Principe d`invariance

smo
osm
Olympiades Suisses de Mathématiques
n1,2,...,4n
a b
|a−b|
a b |a−b|
a≥b|a−b|=a−b
a, b 7→ a−b
−2b
1 + 2 + . . . + 4n= 2n(4n+ 1) ≡0
S

P
P
S S ≥0
a, b, c
(a, b, c) (a−b, b −c, c −a)
(an, bn, cn)n an+bn+cn= 0
n≥1 (an, bn, cn)
An
An+1 =a2
n+1 +b2 n+1 +c2 n+1 = (an−bn)2+ (bn−cn)2+ (cn−an)2
= 2(a2
n+b2 n+c2 n)−2anbn−2bncn−2cnan= 2An−2(anbn+bncn+cnan).
n≥1
0 = (an+bn+cn)2=a2
n+b2 n+c2 n+ 2(anbn+bncn+cnan).
An+1 = 3An,
An+1 = 3nA1a, b, c A1>0
Anan, bn, cn
1,0,1,0,0,0
2×2
a1, a2, . . . , an1−1
a1a2a3a4+a2a3a4a5+. . . +ana1a2a3= 0.
n
3,4,12 a, b
0.6a−0.8b0.8a+ 0.6b4,6,12

n
1,2,...,106
106
1−1
a a
a, b
pgdc(a, b) ppmc(a, b)
4×4
2n n −1
2n
Q, R, S
(x, y)Q, R, S (x−y, y),(x+y, y),(y, x)
(1,2) (19,79) (819,357)
(p, q) (a, b)
3×3 4 ×4
x, y, z
y < 0x, y, z x +y, −y, z +y

10 ×10
1,0,1,0,1,0, . . .
. . . , 0,1,0,1,0,1, . . .
(0,0),(1,0),(2,0),(0,1),(1,1),(0,2)
(0,0)
(x, y) (x, y + 1) (x+ 1, y)
(x, y)
(x, y + 1) (x+ 1, y)
(x, y)y≤0
(0,5)
n2
n×n
n
n≥2n
λ
A B A B A
C B BC/AB =λ
λ M
n
M