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Contests
National and Regional Contests
India Contests
ISI B.Stat Entrance Exam
2010 ISI B.Stat Entrance Exam
2010 ISI B.Stat Entrance Exam
Part of
ISI B.Stat Entrance Exam
Subcontests
(9)
3
1
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Set: one is contained in the union of other two
Let
I
1
,
I
2
,
I
3
I_1, I_2, I_3
I
1
,
I
2
,
I
3
be three open intervals of
R
\mathbb{R}
R
such that none is contained in another. If
I
1
∩
I
2
∩
I
3
I_1\cap I_2 \cap I_3
I
1
∩
I
2
∩
I
3
is non-empty, then show that at least one of these intervals is contained in the union of the other two.
4
1
Hide problems
f'(1)>1 implies f has a fixed point in (0,1)
A real valued function
f
f
f
is defined on the interval
(
−
1
,
2
)
(-1,2)
(
−
1
,
2
)
. A point
x
0
x_0
x
0
is said to be a fixed point of
f
f
f
if
f
(
x
0
)
=
x
0
f(x_0)=x_0
f
(
x
0
)
=
x
0
. Suppose that
f
f
f
is a differentiable function such that
f
(
0
)
>
0
f(0)>0
f
(
0
)
>
0
and
f
(
1
)
=
1
f(1)=1
f
(
1
)
=
1
. Show that if
f
′
(
1
)
>
1
f'(1)>1
f
′
(
1
)
>
1
, then
f
f
f
has a fixed point in the interval
(
0
,
1
)
(0,1)
(
0
,
1
)
.
5
1
Hide problems
(g(h(x))=h(g(x))
Let
A
A
A
be the set of all functions
f
:
R
→
R
f:\mathbb{R} \to \mathbb{R}
f
:
R
→
R
such that
f
(
x
y
)
=
x
f
(
y
)
f(xy)=xf(y)
f
(
x
y
)
=
x
f
(
y
)
for all
x
,
y
∈
R
x,y \in \mathbb{R}
x
,
y
∈
R
.(a) If
f
∈
A
f \in A
f
∈
A
then show that
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
f(x+y)=f(x)+f(y)
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
for all
x
,
y
∈
R
x,y \in \mathbb{R}
x
,
y
∈
R
(b) For
g
,
h
∈
A
g,h \in A
g
,
h
∈
A
, define a function
g
∘
h
g\circ h
g
∘
h
by
(
g
∘
h
)
(
x
)
=
g
(
h
(
x
)
)
(g \circ h)(x)=g(h(x))
(
g
∘
h
)
(
x
)
=
g
(
h
(
x
))
for
x
∈
R
x \in \mathbb{R}
x
∈
R
. Prove that
g
∘
h
g \circ h
g
∘
h
is in
A
A
A
and is equal to
h
∘
g
h \circ g
h
∘
g
.
7
1
Hide problems
Maximize the area of AB'D'
Consider a rectangular sheet of paper
A
B
C
D
ABCD
A
BC
D
such that the lengths of
A
B
AB
A
B
and
A
D
AD
A
D
are respectively
7
7
7
and
3
3
3
centimetres. Suppose that
B
′
B'
B
′
and
D
′
D'
D
′
are two points on
A
B
AB
A
B
and
A
D
AD
A
D
respectively such that if the paper is folded along
B
′
D
′
B'D'
B
′
D
′
then
A
A
A
falls on
A
′
A'
A
′
on the side
D
C
DC
D
C
. Determine the maximum possible area of the triangle
A
B
′
D
′
AB'D'
A
B
′
D
′
.
