Éléments de correction Olympiades Académiques

n2=n+n+· · · +n n n
1
n+1 n+· · · +1 n
| {z }
n
= 1 n2n≥2
n2n
1
2
1
2+1 2= 1 2n+ 2
n1
2+1 3+1 6= 1 2n+ 9 = 2n+ 3 + 6
n
2×24 + 2 = 50 2 ×48 + 2 = 100
2×24 + 9 = 57 2 ×45 + 9 = 99
2×100 4 ×100 · · · 24 2n×100
n

AHD x
2
1
3
x
2=1 3x=2 3
x
2,x 2,1−2×x 2−(1 −x)2 2=1−x2 2
x2+x−1 = 0 0 <x<1x=√5−1
2
(JH) (AC)K(DI)L[AC]
JKC
JK =JC =HB = 1 −x
JL
CI =DJ DC
JL =x2
x2= 1 −x JK =JL K =L
vAvB
T C t
M N
AC =vAT BC =vBT
AM =vAt BN =vBT
MC =vA(T−t)NC =vB(T−t)
CM
CA =CN CB
(MN) (AB)
AMNB
AM =MR BN =NR d MAR =d MRA d NBR =d NRB
d
RAB =d MRA d NRB =d RBA
ABC (AR)b A(BR)b B
R ABC
ABC R M N
(BC)R

C(1,9) = 37 C(1,10) = 46 C(1,13) = 79
C(9,1) = 45 C(10,1) = 55 C(13,1) = 91
C(1, j + 1) = C(1, j) + j C(i+ 1,1) = C(i, 1) + (i+ 1)
j=C(1, j + 1) −C(1, j)
C(1,1) = 1 C(1,2) = 2 C(2,2) = 5 C(2,1) = 3
C(1, j+1) C(1, j)j(j)
C(i+ 1,1) C(i, 1) i+ 1 (i+ 1)
C(1, j) = C(1,1)+1+2+3+· · · + (j−1) = 1 + (j−1)j
2=j2
−j+2
2
C(i, j)C(i−1, j + 1) C(i−2, j + 2) · · · C(1, j +i−1)
C(i, j) = C(1, j +i−1) + (i−1)
C(i, j) = i2+ 2ij +j2−i−3j+ 2
2
C(i, j)C(1, j +i−1) C(1, j +i)C(1,63) = 1954 C(1,64) = 2017
j+i−1 = 63 i−1 = 2008 −1954 = 54 C(55,9) = 2008