MathDB
Problems
Contests
National and Regional Contests
India Contests
India LIMIT
2019 LIMIT
2019 LIMIT Category A
Problem 10
Problem 10
Part of
2019 LIMIT Category A
Problems
(2)
# of solns. to 3^x+4^x=8^x over R
Source: LIMIT 2019 CAS1 P10
4/28/2021
Number of solutions of the equation
3
x
+
4
x
=
8
x
3^x+4^x=8^x
3
x
+
4
x
=
8
x
in reals is
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
0
<span class='latex-bold'>(A)</span>~0
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
0
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
1
<span class='latex-bold'>(B)</span>~1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
2
<span class='latex-bold'>(C)</span>~2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
∞
<span class='latex-bold'>(D)</span>~\infty
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
∞
algebra
equation
number of nondecreasing functions f:[3]->[5]
Source: LIMIT 2019 CAS2 P10
4/28/2021
The number of maps
f
f
f
from
1
,
2
,
3
1,2,3
1
,
2
,
3
into the set
1
,
2
,
3
,
4
,
5
1,2,3,4,5
1
,
2
,
3
,
4
,
5
such that
f
(
i
)
≤
f
(
j
)
f(i)\le f(j)
f
(
i
)
≤
f
(
j
)
whenever
i
≤
j
i\le j
i
≤
j
is
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
60
<span class='latex-bold'>(A)</span>~60
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
60
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
50
<span class='latex-bold'>(B)</span>~50
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
50
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
35
<span class='latex-bold'>(C)</span>~35
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
35
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
30
<span class='latex-bold'>(D)</span>~30
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
30
function
combinatorics