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National and Regional Contests
India Contests
Regional Mathematical Olympiad
2015 India Regional MathematicaI Olympiad
2015 India Regional MathematicaI Olympiad
Part of
Regional Mathematical Olympiad
Subcontests
(8)
8
1
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Sum of any 3 sides of a quad div by 4th=>some 2 are equal
The length of each side of a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
is a positive integer. If the sum of the lengths of any three sides is divisible by the length of the remaining side then prove that some two sides of the quadrilateral have the same length.
7
1
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(1+x)(1+y)(1+z)<=4(1+xyz) with a condition on reals x,y,z
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be real numbers such that
x
2
+
y
2
+
z
2
−
2
x
y
z
=
1
x^2+y^2+z^2-2xyz=1
x
2
+
y
2
+
z
2
−
2
x
yz
=
1
. Prove that
(
1
+
x
)
(
1
+
y
)
(
1
+
z
)
≤
4
+
4
x
y
z
.
(1+x)(1+y)(1+z)\le 4+4xyz.
(
1
+
x
)
(
1
+
y
)
(
1
+
z
)
≤
4
+
4
x
yz
.
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5
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4
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3
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2
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1
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