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Contests
National and Regional Contests
India Contests
ISI B.Stat Entrance Exam
2007 ISI B.Stat Entrance Exam
2007 ISI B.Stat Entrance Exam
Part of
ISI B.Stat Entrance Exam
Subcontests
(10)
2
1
Hide problems
Sketch and find behaviour of y=(e^x)sinx
Use calculus to find the behaviour of the function
y
=
e
x
sin
x
−
∞
<
x
<
+
∞
y=e^x\sin{x} \ \ \ \ \ \ \ -\infty <x< +\infty
y
=
e
x
sin
x
−
∞
<
x
<
+
∞
and sketch the graph of the function for
−
2
π
≤
x
≤
2
π
-2\pi \le x \le 2\pi
−
2
π
≤
x
≤
2
π
. Show clearly the locations of the maxima, minima and points of inflection in your graph.
8
1
Hide problems
Sudoku like grid: selection of n^2 cells
The following figure shows a
3
2
×
3
2
3^2 \times 3^2
3
2
×
3
2
grid divided into
3
2
3^2
3
2
subgrids of size
3
×
3
3 \times 3
3
×
3
. This grid has
81
81
81
cells,
9
9
9
in each subgrid. [asy] draw((0,0)--(9,0)--(9,9)--(0,9)--cycle, linewidth(2)); draw((0,1)--(9,1)); draw((0,2)--(9,2)); draw((0,3)--(9,3), linewidth(2)); draw((0,4)--(9,4)); draw((0,5)--(9,5)); draw((0,6)--(9,6), linewidth(2)); draw((0,7)--(9,7)); draw((0,8)--(9,8)); draw((1,0)--(1,9)); draw((2,0)--(2,9)); draw((3,0)--(3,9), linewidth(2)); draw((4,0)--(4,9)); draw((5,0)--(5,9)); draw((6,0)--(6,9), linewidth(2)); draw((7,0)--(7,9)); draw((8,0)--(8,9)); [/asy] Now consider an
n
2
×
n
2
n^2 \times n^2
n
2
×
n
2
grid divided into
n
2
n^2
n
2
subgrids of size
n
×
n
n \times n
n
×
n
. Find the number of ways in which you can select
n
2
n^2
n
2
cells from this grid such that there is exactly one cell coming from each subgrid, one from each row and one from each column.
7
1
Hide problems
Maximize the volume of the prism
Consider a prism with triangular base. The total area of the three faces containing a particular vertex
A
A
A
is
K
K
K
. Show that the maximum possible volume of the prism is
K
3
54
\sqrt{\frac{K^3}{54}}
54
K
3
and find the height of this largest prism.
4
1
Hide problems
Medians: (2a/3, 2b/3, 4c/5) then triangle is impossible
Show that it is not possible to have a triangle with sides
a
,
b
,
a,b,
a
,
b
,
and
c
c
c
whose medians have length
2
3
a
,
2
3
b
\frac{2}{3}a, \frac{2}{3}b
3
2
a
,
3
2
b
and
4
5
c
\frac{4}{5}c
5
4
c
.
6
1
Hide problems
Set: {f(r,r):r in S}=S
Let
S
=
{
1
,
2
,
⋯
,
n
}
S=\{1,2,\cdots ,n\}
S
=
{
1
,
2
,
⋯
,
n
}
where
n
n
n
is an odd integer. Let
f
f
f
be a function defined on
{
(
i
,
j
)
:
i
∈
S
,
j
∈
S
}
\{(i,j): i\in S, j \in S\}
{(
i
,
j
)
:
i
∈
S
,
j
∈
S
}
taking values in
S
S
S
such that (i)
f
(
s
,
r
)
=
f
(
r
,
s
)
f(s,r)=f(r,s)
f
(
s
,
r
)
=
f
(
r
,
s
)
for all
r
,
s
∈
S
r,s \in S
r
,
s
∈
S
(ii)
{
f
(
r
,
s
)
:
s
∈
S
}
=
S
\{f(r,s): s\in S\}=S
{
f
(
r
,
s
)
:
s
∈
S
}
=
S
for all
r
∈
S
r\in S
r
∈
S
Show that
{
f
(
r
,
r
)
:
r
∈
S
}
=
S
\{f(r,r): r\in S\}=S
{
f
(
r
,
r
)
:
r
∈
S
}
=
S
5
1
Hide problems
Trigo: |cosA(sinA+\sqrt(sin^2(A)+3))|<2
Show that
−
2
≤
cos
θ
(
sin
θ
+
sin
2
θ
+
3
)
≤
2
-2 \le \cos \theta\left(\sin \theta + \sqrt{\sin ^2 \theta +3}\right) \le 2
−
2
≤
cos
θ
(
sin
θ
+
sin
2
θ
+
3
)
≤
2
for all values of
θ
\theta
θ
.
