Exercices - IMOSuisse

smo
osm
Olympiades Suisses de Mathématiques
m > 1a
a, a + 1, a + 2, . . . , a +m−1
m
p p + 4 p+ 14 p
x y x2+y2
4k+ 3
9|a2+ab +b2a b
n n2+ 3n+ 5
a1, a2, . . . , an∈ {−1,1}
a1a2+a2a3+. . . +an−1an+ana1= 0.
4|n
f:N0→N0
f(f(n)) = n+ 1987.
p, q p3−q5= (p+q)2
0< a1< a2< . . . < a101 <5050
ak, al, am, an
5050|(ak+al−am−an).
ϕ
n2n−1

2n+ 1
13 |270 + 370
n3|n·2n−1
m n m < n 1978m1978n
m n m +n
n n 2n!−1
n323 |20n+ 16n−3n−1
x, y, z, t n
nx+ny+nz=nt.
Fn= 22n+ 1
641 |F5
77777
p > 3p2≡1 (24)
p p2+ 2 p3+ 2
n≥2q=p1p2· · · pnn q −1
p p2+ 11
n∈N2n+ 1 3n+ 1 40 |n
19x3−84y2= 1984
a, b, c, d, e ∈Na4+b4+c4+d4=e4
n d1, d2, d3, d4
n
d2
1+d2 2+d2 3+d2 4=n.
d6= 2,5,13 {2,5,13, d}
a, b ab −1
a, b 15a+16b16a−15b
1919

n30 |n5−n
n2n−1n
>1 2n+ 1
m n4+m
n
n > 2 22n−2+ 1
p, q, r p2+q3=r4
n n
312 −1 3k−1k= 1,2,...,11
|3x−2y|= 1.
x, y, z
x4+y4+z4−2x2y2−2y2z2−2z2x2= 189.
4545 + 5454
y2=x3+ 7
n a
a25 −a
n≥1 22n+ 22n−1+ 1 n
3x+ 4y= 5z.
(5125 −1)/(525 −1)
4n+ 2n+ 1 n
p x, y
px−yp= 1.

aabb
111 . . . 11
| {z }
n
n
1,31,331,3331, . . .
1982 |222 . . . 22
N N/11
N
abcd 4·abcd =dcba
(M, N) = (3600,2500)
M
N(M, N)
A A
11 . . . 1
| {z }
2n
= 22 . . . 2
| {z }
n
+(33 . . . 3
| {z }
n
)2.
n n n
n= 1623 , n = 3261 k k |n=⇒k|n n k |99
31
2
n
n−1
b
f:N→Rf(1) = 1
f(n) = 1 + f(n−1
2), n
1 + f(n
2), n
f:N→Nf(1) = 1, f(3) = 3 n∈N

f(2n) = f(n)
f(4n+ 1) = 2f(2n+ 1) −f(n)
f(4n+ 3) = 3f(2n+ 1) −2f(n)
n≤1988 f(n) = n
f:N→Nf(1) = 1 n∈N
3f(n)f(2n+ 1) = f(2n)(1 + 3f(n))
f(2n)<6f(n)
f(k) + f(m) = 293
0≤xi< b i = 0,1, . . . , n xn, xn−1>0a > b xnxn−1. . . x0
a A b B xn−1xn−2. . . x0
a A0b B0A0B < AB0
X a + 2b=n
a, b ∈X n