êÆc�ù
ê�nܯK
o d
2010.07
1. ê�{an}½ÂXe:
a1 = 1, an+1 = d(an) + c, n = 1, 2, · · · .
Ù¥c´(½���ê, d(m)L«m���ê�ê.
y²: 3��êk¦�ê�ak, ak+1, · · ·´±Ïê�.
) ky²XeÚn.
Ún é?¿��êm, þk
d(m) ≤ m
2
+ 1.
Ún�y² w,, 3
[m
2
]
+ 1�m− 1mØ3m��ê, K
m− d(m) ≥ (m− 1)−
([m
2
]
+ 1
)
+ 1 ≥ m
2
− 1.
l
, d(m) ≤ m
2
+ 1. Ún�y.
e¡y², é?¿��ên (n ≥ 2) , þkan ≤ 2c+ 1.
^y{. b�t (t ≥ 2)´¦�an ≥ 2c+2¤á����ê,Kd(at−1)+ c ≥ 2c+2,=d(at−1) ≥
c+ 2. qdd(at−1) ≤ at−1
2
+ 1, �
at−1
2
+ 1 ≥ c+ 2,
=at−1 ≥ 2c+2, ùt��5gñ. l
, é?Û��êi, þkai ∈ {1, 2, · · · , 2c+1}, u´, 73i,
j (i 6= j) , ¦�ai = aj . â4íê��5, ê�{an}7l,å´±Ïê�.
2. �ê�{an}�ê�(=þ�ê�ê�) , ÷vé?¿��ên, þk
(n− 1)an+1 = (n+ 1)an − 2(n− 1).
e2000 | a1999, ¦Ñ����ên (n > 1) , ¦�2000 | an.
) w,, a1 = 0, an+1 =
n+ 1
n− 1an − 2, n ≥ 2. -bn =
an
n− 1 , Knbn+1 = (n + 1)bn − 2, ½
=n(bn+1 − 2) = (n+ 1)(bn − 2). u´, n ≥ 2, þkbn+1 − 2 = n+ 1
n
(bn − 2), K
bn − 2 = n
n− 1 ·
n− 1
n− 2 · · · · ·
3
2
(b2 − 2) =
(
b2
2
− 1
)
n,
l
an = (n − 1)
((a2
2
− 1
)
n+ 2
)
, n ≥ 2. d^an ∈ Z, a2
2
∈ Z. �a2 = 2k, Òkan =
(n − 1)[(k − 1)n + 2], |^2000 | a1999, 1998(1999k − 1997) ≡ 0 (mod 2000). K−2(−k + 3) ≡ 0
(mod 2000), l
k ≡ 3 (mod 1000). ù�, �an = (n− 1)[(1000m+ 2)n+ 2], ùpm ∈ Z, m~ê,
n ≥ 2.
1
dd, 2000 | an�¿^´1000 | (n − 1)(n + 1). l
¦nÛê. �n = 2l − 1, K250 |
l(l + 1). u(l, l + 1) = 1,
250 = 53 × 2, ¤±k + 1 ≥ 125, n ≥ 2× 124 + 1 = 249.
nþ, ¤¦�÷v^����ên = 249.
3. é½���êa, b, b > a > 1, aØU�Øb9½���êê�{bn}∞n=1, ÷vé¤k��
ênkbn+1 ≥ 2bn. ´Äo3��ê�ê�{an}∞n=1¦�é¤k��ên, kan+1 − an ∈ {a, b},
é¤k��êm, l (±Ó) , kam + al 6∈ {bn}∞n=1?
) Y´½�, ·^8B{y²{an}�3.
Äk�a1��ê, ¦�2a1 6∈ {bn}∞n=1
a1 > b − a (~Xdbn → +∞3n0 ∈ N+, ¦bn0 >
b− a+ 1, �a1 = bn0 − 1, Ka1 > b− a
2a1 = 2bn0 − 2 < bn0+1, �2a1 6∈ {bn}∞n=1) .