1
1
Hide problems
Complex numbers: a^2+a+1/a+1/a^2+1=0
Suppose
a
a
a
is a complex number such that
a
2
+
a
+
1
a
+
1
a
2
+
1
=
0
a^2+a+\frac{1}{a}+\frac{1}{a^2}+1=0
a
2
+
a
+
a
1
+
a
2
1
+
1
=
0
If
m
m
m
is a positive integer, find the value of
a
2
m
+
a
m
+
1
a
m
+
1
a
2
m
a^{2m}+a^m+\frac{1}{a^m}+\frac{1}{a^{2m}}
a
2
m
+
a
m
+
a
m
1
+
a
2
m
1
10
1
Hide problems
Set: If n >n_0^2 then n in A
Let
A
A
A
be a set of positive integers satisfying the following properties: (i) if
m
m
m
and
n
n
n
belong to
A
A
A
, then
m
+
n
m+n
m
+
n
belong to
A
A
A
; (ii) there is no prime number that divides all elements of
A
A
A
.(a) Suppose
n
1
n_1
n
1
and
n
2
n_2
n
2
are two integers belonging to
A
A
A
such that
n
2
−
n
1
>
1
n_2-n_1 >1
n
2
−
n
1
>
1
. Show that you can find two integers
m
1
m_1
m
1
and
m
2
m_2
m
2
in
A
A
A
such that
0
<
m
2
−
m
1
<
n
2
−
n
1
0< m_2-m_1 < n_2-n_1
0
<
m
2
−
m
1
<
n
2
−
n
1
(b) Hence show that there are two consecutive integers belonging to
A
A
A
. (c) Let
n
0
n_0
n
0
and
n
0
+
1
n_0+1
n
0
+
1
be two consecutive integers belonging to
A
A
A
. Show that if
n
≥
n
0
2
n\geq n_0^2
n
≥
n
0
2
then
n
n
n
belongs to
A
A
A
.
9
1
Hide problems
Set: there exist some lambda in [0,1]
Let
X
⊂
R
2
X \subset \mathbb{R}^2
X
⊂
R
2
be a set satisfying the following properties: (i) if
(
x
1
,
y
1
)
(x_1,y_1)
(
x
1
,
y
1
)
and
(
x
2
,
y
2
)
(x_2,y_2)
(
x
2
,
y
2
)
are any two distinct elements in
X
X
X
, then
either,
x
1
>
x
2
and
y
1
>
y
2
or,
x
1
<
x
2
and
y
1
<
y
2
\text{ either, }\ \ x_1>x_2 \text{ and } y_1>y_2\\ \text{ or, } \ \ x_1<x_2 \text{ and } y_1<y_2
either,
x
1
>
x
2
and
y
1
>
y
2
or,
x
1
<
x
2
and
y
1
<
y
2
(ii) there are two elements
(
a
1
,
b
1
)
(a_1,b_1)
(
a
1
,
b
1
)
and
(
a
2
,
b
2
)
(a_2,b_2)
(
a
2
,
b
2
)
in
X
X
X
such that for any
(
x
,
y
)
∈
X
(x,y) \in X
(
x
,
y
)
∈
X
,
a
1
≤
x
≤
a
2
and
b
1
≤
y
≤
b
2
a_1\le x \le a_2 \text{ and } b_1\le y \le b_2
a
1
≤
x
≤
a
2
and
b
1
≤
y
≤
b
2
(iii) if
(
x
1
,
y
1
)
(x_1,y_1)
(
x
1
,
y
1
)
and
(
x
2
,
y
2
)
(x_2,y_2)
(
x
2
,
y
2
)
are two elements of
X
X
X
, then for all
λ
∈
[
0
,
1
]
\lambda \in [0,1]
λ
∈
[
0
,
1
]
,
(
λ
x
1
+
(
1
−
λ
)
x
2
,
λ
y
1
+
(
1
−
λ
)
y
2
)
∈
X
\left(\lambda x_1+(1-\lambda)x_2, \lambda y_1 + (1-\lambda)y_2\right) \in X
(
λ
x
1
+
(
1
−
λ
)
x
2
,
λ
y
1
+
(
1
−
λ
)
y
2
)
∈
X
Show that if
(
x
,
y
)
∈
X
(x,y) \in X
(
x
,
y
)
∈
X
, then for some
λ
∈
[
0
,
1
]
\lambda \in [0,1]
λ
∈
[
0
,
1
]
,
x
=
λ
a
1
+
(
1
−
λ
)
a
2
,
y
=
λ
b
1
+
(
1
−
λ
)
b
2
x=\lambda a_1 +(1-\lambda)a_2, y=\lambda b_1 +(1-\lambda)b_2
x
=
λ
a
1
+
(
1
−
λ
)
a
2
,
y
=
λ
b
1
+
(
1
−
λ
)
b
2
3
1
Hide problems
show the following equality involving integration
Let
f
(
u
)
f(u)
f
(
u
)
be a continuous function and, for any real number
u
u
u
, let denote the greatest integer less than or equal to $u$. Show that for any $x>1$,\int_{1}^{x} (+1)f(u)du = 2\sum_{i=1}^{[x]} i \int_{i}^{x} f(u)du$$