MathDB
Problems
Contests
National and Regional Contests
China Contests
China Team Selection Test
2020 China Team Selection Test
4
4
Part of
2020 China Team Selection Test
Problems
(1)
Bounding Diophantine Equation
Source: China Additional TST for IMO 2020, P4
10/20/2020
Show that the following equation has finitely many solutions
(
t
,
A
,
x
,
y
,
z
)
(t,A,x,y,z)
(
t
,
A
,
x
,
y
,
z
)
in positive integers
t
(
1
−
A
−
2
)
(
1
−
x
−
2
)
(
1
−
y
−
2
)
(
1
−
z
−
2
)
=
(
1
+
x
−
1
)
(
1
+
y
−
1
)
(
1
+
z
−
1
)
\sqrt{t(1-A^{-2})(1-x^{-2})(1-y^{-2})(1-z^{-2})}=(1+x^{-1})(1+y^{-1})(1+z^{-1})
t
(
1
−
A
−
2
)
(
1
−
x
−
2
)
(
1
−
y
−
2
)
(
1
−
z
−
2
)
=
(
1
+
x
−
1
)
(
1
+
y
−
1
)
(
1
+
z
−
1
)
number theory
Diophantine equation