Ùg, b�a1, a2, · · · , ak®�½, ÷vai+1 − ai ∈ {a, b} (i = 1, 2, · · · , k − 1)
al + am 6∈ {bn}∞n=1
(1 ≤ l ≤ k, 1 ≤ m ≤ k) . Ïak+1 − ak ∈ {a, b}, �ak+1��kak + aak + bü«U, l
a1 + ak+1, a2 + ak+1, · · · , ak + ak+1, ak+1 + ak+1��ke�ü«U:
(I) a1 + (ak + a), a2 + (ak + a), · · · , ak + (ak + a), 2(ak + a);
(II) a1 + (ak + b), a2 + (ak + b), · · · , ak + (ak + b), 2(ak + b).
e¡·y²(I) (II) ¥�kع{bn}∞n=1¥�. e(I) ¥k{bn}∞n=1¥�bu, ¿
(II)
¥k{bn}∞n=1¥�bv, Kdb < a1 + a, �2(ak + b) < 2(a1 + ak + a) ≤ 2bu, l
bv ≤ 2(ak +
b) < 2bu ≤ bu+1. qbv+1 ≥ 2bv ≥ 2(a1 + ak + b) > 2(a1 + ak + a) > 2(ak + a) ≥ bu, �bu = bv.
qbv = bu < 2(ak + a) < 2(ak + b), l
31 ≤ j ≤ k, ¦�bv = aj + ak + b.
/1: bu = ai + ak + a (1 ≤ i ≤ k) , d, 0 = bu − bv = ai − aj + a− b, l
ai − aj = b− a > 0,
d8Bb�, kai − aj = ca+ db (c, dK�ê) , u´ca+ db = b− a, (1− d)b = (1 + c)a > 0, ¤
±d = 0, b = (1 + c)a, ùa - bgñ.
/2: bu = 2(ak + a), d0 = bu − bv = ak − aj + 2a − b, l
ak − aj = b − 2a, d1 ≤ j ≤ k,
ak − aj ≥ 0. 2d8Bb�ak − aj = c′a + d′b (c′, d′K�ê) , u´c′a + d′b = b − 2a,
(1− d′)b = (2 + c′)a > 0, ¤±d′ = 0, b = (2 + c′)a, gñ.
�(I) (II)¥�kع{bn}∞n=1¥�.Ïd�ak+1 = ak+a½ak+1 = ak+b,¦�ai+ak+1 6∈
{bn}∞n=1 (1 ≤ i ≤ k + 1) , u´·K�y.
4. �p0´½�Ûê, ê�a0, a1, a2, · · ·÷va0 = 0, a1 = 1,
an+1 = 2an + (a − 1)an−1, n =
1, 2, · · · . Ù¥a´��ê.
¦÷v±eü^�a��:
(i) XJp´ê,
p ≤ p0, Kp | ap;
(ii) XJp´ê,
p > p0, Kp - ap.
) k4í'X9a0 = 0, a1 = 1�
an =
1
2
√
a
[(1 +
√
a)n − (1−√a)n].
l
é?ÛÛêp,
ap =
1
2
√
a
[
p∑
k=0
Cpka
k
2 −
p∑
k=0
(−1)kCpka
a
2
]
=
1
2
√
a
(
2p
√
a+ 2C3pa
3
2 + · · ·+ 2Cp−2p a
p−2
2 + 2Cppa
p
2
)
= p+C3pa+ · · ·+Cp−2p a
p−3
2 + a
p−1
2 .
2
Ckp =
p!
k!(p− k)! , u´éu¤k1 ≤ k ≤ p − 1, kp | C
p
k. dd�, p | ap ⇔ p | a
p−1
2 ⇔ p | a. 2
da2 = 2, ¤¦a��´[3, p0]¥¤kê�¦È.
5. Á�ä´Ä3áõ4ABC, ÷vAB, BC, CA�´¤��ê��p���ê(AB <
BC < CA) ,
BC>þ�p94ABC�¡Èþ��ê. ¿Ñy².
) ·òy², 3áõ÷v^�4ABC.
