Sélection OIM 2013

ABC AB BC CA X Y
Z I1I2I3
AXZ BY X CZY (α, β, γ)
(∠BAC, ∠CBA, ∠ACB)ABC
I1I2I3
P u, v ∈N
u22013 −v22013 |P(u)−P(v).
Q P (x) = Q(x22013 )
n m
x1, . . . , xna
b x1, . . . , xn
a·b≥m
x1, . . . , xn
xi1≤xi2≤. . . ≤xikxi1≥xi2≥. . . ≥xik1≤i1< i2<··· < ik≤n

p x, y, z < p
x3y3z3p x2+y2+z2
x+y+z
A n ≥1A1, . . . , Ak
A
k

n≥2n×n
2×2 2 ×2
2×2
p q
p2+p+ 1 = q3.
f:R>0×R>0→R>0
ffx
y, y, z=x·f(1, z)x, y, z ∈R>0
g:R>0→R>0g(x) = f(x, x)x∈R>0

k > 0x, y, z > 0x+y+z≤kxyz
1
x+y+√xy +1 y+z+√yz +1
z+x+√zx ≤r3
k.
ABC ∠ACB ≥90◦k AB
O ABC AC BC M N
MN k X Y ∠XOY =∠ACB
n
X
d|n
d4=n4+n3+n2+n+ 1.