MathDB
Problems
Contests
National and Regional Contests
India Contests
India IMO Training Camp
2016 India IMO Training Camp
2016 India IMO Training Camp
Part of
India IMO Training Camp
Subcontests
(3)
2
3
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Powers with same first digit
Given that
n
n
n
is a natural number such that the leftmost digits in the decimal representations of
2
n
2^n
2
n
and
3
n
3^n
3
n
are the same, find all possible values of the leftmost digit.
Another FE
Find all functions
f
:
R
→
R
f:\mathbb R\to\mathbb R
f
:
R
→
R
such that
f
(
x
2
+
x
f
(
y
)
)
=
x
f
(
x
+
y
)
f\left( x^2+xf(y)\right)=xf(x+y)
f
(
x
2
+
x
f
(
y
)
)
=
x
f
(
x
+
y
)
for all reals
x
,
y
x,y
x
,
y
.
Functional Equation
Find all functions
f
:
R
→
R
f:\mathbb{R}\to\mathbb{R}
f
:
R
→
R
such that
f
(
x
3
+
f
(
y
)
)
=
x
2
f
(
x
)
+
y
,
f\left(x^3+f(y)\right)=x^2f(x)+y,
f
(
x
3
+
f
(
y
)
)
=
x
2
f
(
x
)
+
y
,
for all
x
,
y
∈
R
.
x,y\in\mathbb{R}.
x
,
y
∈
R
.
(Here
R
\mathbb{R}
R
denotes the set of all real numbers.)
1
5
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3
5
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