�AB = a− d, BC = a, CA = a+ d, Ù¥a, d ∈ N+, a > d, 4ABC�¡ÈS, BC>þ�pha,
â°Ô-Ê�úª, �
S =
√
3a
2
·
(a
2
+ d
)
· a
2
·
(a
2
− d
)
=
1
2
√
3
[(a
2
)2
− d2
]
.
eS��ê, Ka7óê, -a = 2x(x ∈ N+), KS = x√3(x2 − d2), u´
ha =
2S
a
=
√
3(x2 − d2),
=h2a = 3(x
2 − d2), l
, ha73��ê. �ha = 3y (y ∈ N+) , Kx2 − 3y2 = d2, Ø�d = 1, K
x2 − 3y2 = 1, (1)
u´(x, y) = (2, 1)´§(1) �). -(2 +
√
3)n = xn + yn
√
3 (ùp, xnÚynþ��ê, n ∈ N+)
, K(2 − √3)n = xn − yn
√
3, u´x2n − 3y2n = 1, =(x, y) = (xn, yn) (n ∈ N+) Ñ´(1) ���ê).
dxn+1 + yn+1
√
3 = (2 +
√
3)n+1 = (2 +
√
3)(xn + yn
√
3) = (2xn + 3yn) + (xn + 2yn)
√
3, �{
xn+1 = 2xn + 3yn,
yn+1 = xn + 2yn,
x1 = 2, y1 = 1. �yn, �xn+2 = 4xn+1 − xn, x1 = 2, x2 = 7. u´�AB = 2xn − 1, BC = 2xn,
CA = 2xn + 1, n = 1, 2, · · · , KAB, BC, CA´¤��ê��p���ê(AB < BC < CA) ,
4ABC�¡ÈS = xn
√
3(x2n − 1) = 3xnyn, BC>þ�pha = 3ynþ��ê. Ïd, 3áõ
÷v^�4ABC.
6. ®ê�a1 = 20, a2 = 30, an+2 = 3an+1 − an (n ≥ 1) . ¦¤k���ên, ¦�1 + 5anan+1´�
�²ê.
) �bn = an + an+1, cn = 1 + 5anan+1, K5an+1 = bn+1 + bn, an+2 − an = bn+1 − bn, u´,
cn+1− cn = 5an+1(an+2− an) = b2n+1− b2n. l
, cn+1− b2n+1 = cn− b2n = · · · = c1− b21 = 501 = 3× 167.
�3��ên, m, ¦�cn = m2, Kk(m+ bn)(m− bn) = 3× 167.
em+ bn = 167, m− bn = 3, Km = 85, cn = 852.
em+ bn = 501, m− bn = 1, Km = 251, cn = 2512.
ê�{an}î4O,Kê�{cn}î4O,qc1 = 1+ 5× 20× 30 < 852 < c2 = 1+ 5× 30× 70 <
c3 = 1 + 5× 70× 180 = 2512. Ïd, ÷v^�nk, =n = 3.
7. ®ê�{cn}÷vc0 = 1, c1 = 0, c2 = 2005, cn+2 = −3cn − 4cn−1 + 2008 (n = 1, 2, 3, · · · ) .
Pan = 5(cn+2 − cn)(502− cn−1 − cn−2) + 4n × 2004× 501 (n = 2, 3, · · · ) . ¯: én > 2, an´Äþ²
ê, `²nd.
) dcn+2 − cn = −4(cn + cn−1) + 2008an = −20(cn + cn−1 − 502)(502 − cn−1 − cn−2) +
4n+1 × 5012. -dn = cn − 251, Kdn+2 = −3dn − 4dn−1, d0 = −250, d1 = −251, d2 = 1754. an =
3
20(dn + dn−1)(dn−1 + dn−2) + 4n+1 × 5012. -wn = dn + dn+1, Kwn+2 = wn+1 − 4wn, w1 = −501,
w2 = 1503, an = wnwn−1 + 4n+1 × 5012. -wn = 501Tn, KTn+2 = Tn+1 − 4Tn, T1 = −1, T2 = 3, ½
ÂT0 = −1, an = 5012 × 22(5TnTn−1 + 4n).
�x2 − x + 4 = 0�üα, β, Kα + β = 1, αβ = 4, Tn − αTn−1 = β(Tn−1 − αTn−1) =
· · · = βn−1(T1 − αT0) = −βn−1(α − 1), Tn − βTn−1 = −αn−1(β − 1), (Tn − αTn−1)(Tn − βTn−1) =
(αβ)n−1(α+ 1)(β + 1) = 4n. =T 2n − TnTn−1 + 4T 2n−1 = 4n.
l
, an = 5012 × 22(5TnTn−1 + T 2n − TnTn−1 + 4T 2n−1) = 10022(Tn + 2Tn−1)2.
