µ5 dkjý L ýï, =k 15% ÓÆ é. þã){ 1 : Äk$^ Abel C?1ðC/, C/ ªfØU ŒŠ, ^^ ò { (Ï ØUä½ a n 1 a n ¹ S êª. þã){ 2 : ØU é S ^8B 1, ^8Bb(Jfu(Ø), æ^\r 2 8B{, ==y² r :(J T n n. ùbò K=z y² ( ) 2 4

5 ª. ( ) ªy²XJ?nØ ƒ, XJò a n 1 wšìc, ò = z 'u a n 1 gøªòøj?n. kõ ÓÆæ^þã\r8B{5 y²k. K 1.3 {x i } i=1 ê S, é z é ê m, n k x mn x m(n+1). y²: 3 ê i x i y² ê i < j /ëï0, e3ê m, n i = mn, j = m(n + 1). dk, e i, j ëï, K x i x j. y(ø, Iy {x i } i=1 k 2017 ØÓ. d, Iy²3 {x i } i=1 fs x i 1,, x i2017 i 1,, i 2017?Ûü þ ëï. e, ^8B {y² (Ø: é? ê k 2, ê8 N UÀÑ k ØÓê i 1,, i k Ù?ü þ ëï. k = 2 ž, (Øw, á. b(øé k á, =3üüëÏ k ê i 1, i 2,, i k. yä k + 1 œ¹. 5 é? ê i < j, i, j ëï = j i i. ùxj i, j ëï, Ké? v i a ê a k a + i a + j ëï. yp A = i 1 i 2 i k, Kd8Bbe k +1 ê: A, A+i 1,, A+i k?ûü þ ëï. ù`²(øé k + 1 á. µ5 dk JÝ K. ŸþŒ=z e K: é? ê k, 3 N k f8 S vé S? ü i < j k j i i. ù éž@c aq K,?nÃ{ c USAMO K: é? ê k, 3 N k f8 S é? i, j S, i j k (j i) 2 ij. ùa35 K^8BE {?n k. 5 ت /ª, Ïd 38BbÄ:þ ^² Cz5?1E,,ÀJв þ '. K 1.4 A Ú B ü k 8, ve5ÿn f: A A ê: 3ü N g: A B Ú h: B A g(h(x)) = x, x B h(g(x)) = f(x), x A. 5 êæ#(

6 ) Y ( A B ) B A B. d g(h(x)) = x, x B h ü, g, l B A. d h(g(x)) = f(x), x A á f Š 5 f(h(x)) = h(g(h(x))) = h(x), x B, f Œ±L«Xe/ª x, x h(b) f(x) =, ϕ(x), x A \ h(b) Ù ϕ A \ h(b) h(b)? N. h(b) = {h(x) : x B}. e `² kul«e/ª f ½3N g: A B Ú h: B A v^ : Ù ϕ A \ A A? N. x, x A A, A = B f(x) =, ( ) ϕ(x), x A \ A B A þ N l(x). þ, h 1 (x) A A þðón, - h : B A A : h(x) = h 1 (l(x)) = l(x), x B. 2N g 1 : A A : x, x A g 1 (x) =, ϕ(x), x A \ A -N g : A B : g(x) = l 1 (g 1 (x)), x A. y g(h(x)) = x, x B h(g(x)) = f(x), x A. q ( ) ª A Àk ( A B ) «{, ϕ(x) Àk B A B «{, /X ( ) ª f ê ( A B ) B A B, ù v^ f ê. µ5 dk JK. =k 2.8% < édk. ka Æ)ò f(x) Xe/ª x, x h(b) f(x) =, ϕ(x), x A \ h(b) Ù ϕ A \ h(b) A? N, lä½ù f ê ( A B ) A A B, 6

7 ù Ø, ÙÏ Ñ?Ø f(x) Š.,, dkæ)~, Ø, uy f(x) äk/ª Bêþ Ñ f OŽ(J, B).. þ, I y²äk/ª ( ) f ½Œ± é v g Ú h, ù î>5ä. K 2.1 n ê, n 2, y²: n ( ) < 2. k y² 5 é 0 < x < 1 k 1 + x < 1. 1 x n ( ) < 1 k n (1 1 2 k ). (1) d Bernoulli ت n (1 1 n 1 2 ) 1 1 k 2 > 1 4 k = 1 2. (2). nü (1), (2) á(ø. µ5 dk {ük, ŒõêÆ) édk. i) þ ){' XÛ ^ Bernoulli ت, Ïd (1) ª= 7 ii) ØÆ)^\r8B {y² n (1 + 1 ) < k 2 n 2. iii) dk Œ±^Í Ûت: ln(1 + x) < x, x (0, ) 5y². K 2.2 I š ABC S%, Sƒ I BC, CA ƒ: O D, E, H ABI R%, K = AI BH, L = BI AH. y²: I DKH 9 ELH n:. 7 êæ#(

8 y² d^ Œ, IDB = 90 = IKB, ± B, K, D, I o:. q ALB = 90 = AKB, ± B, K, L, A o:. Ïd BKD = 180 BID = 180 ( ABC) = 180 BAL = BKL. Ïd, K, D, L. d A, E, L, I o: ( AEI = ALI = 90 ) k ALE = AIE = CAB = 90 BAI = 90 BAK = ABK. qd L, A, B, K o:, ABK = 180 ALK. ± K, E, L. K, D, E, L o:. S DKH Ú ELH 1 :. K DSE = 360 DSH HSE = DKH HLE = LKH + HLK = 180 KHL. q H, L, I, K o:, 180 KHL = KIL = AIB = 180 IBA IAB = CBA 1 2 CAB. ± DSE = CBA 1 2 CAB.,, Sƒ I > AB ƒu: F. K A, F, I, E 9 B, F, I, D O o:. Ïd DF E = DF I + IF E = DBI + IAE = 1 2 CBA CAB. DF E + DSE = 180. ù`² S 3 DEF þ, = ABC Sƒþ :. ùòy² I DKH 9 ELH : S. 8

