Exercices de préparation au concours général de

a b (zn)zn+1 =azn+b
1995 (un)
u0≥0n∈Nun+1 =√un+1
n+ 1
d(n)n(un)
un=
n
X
k=1
d(k).
un
¡1 + a
n¢n−ea.
fn(x) = |x(x−1)(x−2) ···(x−n)|Mn= sup
J1,...,nK|fn|.
Mn

p≥5 24 p2−1
n2n+ 1 3
(n, m)∈N23n−2m= 1
Fn= 2(2n)−1n6=m FnFm
n p ¡n
p−1¢ ¡n p¢ ¡ n p+1¢
1991 (xn)
n
x3
0+x3 1+··· +x3 n= (x0+x1+··· +xn)2.
n m
x0+x1+··· +xn=m(m+ 1)
2.
n p Sn,p = 1p+ 2p+···+np
p n Sn,p
1999 N
(n+ 3)n=
n+2
X
k=3
kn.
µn
0¶+µn 3¶+µn 6¶+··· .

¡2n
k¢1≤k≤2n−1
1992
101992
1083 + 7.
S(n, p)n
p
S(n, n)S(p+ 1, p)
S(n, p) =
n
X
k=1 µn
k¶S(n−k, p −1).
p
X
k=0 µp
k¶S(n, k) = pn.
An{1, . . . , n}
An=An−1+An−2.
An
2000 b n
s
r
p
p b, n, r s
s r p

A, B, C a, b, c ∈C
ABC a2+b2+c2=ab +ac +bc
a, b, c ∈R3
a, b, c
a, b, c
1992 (C) 1
ABC (C)AB2+BC2+CA2
ABCD (C)AB2+AC2+
AD2+BC2+BD2+CD2
1993
A B
MAB
MAB
V a b c d
L=a+b+c+d
V
L3.
1997 A B
C(AB)G ABC I
α0< α < p ΓC(−→ CA, −−→ CB) = α+ 2kπ
CΓG I
π
3< α < π C ΓGI
f(α)GI f(α)a=AB
α f(α)α]π
3;π[