8. ê�{an}Ú{bn}÷va0 = b0 = 1, an+1 = αan + βbn, bn+1 = βan + γbn, Ù¥α, β, γ��ê,
α < γ,
αγ = β2 + 1.
y²: é?¿K�ên, an + bn7Lü��ê�²Ú.
) ky²üÚn.
Ún1 e��êm, n, k÷vmn = k2 + 1, K3x1, x2, y1, y2 ∈ N, ¦±enªÓ¤á:
m = x21 + y
2
1 , n = x
2
2 + y
2
2 , k = x1x2 + y1y2.
Ún1�y² Ø�m ≤ n, ék^êÆ8B{.
�k = 1, dmn = 2, km = 1, n = 2, Km = 02 +12, n = 12 +12, k = 0× 1+ 1× 1, =k < 2(
ؤá.
��k < r (r ≥ 2) (ؤá.
�k = r, d
mn = r2 + 1, (1)
Kn ≥ r + 1, m = r
2 + 1
n
≤ r
2 + 1
r + 1
<
r2 + r
r + 1
= r.
-n = r + s, m = r − t (s, t ∈ N+) , ª(1) ¤(r − t)(r + s) = r2 + 1, =rs− ts− tr = 1. ü>Ó
\t2, �(r − t)(s− t) = t2 + 1.
r − t = m > 0, Ks− t > 0, t < r.
d8Bb�, 3m1,m2, n1, n2 ∈ N, ¦r − t = m21 + n21, s − t = m22 + n22, t = m1m2 + n1n2.
=m = r − t = m21 + n21, n = r + s = (r − t) + (s− t) + 2t = (m1 +m2)2 + (n1 + n2)2, r = (r − t)− t =
m1(m1 +m2) + n1(n1 + n2).
ePx1 = m1, y1 = n1, x2 = m1 +m2, y2 = n1 + n2, K3ª(1) ¥km = x21 + y
2
1 , n = x
2
2 + y
2
2 ,
k = x1x2 + y1y2 (x1, x2, y1, y2 ∈ N) . Ïd, �k = r(ؤá. dêÆ8B{, y�Ún1¤á.
Ún2 é?¿�m,n ∈ N, k
am+n + bm+n = aman + bmbn. (2)
Ún2�y² �½Cþm+ n, Iyéu÷v0 ≤ k ≤ m+ n�z�êk, k
am+n + bm+n = akam+n−k + bkbm+n−k. (3)
ék^êÆ8B{.
�k = 0, (Øw,¤á. �k = 1, duam+n = αam+n−1 + βbm+n−1, bm+n = βam+n−1 +
γbm+n−1, Kam+n + bm+n = (α+ β)am+n−1 + (β + γ)bm+n−1 = a1am+n−1 + b1bm+n−1.
��k = p, am+n+bm+n = apam+n−p+bpbm+n−p. �k = p+1,kam+n+bm+n = apam+n−p+
bpbm+n−p = ap(αam+n−p−1 + βbm+n−p−1) + bp(βam+n−p−1 + γbm+n−p−1) = (αap + βbp)am+n−p−1 +
(βap + γbp)bm+n−p−1 = ap+1am+n−p−1 + bp+1bm+n−p−1. l
ª(3) ¤á.
3ª(3) ¥-k = m, Kam+n + bm+n = aman + bmbn.
4
£��K, 3ª(2) ¥, -m = n, Ka2n + b2n = a2n + b
2
n.
�m = n + 1, dÚn1ka2n+1 + b2n+1 = an+1an + bn+1bn = (αan + βbn)an + (βan + γbn)bn =
αa2n+γb
2
n+2βanbn = (x
2
1+ y
2
1)a
2
n+(x
2
2+ y
2
2)b
2
n+2(x1x2+ y1y2)anbn = (x1an+x2bn)
2+(y1an+ y2bn)
2.
Ïd, (Ø�y.
5
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