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Taiwan National Olympiad
2006 Taiwan National Olympiad
2
Mod problem: xy = a mod z and cyclic
Mod problem: xy = a mod z and cyclic
Source: Taiwan NMO 2006
March 21, 2006
modular arithmetic
number theory unsolved
number theory
Problem Statement
x
,
y
,
z
,
a
,
b
,
c
x,y,z,a,b,c
x
,
y
,
z
,
a
,
b
,
c
are positive integers that satisfy
x
y
≡
a
(
m
o
d
z
)
xy \equiv a \pmod z
x
y
≡
a
(
mod
z
)
,
y
z
≡
b
(
m
o
d
x
)
yz \equiv b \pmod x
yz
≡
b
(
mod
x
)
,
z
x
≡
c
(
m
o
d
y
)
zx \equiv c \pmod y
z
x
≡
c
(
mod
y
)
. Prove that
min
{
x
,
y
,
z
}
≤
a
b
+
b
c
+
c
a
\min{\{x,y,z\}} \le ab+bc+ca
min
{
x
,
y
,
z
}
≤
ab
+
b
c
+
c
a
.
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