9 µ5 dkjýøœ, þ ){ ^ E, L, D, K o:,? d'x(j. þ, H, S, I, F o:. K 2.3 X k 8. X = A 1 A 10, X = B 1 B 10 vxe5ÿü y: e A i B j =, 1 i, j 10, K A i B j 10. X Š. ) X Š 50. ky² X 50. Ä8Ü A 1,, A 10, B 1,, B 10 ƒ ê 8Ü, Ø A 1. P A 1 = a, K A 1 õ B 1,, B 10 a 8܃. Ø A 1 B i, i = 1,, k A 1 B i =, i = k + 1,, 10, Ù k a. A 1 B i 10, i = k + 1,, 10. lé i k + 1 k B i 10 A 1 = 10 a. d A 1 5 B 1,..., B k ƒ êþøu 5. l X = B 1 B 10 = B B k + B k B 10 k a + (10 k)(10 a) = (5 k)(5 a). i) e a 5, K k 5, dždþª X 50; ii) e a > 5, d A 1 A 1,, A 10 ƒ ê 8Ü X 10a > 50. X 50.,, X U 50, ~X, A 1 = B 1 = {1, 2, 3, 4, 5}, A 2 = B 2 = {6, 7, 8, 9, 10},, A 10 = B 10 = {46, 47, 48, 49, 50}. w, v^, ùž X = {1, 2,, 50}. µ5 dk,e A i B j =, K B j 10 A i, ù تŒ±O A i ؃ B j ƒ êe.. Ïg,Ž^4àn5O{e B j. K 2.4 m ½ê, d Ïf. {a i } i=0 Ú {b i } i=0 ü dê ê, v: 3ê i, j, k, l (a i, b j ) = 1, (a k, b l ) = m. y²: 3ê t, s (a t, b s ) = d. y{ 5 m ŒÅgر ezƒïf d, ù? ƒïf p, 3ê α, β (a α, b β ) = m p. Iyé m 9 êæ#(

10 Ø. du (a k, b l ) = m. a k p, b l p 7k ØU m Ø. Ø a k p ØU m ê {a i }, {b j } ú O u, v, K a i = a 0 + iu, b j = b 0 + jv. e ü«œ¹: 1) e p v. - α = k, β = l + a k p. ùž b β = b l + a kv p. 5 m p a k, m p b β, q p v m úïf, m Ø úïf. a kp, ± m b β. ù`² m p a α, b β e a α, b β 3ØÓu p ƒúïf q, K q a α, q b β, q a αp. q (a α, b β aα p v), = q (a k, b l ) = m. q p q, q m p. (a α, b β ) = m p. 2) e p v. ky p u ( ). þ, b p u, d (a k, b l ) = m p a k ku = a 0, p b l lu = b 0. Ïd p a i, p b j, ù (a i, b j ) = 1 gñ! ( ) y. ê s, b l p s U m p Ø, ØU m Ø. - α = k + b l p s, β = l. ùž a α = a k + b l p s u, b β = b l. 5 m p a k, m p b l, m p b l p s, ± m p p u, ± m a α, l m Ø a α, b β úïf. a α b β úïf. q m b l p s, ƒê q a α, b β ØÓu p úïf, K q b β, q (a α b l p s u) = a k. = q (a k, b l ) = m. q q p, ± q m p. ù`² (a α, b β ) = m p. d 1), 2) (Ø á. y{ ( är Æ㈠) a n = a 0 + nd 1, b n = b 0 + nd 2. 5 m Œ±ëYر ezƒïf d, Iy²é m? ƒ Ïf p, 3ê t, s (a t, b s ) = m. p ky²xeún: Ún a, b, c N, e (a, b) = 1, K3 n N, (an + b, c) = 1. Úny²?ƒê q c, e q a, - n q = 1; e q a,? n q b (mod q). a l n n q (mod q) ž,k q an + b. q H{ c kƒïfž, d I {½n, ù n N 3, (an + b, c) = 1. K. - m = pd. Ï (a k, b l ) = m, Œ a k = pdx, b l = pdy, Kk (x, y) =

11 Ï (a i, b j ) = 1, p a i ½ p b j. Ø p a i, K p a k a i. ± p (k i)d 1, lk p d 1. r = (d 1, x), qdu p d 1, (x, y) = 1, (r, py) = 1, du ( d 1 r, x ) = 1, ± r ( d 1 r, px r ) = 1. 3Ún a = d 1 r, b = px r, c = py, K (a, b) = 1. 3 n N, (an + b, c) = 1. l ( d 1 r n + px r, py) = 1. q (r, py) = 1, (d 1n + px, py) = 1. - t = k + nd, s = l, lk a t = a k + ndd 1 = d(d 1 n + px), b s = dpy, džk (a t, b s ) = d(d 1 n + px, py) = d. Ky. µ5 y² Úy²Ñy²d K, = Iy p ƒêœ/. Ø Ó, y² æ^ a E {; y² : y3 n N, (d 1 n + px, py) = 1, Ž^ I {½ny²Ún. 11 êæ#(