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International Contests
JBMO ShortLists
2007 JBMO Shortlist
2
2
Part of
2007 JBMO Shortlist
Problems
(2)
Adding 1
Source: JBMO Shortlist 2007
5/12/2016
A2
\boxed{\text{A2}}
A2
Prove that for all Positive reals
a
,
b
,
c
a,b,c
a
,
b
,
c
a
2
−
b
c
2
a
2
+
b
c
+
b
2
−
c
a
2
b
2
+
c
a
+
c
2
−
a
b
2
c
2
+
a
b
≤
0
\frac{a^2-bc}{2a^2+bc}+\frac{b^2-ca}{2b^2+ca}+\frac{c^2-ab}{2c^2+ab}\leq 0
2
a
2
+
b
c
a
2
−
b
c
+
2
b
2
+
c
a
b
2
−
c
a
+
2
c
2
+
ab
c
2
−
ab
≤
0
inequalities
algebra
x^2006 - 4y^2006 - 2006 = 4y^2007 + 2007y
Source: JBMO Shortlist 2007 N2
10/14/2017
Prove that the equation
x
2006
−
4
y
2006
−
2006
=
4
y
2007
+
2007
y
x^{2006} - 4y^{2006} -2006 = 4y^{2007} + 2007y
x
2006
−
4
y
2006
−
2006
=
4
y
2007
+
2007
y
has no solution in the set of the positive integers.
JBMO
number theory