MathDB
Problems
Contests
National and Regional Contests
India Contests
ISI Entrance Examination
2018 ISI Entrance Examination
2018 ISI Entrance Examination
Part of
ISI Entrance Examination
Subcontests
(8)
8
1
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ISI 2018 #8
Let
n
⩾
3
n\geqslant 3
n
⩾
3
. Let
A
=
(
(
a
i
j
)
)
1
⩽
i
,
j
⩽
n
A=((a_{ij}))_{1\leqslant i,j\leqslant n}
A
=
((
a
ij
)
)
1
⩽
i
,
j
⩽
n
be an
n
×
n
n\times n
n
×
n
matrix such that
a
i
j
∈
{
−
1
,
1
}
a_{ij}\in\{-1,1\}
a
ij
∈
{
−
1
,
1
}
for all
1
⩽
i
,
j
⩽
n
1\leqslant i,j\leqslant n
1
⩽
i
,
j
⩽
n
. Suppose that
a
k
1
=
1
for all
1
⩽
k
⩽
n
a_{k1}=1~~\text{for all}~1\leqslant k\leqslant n
a
k
1
=
1
for all
1
⩽
k
⩽
n
and
∑
k
=
1
n
a
k
i
a
k
j
=
0
for all
i
≠
j
~~\sum_{k=1}^n a_{ki}a_{kj}=0~~\text{for all}~i\neq j
∑
k
=
1
n
a
ki
a
kj
=
0
for all
i
=
j
. Show that
n
n
n
is a multiple of
4
4
4
.
6
1
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Isi 2018 problem 6
Let,
a
≥
b
≥
c
>
0
a\geq b\geq c >0
a
≥
b
≥
c
>
0
be real numbers such that for all natural number
n
n
n
, there exist triangles of side lengths
a
n
,
b
n
,
c
n
a^{n} , b^{n} ,c^{n}
a
n
,
b
n
,
c
n
. Prove that the triangles are isosceles.
5
1
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ISI 2018 #5
Let
f
:
R
→
R
f:\mathbb{R}\to\mathbb{R}
f
:
R
→
R
be a differentiable function such that its derivative
f
′
f'
f
′
is a continuous function. Moreover, assume that for all
x
∈
R
x\in\mathbb{R}
x
∈
R
,
0
⩽
∣
f
′
(
x
)
∣
⩽
1
2
0\leqslant \vert f'(x)\vert\leqslant \frac{1}{2}
0
⩽
∣
f
′
(
x
)
∣
⩽
2
1
Define a sequence of real numbers
{
a
n
}
n
∈
N
\{a_n\}_{n\in\mathbb{N}}
{
a
n
}
n
∈
N
by :
a
1
=
1
and
a
n
+
1
=
f
(
a
n
)
for all
n
∈
N
a_1=1~~\text{and}~~a_{n+1}=f(a_n)~\text{for all}~n\in\mathbb{N}
a
1
=
1
and
a
n
+
1
=
f
(
a
n
)
for all
n
∈
N
Prove that there exists a positive real number
M
M
M
such that for all
n
∈
N
n\in\mathbb{N}
n
∈
N
,
∣
a
n
∣
⩽
M
\vert a_n\vert \leqslant M
∣
a
n
∣
⩽
M
7
1
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ISI Problem 7 2018
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
are natural numbers such that
a
2
+
b
2
=
c
2
a^{2}+b^{2}=c^{2}
a
2
+
b
2
=
c
2
and
c
−
b
=
1
c-b=1
c
−
b
=
1
Prove that
(
i
)
(i)
(
i
)
a
a
a
is odd.
(
i
i
)
(ii)
(
ii
)
b
b
b
is divisible by
4
4
4
(
i
i
i
)
(iii)
(
iii
)
a
b
+
b
a
a^{b}+b^{a}
a
b
+
b
a
is divisible by
c
c
c
4
1
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ISI 2018 #4
Let
f
:
(
0
,
∞
)
→
R
f:(0,\infty)\to\mathbb{R}
f
:
(
0
,
∞
)
→
R
be a continuous function such that for all
x
∈
(
0
,
∞
)
x\in(0,\infty)
x
∈
(
0
,
∞
)
,
f
(
2
x
)
=
f
(
x
)
f(2x)=f(x)
f
(
2
x
)
=
f
(
x
)
Show that the function
g
g
g
defined by the equation
g
(
x
)
=
∫
x
2
x
f
(
t
)
d
t
t
for
x
>
0
g(x)=\int_{x}^{2x} f(t)\frac{dt}{t}~~\text{for}~x>0
g
(
x
)
=
∫
x
2
x
f
(
t
)
t
d
t
for
x
>
0
is a constant function.
3
1
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ISI 2018 #3
Let
f
:
R
→
R
f:\mathbb{R}\to\mathbb{R}
f
:
R
→
R
be a continuous function such that for all
x
∈
R
x\in\mathbb{R}
x
∈
R
and for all
t
⩾
0
t\geqslant 0
t
⩾
0
,
f
(
x
)
=
f
(
e
t
x
)
f(x)=f(e^tx)
f
(
x
)
=
f
(
e
t
x
)
Show that
f
f
f
is a constant function.
2
1
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ISI 2018 #2
Suppose that
P
Q
PQ
PQ
and
R
S
RS
RS
are two chords of a circle intersecting at a point
O
O
O
. It is given that
P
O
=
3
cm
PO=3 \text{cm}
PO
=
3
cm
and
S
O
=
4
cm
SO=4 \text{cm}
SO
=
4
cm
. Moreover, the area of the triangle
P
O
R
POR
POR
is
7
cm
2
7 \text{cm}^2
7
cm
2
. Find the area of the triangle
Q
O
S
QOS
QOS
.
1
1
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ISI 2018 #1
Find all pairs
(
x
,
y
)
(x,y)
(
x
,
y
)
with
x
,
y
x,y
x
,
y
real, satisfying the equations
sin
(
x
+
y
2
)
=
0
,
∣
x
∣
+
∣
y
∣
=
1
\sin\bigg(\frac{x+y}{2}\bigg)=0~,~\vert x\vert+\vert y\vert=1
sin
(
2
x
+
y
)
=
0
,
∣
x
∣
+
∣
y
∣
=
1