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Problems
Contests
National and Regional Contests
India Contests
ISI Entrance Examination
2022 ISI Entrance Examination
2022 ISI Entrance Examination
Part of
ISI Entrance Examination
Subcontests
(9)
9
1
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B.Stat & B.Math 2022 - Q9
Find the smallest positive real number
k
k
k
such that the following inequality holds
∣
z
1
+
…
+
z
n
∣
⩾
1
k
(
∣
z
1
∣
+
…
+
∣
z
n
∣
)
.
\left|z_{1}+\ldots+z_{n}\right| \geqslant \frac{1}{k}\big(\left|z_{1}\right|+\ldots+\left|z_{n}\right|\big) .
∣
z
1
+
…
+
z
n
∣
⩾
k
1
(
∣
z
1
∣
+
…
+
∣
z
n
∣
)
.
for every positive integer
n
⩾
2
n \geqslant 2
n
⩾
2
and every choice
z
1
,
…
,
z
n
z_{1}, \ldots, z_{n}
z
1
,
…
,
z
n
of complex numbers with non-negative real and imaginary parts. [Hint: First find
k
k
k
that works for
n
=
2
n=2
n
=
2
. Then show that the same
k
k
k
works for any
n
⩾
2
n \geqslant 2
n
⩾
2
.]
8
1
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B.Stat & B.Math 2022 - Q8
Find the minimum value of
∣
sin
x
+
cos
x
+
tan
x
+
cot
x
+
sec
x
+
cosec
x
∣
\big|\sin x+\cos x+\tan x+\cot x+\sec x+\operatorname{cosec}x\big|
sin
x
+
cos
x
+
tan
x
+
cot
x
+
sec
x
+
cosec
x
for real numbers
x
x
x
not multiple of
π
2
\frac{\pi}{2}
2
π
.
7
1
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B.Stat & B.Math 2022 - Q7
Let
P
(
x
)
=
1
+
2
x
+
7
x
2
+
13
x
3
,
x
∈
R
.
P(x)=1+2 x+7 x^{2}+13 x^{3}~,\qquad x \in \mathbb{R} .
P
(
x
)
=
1
+
2
x
+
7
x
2
+
13
x
3
,
x
∈
R
.
Calculate for all
x
∈
R
,
x \in \mathbb{R},
x
∈
R
,
lim
n
→
∞
(
P
(
x
n
)
)
n
\lim _{n \rightarrow \infty}\left(P\left(\frac{x}{n}\right)\right)^{n}
n
→
∞
lim
(
P
(
n
x
)
)
n
6
1
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B.Stat & B.Math 2022 - Q6
Consider a sequence
P
1
,
P
2
,
…
P_{1}, P_{2}, \ldots
P
1
,
P
2
,
…
of points in the plane such that
P
1
,
P
2
,
P
3
P_{1}, P_{2}, P_{3}
P
1
,
P
2
,
P
3
are non-collinear and for every
n
≥
4
,
P
n
n \geq 4, P_{n}
n
≥
4
,
P
n
is the midpoint of the line segment joining
P
n
−
2
P_{n-2}
P
n
−
2
and
P
n
−
3
P_{n-3}
P
n
−
3
. Let
L
L
L
denote the line segment joining
P
1
P_{1}
P
1
and
P
5
P_{5}
P
5
. Prove the following:[*] The area of the triangle formed by the points
P
n
,
P
n
−
1
,
P
n
−
2
P_{n}, P_{n-1}, P_{n-2}
P
n
,
P
n
−
1
,
P
n
−
2
converges to zero as
n
n
n
goes to infinity.[*] The point
P
9
P_{9}
P
9
lies on
L
L
L
.
5
1
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B.Stat & B.Math 2022 - Q5
For any positive integer
n
n
n
, and
i
=
1
,
2
i=1,2
i
=
1
,
2
, let
f
i
(
n
)
f_{i}(n)
f
i
(
n
)
denote the number of divisors of
n
n
n
of the form
3
k
+
i
3 k+i
3
k
+
i
(including
1
1
1
and
n
n
n
). Define, for any positive integer
n
n
n
,
f
(
n
)
=
f
1
(
n
)
−
f
2
(
n
)
f(n)=f_{1}(n)-f_{2}(n)
f
(
n
)
=
f
1
(
n
)
−
f
2
(
n
)
Find the value of
f
(
5
2022
)
f\left(5^{2022}\right)
f
(
5
2022
)
and
f
(
2
1
2022
)
f\left(21^{2022}\right)
f
(
2
1
2022
)
.
4
1
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B.Stat & B.Math 2022 - Q4
Let
P
(
x
)
P(x)
P
(
x
)
be an odd degree polynomial in
x
x
x
with real coefficients. Show that the equation
P
(
P
(
x
)
)
=
0
P(P(x))=0
P
(
P
(
x
))
=
0
has at least as many distinct real roots as the equation
P
(
x
)
=
0
P(x)=0
P
(
x
)
=
0
.
3
1
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B.Stat & B.Math 2022 - Q3
Consider the parabola
C
:
y
2
=
4
x
C: y^{2}=4 x
C
:
y
2
=
4
x
and the straight line
L
:
y
=
x
+
2
L: y=x+2
L
:
y
=
x
+
2
. Let
P
P
P
be a variable point on
L
L
L
. Draw the two tangents from
P
P
P
to
C
C
C
and let
Q
1
Q_{1}
Q
1
and
Q
2
Q_{2}
Q
2
denote the two points of contact on
C
C
C
. Let
Q
Q
Q
be the mid-point of the line segment joining
Q
1
Q_{1}
Q
1
and
Q
2
Q_{2}
Q
2
. Find the locus of
Q
Q
Q
as
P
P
P
moves along
L
L
L
.
2
1
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B.Stat & B.Math 2022 - Q2
Consider the function
f
(
x
)
=
∑
k
=
1
m
(
x
−
k
)
4
,
x
∈
R
f(x)=\sum_{k=1}^{m}(x-k)^{4}~, \qquad~ x \in \mathbb{R}
f
(
x
)
=
k
=
1
∑
m
(
x
−
k
)
4
,
x
∈
R
where
m
>
1
m>1
m
>
1
is an integer. Show that
f
f
f
has a unique minimum and find the point where the minimum is attained.
1
1
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B.Stat & B.Math 2022 - Q1
Consider a board having 2 rows and
n
n
n
columns. Thus there are
2
n
2n
2
n
cells in the board. Each cell is to be filled in by
0
0
0
or
1
1
1
.[*] In how many ways can this be done such that each row sum and each column sum is even?[*] In how many ways can this be done such that each row sum and each column sum is odd?