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Contests
National and Regional Contests
India Contests
ISI Entrance Examination
2020 ISI Entrance Examination
2020 ISI Entrance Examination
Part of
ISI Entrance Examination
Subcontests
(8)
8
1
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ISI 2020 : Problem 8
A finite sequence of numbers
(
a
1
,
⋯
,
a
n
)
(a_1,\cdots,a_n)
(
a
1
,
⋯
,
a
n
)
is said to be alternating if
a
1
>
a
2
,
a
2
<
a
3
,
a
3
>
a
4
,
a
4
<
a
5
,
⋯
a_1>a_2~,~a_2<a_3~,~a_3>a_4~,~a_4<a_5~,~\cdots
a
1
>
a
2
,
a
2
<
a
3
,
a
3
>
a
4
,
a
4
<
a
5
,
⋯
or
a
1
<
a
2
,
a
2
>
a
3
,
a
3
<
a
4
,
a
4
>
a
5
,
⋯
\text{or ~}~~a_1<a_2~,~a_2>a_3~,~a_3<a_4~,~a_4>a_5~,~\cdots
or
a
1
<
a
2
,
a
2
>
a
3
,
a
3
<
a
4
,
a
4
>
a
5
,
⋯
How many alternating sequences of length
5
5
5
, with distinct numbers
a
1
,
⋯
,
a
5
a_1,\cdots,a_5
a
1
,
⋯
,
a
5
can be formed such that
a
i
∈
{
1
,
2
,
⋯
,
20
}
a_i\in\{1,2,\cdots,20\}
a
i
∈
{
1
,
2
,
⋯
,
20
}
for
i
=
1
,
⋯
,
5
i=1,\cdots,5
i
=
1
,
⋯
,
5
?
7
1
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ISI 2020 : Problem 7
Consider a right-angled triangle with integer-valued sides
a
<
b
<
c
a<b<c
a
<
b
<
c
where
a
,
b
,
c
a,b,c
a
,
b
,
c
are pairwise co-prime. Let
d
=
c
−
b
d=c-b
d
=
c
−
b
. Suppose
d
d
d
divides
a
a
a
. Then (a) Prove that
d
⩽
2
d\leqslant 2
d
⩽
2
. (b) Find all such triangles (i.e. all possible triplets
a
,
b
,
c
a,b,c
a
,
b
,
c
) with perimeter less than
100
100
100
.
3
1
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ISI 2020 : Problem 3
Let
A
A
A
and
B
B
B
be variable points on
x
−
x-
x
−
axis and
y
−
y-
y
−
axis respectively such that the line segment
A
B
AB
A
B
is in the first quadrant and of a fixed length
2
d
2d
2
d
. Let
C
C
C
be the mid-point of
A
B
AB
A
B
and
P
P
P
be a point such that(a)
P
P
P
and the origin are on the opposite sides of
A
B
AB
A
B
and,(b)
P
C
PC
PC
is a line segment of length
d
d
d
which is perpendicular to
A
B
AB
A
B
.Find the locus of
P
P
P
.
6
1
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ISI 2020 : Problem 6
Prove that the family of curves
x
2
a
2
+
λ
+
y
2
b
2
+
λ
=
1
\frac{x^2}{a^2+\lambda}+\frac{y^2}{b^2+\lambda}=1
a
2
+
λ
x
2
+
b
2
+
λ
y
2
=
1
satisfies
d
y
d
x
(
a
2
−
b
2
)
=
(
x
+
y
d
y
d
x
)
(
x
d
y
d
x
−
y
)
\frac{dy}{dx}(a^2-b^2)=\left(x+y\frac{dy}{dx}\right)\left(x\frac{dy}{dx}-y\right)
d
x
d
y
(
a
2
−
b
2
)
=
(
x
+
y
d
x
d
y
)
(
x
d
x
d
y
−
y
)
5
1
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ISI 2020 : Problem 5
Prove that the largest pentagon (in terms of area) that can be inscribed in a circle of radius
1
1
1
is regular (i.e., has equal sides).
2
1
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ISI 2020 : Problem 2
Let
a
a
a
be a fixed real number. Consider the equation
(
x
+
2
)
2
(
x
+
7
)
2
+
a
=
0
,
x
∈
R
(x+2)^{2}(x+7)^{2}+a=0, x \in R
(
x
+
2
)
2
(
x
+
7
)
2
+
a
=
0
,
x
∈
R
where
R
R
R
is the set of real numbers. For what values of
a
a
a
, will the equ have exactly one double-root?
4
1
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ISI 2020 : Problem 4
Let a real-valued sequence
{
x
n
}
n
⩾
1
\{x_n\}_{n\geqslant 1}
{
x
n
}
n
⩾
1
be such that
lim
n
→
∞
n
x
n
=
0
\lim_{n\to\infty}nx_n=0
n
→
∞
lim
n
x
n
=
0
Find all possible real values of
t
t
t
such that
lim
n
→
∞
x
n
(
log
n
)
t
=
0
\lim_{n\to\infty}x_n\big(\log n\big)^t=0
lim
n
→
∞
x
n
(
lo
g
n
)
t
=
0
.
1
1
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ISI 2020 : Problem 1
Let
i
i
i
be a root of the equation
x
2
+
1
=
0
x^2+1=0
x
2
+
1
=
0
and let
ω
\omega
ω
be a root of the equation
x
2
+
x
+
1
=
0
x^2+x+1=0
x
2
+
x
+
1
=
0
. Construct a polynomial
f
(
x
)
=
a
0
+
a
1
x
+
⋯
+
a
n
x
n
f(x)=a_0+a_1x+\cdots+a_nx^n
f
(
x
)
=
a
0
+
a
1
x
+
⋯
+
a
n
x
n
where
a
0
,
a
1
,
⋯
,
a
n
a_0,a_1,\cdots,a_n
a
0
,
a
1
,
⋯
,
a
n
are all integers such that
f
(
i
+
ω
)
=
0
f(i+\omega)=0
f
(
i
+
ω
)
=
0
.