MathLabs
Language
Tiếng Việt
English
日本語
简体中文
← Back
Competitions
›
Asian Pacific Mathematics Olympiad
›
2011
›
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 3 of 5: Cycle again
Previous step
Next step
c
2
+
a
+
b
≥
(
c
+
1
)
2
c^2+a+b\ge(c+1)^2
c
2
+
a
+
b
≥
(
c
+
1
)
2
Detailed analysis
Likewise,
a
+
b
≥
2
c
+
1
a+b\ge2c+1
a
+
b
≥
2
c
+
1
.
Home
Library
Great problems
Quiz
Mathematicians
Competitions