MathDB
Problems
Contests
National and Regional Contests
India Contests
India IMO Training Camp
2012 India IMO Training Camp
2012 India IMO Training Camp
Part of
India IMO Training Camp
Subcontests
(3)
3
3
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Prove f(x+y+xy)=f(x)+f(y)+f(xy) implies f(x+y)=f(x)+f(y)
Let
f
:
R
⟶
R
f:\mathbb{R}\longrightarrow \mathbb{R}
f
:
R
⟶
R
be a function such that
f
(
x
+
y
+
x
y
)
=
f
(
x
)
+
f
(
y
)
+
f
(
x
y
)
f(x+y+xy)=f(x)+f(y)+f(xy)
f
(
x
+
y
+
x
y
)
=
f
(
x
)
+
f
(
y
)
+
f
(
x
y
)
for all
x
,
y
∈
R
x, y\in\mathbb{R}
x
,
y
∈
R
. Prove that
f
f
f
satisfies
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
f(x+y)=f(x)+f(y)
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
for all
x
,
y
∈
R
x, y\in\mathbb{R}
x
,
y
∈
R
.
Number of 6-tuples satisfying conditions
How many
6
6
6
-tuples
(
a
,
b
,
c
,
d
,
e
,
f
)
(a, b, c, d, e, f)
(
a
,
b
,
c
,
d
,
e
,
f
)
of natural numbers are there for which
a
>
b
>
c
>
d
>
e
>
f
a>b>c>d>e>f
a
>
b
>
c
>
d
>
e
>
f
and
a
+
f
=
b
+
e
=
c
+
d
=
30
a+f=b+e=c+d=30
a
+
f
=
b
+
e
=
c
+
d
=
30
are simultaneously true?
Determine function given two inequalities
Let
R
+
\mathbb{R}^{+}
R
+
denote the set of all positive real numbers. Find all functions
f
:
R
+
⟶
R
f:\mathbb{R}^{+}\longrightarrow \mathbb{R}
f
:
R
+
⟶
R
satisfying
f
(
x
)
+
f
(
y
)
≤
f
(
x
+
y
)
2
,
f
(
x
)
x
+
f
(
y
)
y
≥
f
(
x
+
y
)
x
+
y
,
f(x)+f(y)\le \frac{f(x+y)}{2}, \frac{f(x)}{x}+\frac{f(y)}{y}\ge \frac{f(x+y)}{x+y},
f
(
x
)
+
f
(
y
)
≤
2
f
(
x
+
y
)
,
x
f
(
x
)
+
y
f
(
y
)
≥
x
+
y
f
(
x
+
y
)
,
for all
x
,
y
∈
R
+
x, y\in \mathbb{R}^{+}
x
,
y
∈
R
+
.
2
6
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