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Problems
Contests
National and Regional Contests
India Contests
ISI Entrance Examination
2017 ISI Entrance Examination
2017 ISI Entrance Examination
Part of
ISI Entrance Examination
Subcontests
(8)
8
1
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Two polynomials P(x) and Q(x)
Let
k
,
n
k,n
k
,
n
and
r
r
r
be positive integers.(a) Let
Q
(
x
)
=
x
k
+
a
1
x
k
+
1
+
⋯
+
a
n
x
k
+
n
Q(x)=x^k+a_1x^{k+1}+\cdots+a_nx^{k+n}
Q
(
x
)
=
x
k
+
a
1
x
k
+
1
+
⋯
+
a
n
x
k
+
n
be a polynomial with real coefficients. Show that the function
Q
(
x
)
x
k
\frac{Q(x)}{x^k}
x
k
Q
(
x
)
is strictly positive for all real
x
x
x
satisfying
0
<
∣
x
∣
<
1
1
+
∑
i
=
1
n
∣
a
i
∣
0<|x|<\frac1{1+\sum\limits_{i=1}^n |a_i|}
0
<
∣
x
∣
<
1
+
i
=
1
∑
n
∣
a
i
∣
1
(b) Let
P
(
x
)
=
b
0
+
b
1
x
+
⋯
+
b
r
x
r
P(x)=b_0+b_1x+\cdots+b_rx^r
P
(
x
)
=
b
0
+
b
1
x
+
⋯
+
b
r
x
r
be a non zero polynomial with real coefficients. Let
m
m
m
be the smallest number such that
b
m
≠
0
b_m \neq 0
b
m
=
0
. Prove that the graph of
y
=
P
(
x
)
y=P(x)
y
=
P
(
x
)
cuts the
x
x
x
-axis at the origin (i.e.,
P
P
P
changes signs at
x
=
0
x=0
x
=
0
) if and only if
m
m
m
is an odd integer.
7
1
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Find the number of all such permutations P
Let
A
=
{
1
,
2
,
…
,
n
}
A=\{1,2,\ldots,n\}
A
=
{
1
,
2
,
…
,
n
}
. For a permutation
P
=
(
P
(
1
)
,
P
(
2
)
,
…
,
P
(
n
)
)
P=(P(1), P(2), \ldots, P(n))
P
=
(
P
(
1
)
,
P
(
2
)
,
…
,
P
(
n
))
of the elements of
A
A
A
, let
P
(
1
)
P(1)
P
(
1
)
denote the first element of
P
P
P
. Find the number of all such permutations
P
P
P
so that for all
i
,
j
∈
A
i,j \in A
i
,
j
∈
A
:(a) if
i
<
j
<
P
(
1
)
i < j<P(1)
i
<
j
<
P
(
1
)
, then
j
j
j
appears before
i
i
i
in
P
P
P
; and(b) if
P
(
1
)
<
i
<
j
P(1)<i<j
P
(
1
)
<
i
<
j
, then
i
i
i
appears before
j
j
j
in
P
P
P
.
6
1
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4+p_1p_2, 4+p_1p_3 are perfect squares
Let
p
1
,
p
2
,
p
3
p_1,p_2,p_3
p
1
,
p
2
,
p
3
be primes with
p
2
≠
p
3
p_2\neq p_3
p
2
=
p
3
such that
4
+
p
1
p
2
4+p_1p_2
4
+
p
1
p
2
and
4
+
p
1
p
3
4+p_1p_3
4
+
p
1
p
3
are perfect squares. Find all possible values of
p
1
,
p
2
,
p
3
p_1,p_2,p_3
p
1
,
p
2
,
p
3
.
5
1
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g(n)= Product of digits of n
Let
g
:
N
→
N
g:\mathbb{N} \to \mathbb{N}
g
:
N
→
N
with
g
(
n
)
g(n)
g
(
n
)
being the product of the digits of
n
n
n
.(a) Prove that
g
(
n
)
≤
n
g(n) \le n
g
(
n
)
≤
n
for all
n
∈
N
n\in \mathbb{N}
n
∈
N
(b) Find all
n
∈
N
n\in \mathbb{N}
n
∈
N
for which
n
2
−
12
n
+
36
=
g
(
n
)
n^2-12n+36=g(n)
n
2
−
12
n
+
36
=
g
(
n
)
4
1
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Find the area of the region R
Let
S
S
S
be a square formed by the four vertices
(
1
,
1
)
,
(
1.
−
1
)
,
(
−
1
,
1
)
(1,1),(1.-1),(-1,1)
(
1
,
1
)
,
(
1.
−
1
)
,
(
−
1
,
1
)
and
(
−
1
,
−
1
)
(-1,-1)
(
−
1
,
−
1
)
. Let the region
R
R
R
be the set of points inside
S
S
S
which are closer to the center than any of the four sides. Find the area of the region
R
R
R
.
3
1
Hide problems
Find $f'(1)$ and evaluate a limit
Suppose
f
:
R
→
R
f:\mathbb{R} \to \mathbb{R}
f
:
R
→
R
is a function given by f(x) =\begin{cases} 1 & \mbox{if} \ x=1 \\ e^{(x^{10}-1)}+(x-1)^2\sin\frac1{x-1} & \mbox{if} \ x\neq 1\end{cases}(a) Find
f
′
(
1
)
f'(1)
f
′
(
1
)
(b) Evaluate
lim
u
→
∞
[
100
u
−
u
∑
k
=
1
100
f
(
1
+
k
u
)
]
\displaystyle \lim_{u\to\infty} \left[100u-u\sum_{k=1}^{100} f\left(1+\frac{k}{u}\right)\right]
u
→
∞
lim
[
100
u
−
u
k
=
1
∑
100
f
(
1
+
u
k
)
]
.
2
1
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Length of BC must be $(12+9\sqrt{15})/5$
Consider a circle of radius
6
6
6
. Let
B
,
C
,
D
B,C,D
B
,
C
,
D
and
E
E
E
be points on the circle such that
B
D
BD
B
D
and
C
E
CE
CE
, when extended, intersect at
A
A
A
. If
A
D
AD
A
D
and
A
E
AE
A
E
have length
5
5
5
and
4
4
4
respectively, and
D
B
C
DBC
D
BC
is a right angle, then show that the length of
B
C
BC
BC
is
12
+
9
15
5
\frac{12+9\sqrt{15}}{5}
5
12
+
9
15
.
1
1
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$\tan(n\theta)$ is rational
Let the sequence
{
a
n
}
n
≥
1
\{a_n\}_{n\ge 1}
{
a
n
}
n
≥
1
be defined by
a
n
=
tan
(
n
θ
)
a_n=\tan(n\theta)
a
n
=
tan
(
n
θ
)
where
tan
θ
=
2
\tan\theta =2
tan
θ
=
2
. Show that for all
n
n
n
,
a
n
a_n
a
n
is a rational number which can be written with an odd denominator.