MathDB
Problems
Contests
International Contests
Caucasus Mathematical Olympiad
2022 Caucasus Mathematical Olympiad
3
3
Part of
2022 Caucasus Mathematical Olympiad
Problems
(1)
Set of numbers and function a+b-gcd(a;b)
Source: VII Caucasus Mathematical Olympiad
3/13/2022
Pete wrote down
21
21
21
pairwise distinct positive integers, each not greater than
1
,
000
,
000
1,000,000
1
,
000
,
000
. For every pair
(
a
,
b
)
(a, b)
(
a
,
b
)
of numbers written down by Pete, Nick wrote the number
F
(
a
;
b
)
=
a
+
b
−
gcd
(
a
;
b
)
F(a;b)=a+b -\gcd(a;b)
F
(
a
;
b
)
=
a
+
b
−
g
cd
(
a
;
b
)
on his piece of paper. Prove that one of Nick’s numbers differs from all of Pete’s numbers.
number theory
GCD