Exercices de Khôlles de Mathématiques, premier

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E A B E φ
φ:P(E)→P(A)×P(B)
X7→ (X∩A, X ∩B)
φ
φ
P2P
R
aRb ⇔a+b
2
R
n

2
f g RMax(f, g)R
x1, x2, . . . , xny1, y2. . . yn
x1≤x2≤ ··· ≤ xny1≤y2≤ ··· ≤ yn
1
n n
X
i=1
xi! n
X
i=1
yi!≤
n
X
i=1
xiyi.
x1, x2, . . . , xn
n
X
i=1
xi! n
X
i=1
1
xi!≥n2.
R
α, β, γ α +β+γ=π
1−cos(α) + cos(β) + cos(γ).

3
R R (x, y)∈
R2
Zx+y
x−y
f(t)dt =f(x)f(y).
4
∗[a, b]⊃[a1, b1]⊃ ··· ⊃ [an, bn]⊃ ··· lim
+∞an−bn= 0
llim an= lim bn=l
(Cn)CnCnrn
Cn+1 Cnrn+1 ≤rn
2
Cn
ABC [BC]A00
\
ABC
\
BCA [BC]A00 (AB)B00 (AC)C00
AB00 ABC
A0ABC [BC] [BC] [A00A0]

5
a, b, c P (x) = x3+ax2+bx +c
(x, y)y2=P(x)
a, b, c E
P α
ER R
P α
β α > β
Ex=β x =α
E
EQ, R x(t) =
Q(t)y(t) = R(t)P(x)P(x) = (x−β)(x−α)2
x(t)Q