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Asian Pacific Mathematics Olympiad
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2011
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Problem 1
Problem 1
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive integers. Prove that it is impossible for all three numbers
a
2
+
b
+
c
a^2+b+c
a
2
+
b
+
c
,
b
2
+
c
+
a
b^2+c+a
b
2
+
c
+
a
, and
c
2
+
a
+
b
c^2+a+b
c
2
+
a
+
b
to be perfect squares.
Step 2 of 5: Cycle
Previous step
Next step
b
2
+
c
+
a
≥
(
b
+
1
)
2
b^2+c+a\ge(b+1)^2
b
2
+
c
+
a
≥
(
b
+
1
)
2
Detailed analysis
Likewise,
c
+
a
≥
2
b
+
1
c+a\ge2b+1
c
+
a
≥
2
b
+
1
.
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