9
1
Hide problems
d(A,B)=d(f(A),f(B))
Let
f
:
R
2
→
R
2
f: \mathbb{R}^2 \to \mathbb{R}^2
f
:
R
2
→
R
2
be a function having the following property: For any two points
A
A
A
and
B
B
B
in
R
2
\mathbb{R}^2
R
2
, the distance between
A
A
A
and
B
B
B
is the same as the distance between the points
f
(
A
)
f(A)
f
(
A
)
and
f
(
B
)
f(B)
f
(
B
)
.Denote the unique straight line passing through
A
A
A
and
B
B
B
by
l
(
A
,
B
)
l(A,B)
l
(
A
,
B
)
(a) Suppose that
C
,
D
C,D
C
,
D
are two fixed points in
R
2
\mathbb{R}^2
R
2
. If
X
X
X
is a point on the line
l
(
C
,
D
)
l(C,D)
l
(
C
,
D
)
, then show that
f
(
X
)
f(X)
f
(
X
)
is a point on the line
l
(
f
(
C
)
,
f
(
D
)
)
l(f(C),f(D))
l
(
f
(
C
)
,
f
(
D
))
.(b) Consider two more point
E
E
E
and
F
F
F
in
R
2
\mathbb{R}^2
R
2
and suppose that
l
(
E
,
F
)
l(E,F)
l
(
E
,
F
)
intersects
l
(
C
,
D
)
l(C,D)
l
(
C
,
D
)
at an angle
α
\alpha
α
. Show that
l
(
f
(
C
)
,
f
(
D
)
)
l(f(C),f(D))
l
(
f
(
C
)
,
f
(
D
))
intersects
l
(
f
(
E
)
,
f
(
F
)
)
l(f(E),f(F))
l
(
f
(
E
)
,
f
(
F
))
at an angle
α
\alpha
α
. What happens if the two lines
l
(
C
,
D
)
l(C,D)
l
(
C
,
D
)
and
l
(
E
,
F
)
l(E,F)
l
(
E
,
F
)
do not intersect? Justify your answer.
6
1
Hide problems
n^2+(n+1)^4=5(n+2)^3.... isi (bs) 2010 #6
Consider the equation
n
2
+
(
n
+
1
)
4
=
5
(
n
+
2
)
3
n^2+(n+1)^4=5(n+2)^3
n
2
+
(
n
+
1
)
4
=
5
(
n
+
2
)
3
(a) Show that any integer of the form
3
m
+
1
3m+1
3
m
+
1
or
3
m
+
2
3m+2
3
m
+
2
can not be a solution of this equation.(b) Does the equation have a solution in positive integers?
2
1
Hide problems
a+b+c+d=21... isi(bs) 2010 #2
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be distinct digits such that the product of the
2
2
2
-digit numbers
a
b
‾
\overline{ab}
ab
and
c
b
‾
\overline{cb}
c
b
is of the form
d
d
d
‾
\overline{ddd}
ddd
. Find all possible values of
a
+
b
+
c
+
d
a+b+c+d
a
+
b
+
c
+
d
.
1
1
Hide problems
Rearrangement inequality
Let
a
1
,
a
2
,
⋯
,
a
n
a_1,a_2,\cdots, a_n
a
1
,
a
2
,
⋯
,
a
n
and
b
1
,
b
2
,
⋯
,
b
n
b_1,b_2,\cdots, b_n
b
1
,
b
2
,
⋯
,
b
n
be two permutations of the numbers
1
,
2
,
⋯
,
n
1,2,\cdots, n
1
,
2
,
⋯
,
n
. Show that
∑
i
=
1
n
i
(
n
+
1
−
i
)
≤
∑
i
=
1
n
a
i
b
i
≤
∑
i
=
1
n
i
2
\sum_{i=1}^n i(n+1-i) \le \sum_{i=1}^n a_ib_i \le \sum_{i=1}^n i^2
i
=
1
∑
n
i
(
n
+
1
−
i
)
≤
i
=
1
∑
n
a
i
b
i
≤
i
=
1
∑
n
i
2
10
1
Hide problems
100 people with 100 seats
There are
100
100
100
people in a queue waiting to enter a hall. The hall has exactly
100
100
100
seats numbered from
1
1
1
to
100
100
100
. The first person in the queue enters the hall, chooses any seat and sits there. The
n
n
n
-th person in the queue, where
n
n
n
can be
2
,
.
.
.
,
100
2, . . . , 100
2
,
...
,
100
, enters the hall after
(
n
−
1
)
(n-1)
(
n
−
1
)
-th person is seated. He sits in seat number
n
n
n
if he finds it vacant; otherwise he takes any unoccupied seat. Find the total number of ways in which
100
100
100
seats can be filled up, provided the
100
100
100
-th person occupies seat number
100
100
100
.