MathDB
Problems
Contests
National and Regional Contests
India Contests
ISI Entrance Examination
2019 ISI Entrance Examination
2019 ISI Entrance Examination
Part of
ISI Entrance Examination
Subcontests
(8)
8
1
Hide problems
ISI 2019 : Problem #8
Consider the following subsets of the plane:
C
1
=
{
(
x
,
y
)
:
x
>
0
,
y
=
1
x
}
C_1=\Big\{(x,y)~:~x>0~,~y=\frac1x\Big\}
C
1
=
{
(
x
,
y
)
:
x
>
0
,
y
=
x
1
}
and
C
2
=
{
(
x
,
y
)
:
x
<
0
,
y
=
−
1
+
1
x
}
C_2=\Big\{(x,y)~:~x<0~,~y=-1+\frac1x\Big\}
C
2
=
{
(
x
,
y
)
:
x
<
0
,
y
=
−
1
+
x
1
}
Given any two points
P
=
(
x
,
y
)
P=(x,y)
P
=
(
x
,
y
)
and
Q
=
(
u
,
v
)
Q=(u,v)
Q
=
(
u
,
v
)
of the plane, their distance
d
(
P
,
Q
)
d(P,Q)
d
(
P
,
Q
)
is defined by
d
(
P
,
Q
)
=
(
x
−
u
)
2
+
(
y
−
v
)
2
d(P,Q)=\sqrt{(x-u)^2+(y-v)^2}
d
(
P
,
Q
)
=
(
x
−
u
)
2
+
(
y
−
v
)
2
Show that there exists a unique choice of points
P
0
∈
C
1
P_0\in C_1
P
0
∈
C
1
and
Q
0
∈
C
2
Q_0\in C_2
Q
0
∈
C
2
such that d(P_0,Q_0)\leqslant d(P,Q) \forall ~P\in C_1~\text{and}~Q\in C_2.
7
1
Hide problems
ISI 2019 : Problem #7
Let
f
f
f
be a polynomial with integer coefficients. Define
a
1
=
f
(
0
)
,
a
2
=
f
(
a
1
)
=
f
(
f
(
0
)
)
,
a_1 = f(0)~,~a_2 = f(a_1) = f(f(0))~,
a
1
=
f
(
0
)
,
a
2
=
f
(
a
1
)
=
f
(
f
(
0
))
,
and
a
n
=
f
(
a
n
−
1
)
~a_n = f(a_{n-1})
a
n
=
f
(
a
n
−
1
)
for
n
⩾
3
n \geqslant 3
n
⩾
3
.If there exists a natural number
k
⩾
3
k \geqslant 3
k
⩾
3
such that
a
k
=
0
a_k = 0
a
k
=
0
, then prove that either
a
1
=
0
a_1=0
a
1
=
0
or
a
2
=
0
a_2=0
a
2
=
0
.
6
1
Hide problems
ISI 2019 : Problem #6
For all natural numbers
n
n
n
, let A_n=\sqrt{2-\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}} \text{(n many radicals)} (a) Show that for
n
⩾
2
n\geqslant 2
n
⩾
2
,
A
n
=
2
sin
π
2
n
+
1
A_n=2\sin\frac{\pi}{2^{n+1}}
A
n
=
2
sin
2
n
+
1
π
(b) Hence or otherwise, evaluate the limit
lim
n
→
∞
2
n
A
n
\lim_{n\to\infty} 2^nA_n
n
→
∞
lim
2
n
A
n
5
1
Hide problems
ISI 2019 : Problem #5
A subset
S
\bf{S}
S
of the plane is called convex if given any two points
x
x
x
and
y
y
y
in
S
\bf{S}
S
, the line segment joining
x
x
x
and
y
y
y
is contained in
S
\bf{S}
S
. A quadrilateral is called convex if the region enclosed by the edges of the quadrilateral is a convex set. Show that given a convex quadrilateral
Q
Q
Q
of area
1
1
1
, there is a rectangle
R
R
R
of area
2
2
2
such that
Q
Q
Q
can be drawn inside
R
R
R
.
4
1
Hide problems
ISI 2019 : Problem #4
Let
f
:
R
→
R
f:\mathbb{R}\to\mathbb{R}
f
:
R
→
R
be a twice differentiable function such that
1
2
y
∫
x
−
y
x
+
y
f
(
t
)
d
t
=
f
(
x
)
∀
x
∈
R
&
y
>
0
\frac{1}{2y}\int_{x-y}^{x+y}f(t)\, dt=f(x)\qquad\forall~x\in\mathbb{R}~\&~y>0
2
y
1
∫
x
−
y
x
+
y
f
(
t
)
d
t
=
f
(
x
)
∀
x
∈
R
&
y
>
0
Show that there exist
a
,
b
∈
R
a,b\in\mathbb{R}
a
,
b
∈
R
such that
f
(
x
)
=
a
x
+
b
f(x)=ax+b
f
(
x
)
=
a
x
+
b
for all
x
∈
R
x\in\mathbb{R}
x
∈
R
.
3
1
Hide problems
ISI 2019 : Problem #3
Let
Ω
=
{
z
=
x
+
i
y
∈
C
:
∣
y
∣
⩽
1
}
\Omega=\{z=x+iy~\in\mathbb{C}~:~|y|\leqslant 1\}
Ω
=
{
z
=
x
+
i
y
∈
C
:
∣
y
∣
⩽
1
}
. If
f
(
z
)
=
z
2
+
2
f(z)=z^2+2
f
(
z
)
=
z
2
+
2
, then draw a sketch of
f
(
Ω
)
=
{
f
(
z
)
:
z
∈
Ω
}
f\Big(\Omega\Big)=\{f(z):z\in\Omega\}
f
(
Ω
)
=
{
f
(
z
)
:
z
∈
Ω
}
Justify your answer.
2
1
Hide problems
ISI 2019 : Problem #2
Let
f
:
(
0
,
∞
)
→
R
f:(0,\infty)\to\mathbb{R}
f
:
(
0
,
∞
)
→
R
be defined by
f
(
x
)
=
lim
n
→
∞
cos
n
(
1
n
x
)
f(x)=\lim_{n\to\infty}\cos^n\bigg(\frac{1}{n^x}\bigg)
f
(
x
)
=
n
→
∞
lim
cos
n
(
n
x
1
)
(a) Show that
f
f
f
has exactly one point of discontinuity. (b) Evaluate
f
f
f
at its point of discontinuity.
1
1
Hide problems
ISI 2019 : Problem #1
Prove that the positive integers
n
n
n
that cannot be written as a sum of
r
r
r
consecutive positive integers, with
r
>
1
r>1
r
>
1
, are of the form
n
=
2
l
n=2^l~
n
=
2
l
for some
l
⩾
0
l\geqslant 0
l
⩾
0
.