Exercices - IMOSuisse

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Olympiades Suisses de Mathématiques
n n n2+ 3n+ 27
5·17 |52·17 + 3 ·5·9+5·3·8
n(n+m)|3mn2+amn2+ 3n3+an3
n n + 1 |n2+ 1
n n
n2n3n
5n

n
21n+ 4
14n+ 3
(a, b)
ppmc(a, b) = 10 pgdc(a, b)
n > 6>1
a b a ·b a b
apgdc(a, b)
m n m
n
(x, y)x≤y
pgdc(x, y) = 5! ppmc(x, y) = 50!
(x, y)
1
x+2 y= 1.

(x, y, z)
1
x+2 y−3 z= 1.
n n2+ 1 n7+ 13
y2=x(x+ 1)(x+ 2)(x+ 3)
x
x! = x2+ 11x−36
(a, b)a2b+a+b
ab2+b+ 7
(m, n)
(m+ 1)(n+ 2) mn
a, b n
a! + b! = 2n
m, n
m3+mn +n3mn(m+n)
n n3
n

(p, q, r)
|p−q|,|q−r|,|r−p|
n
n n
(m, n)
(m+ 1)! + (n+ 1)! = m2n2