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National and Regional Contests
India Contests
ISI B.Stat Entrance Exam
2005 ISI B.Stat Entrance Exam
2005 ISI B.Stat Entrance Exam
Part of
ISI B.Stat Entrance Exam
Subcontests
(10)
5
1
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sum of reciprocals of angles > 1/45 ....[ISI(BS) '05 #5]
Consider an acute angled triangle
P
Q
R
PQR
PQR
such that
C
,
I
C,I
C
,
I
and
O
O
O
are the circumcentre, incentre and orthocentre respectively. Suppose
∠
Q
C
R
,
∠
Q
I
R
\angle QCR, \angle QIR
∠
QCR
,
∠
Q
I
R
and
∠
Q
O
R
\angle QOR
∠
QOR
, measured in degrees, are
α
,
β
\alpha, \beta
α
,
β
and
γ
\gamma
γ
respectively. Show that
1
α
+
1
β
+
1
γ
>
1
45
\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}>\frac{1}{45}
α
1
+
β
1
+
γ
1
>
45
1
10
1
Hide problems
number of regions formed by partition in a triangle
Let
A
B
C
ABC
A
BC
be a triangle. Take
n
n
n
point lying on the side
A
B
AB
A
B
(different from
A
A
A
and
B
B
B
) and connect all of them by straight lines to the vertex
C
C
C
. Similarly, take
n
n
n
points on the side
A
C
AC
A
C
and connect them to
B
B
B
. Into how many regions is the triangle
A
B
C
ABC
A
BC
partitioned by these lines?Further, take
n
n
n
points on the side
B
C
BC
BC
also and join them with
A
A
A
. Assume that no three straight lines meet at a point other than
A
,
B
A,B
A
,
B
and
C
C
C
. Into how many regions is the triangle
A
B
C
ABC
A
BC
partitioned now?
9
1
Hide problems
every point is colored red or blue
Suppose that to every point of the plane a colour, either red or blue, is associated.(a) Show that if there is no equilateral triangle with all vertices of the same colour then there must exist three points
A
,
B
A,B
A
,
B
and
C
C
C
of the same colour such that
B
B
B
is the midpoint of
A
C
AC
A
C
.(b) Show that there must be an equilateral triangle with all vertices of the same colour.
8
1
Hide problems
Are the following function multiplicative?
A function
f
(
n
)
f(n)
f
(
n
)
is defined on the set of positive integers is said to be multiplicative if
f
(
m
n
)
=
f
(
m
)
f
(
n
)
f(mn)=f(m)f(n)
f
(
mn
)
=
f
(
m
)
f
(
n
)
whenever
m
m
m
and
n
n
n
have no common factors greater than
1
1
1
. Are the following functions multiplicative? Justify your answer.(a)
g
(
n
)
=
5
k
g(n)=5^k
g
(
n
)
=
5
k
where
k
k
k
is the number of distinct primes which divide
n
n
n
.(b)
h
(
n
)
=
{
0
if
n
is divisible by
k
2
for some integer
k
>
1
1
otherwise
h(n)=\begin{cases} 0 & \text{if} \ n \ \text{is divisible by} \ k^2 \ \text{for some integer} \ k>1 \\ 1 & \text{otherwise} \end{cases}
h
(
n
)
=
{
0
1
if
n
is divisible by
k
2
for some integer
k
>
1
otherwise
6
1
Hide problems
g(x)=f(h(x)), h is increasing
Let
f
f
f
be a function defined on
(
0
,
∞
)
(0, \infty )
(
0
,
∞
)
as follows:
f
(
x
)
=
x
+
1
x
f(x)=x+\frac1x
f
(
x
)
=
x
+
x
1
Let
h
h
h
be a function defined for all
x
∈
(
0
,
1
)
x \in (0,1)
x
∈
(
0
,
1
)
as
h
(
x
)
=
x
4
(
1
−
x
)
6
h(x)=\frac{x^4}{(1-x)^6}
h
(
x
)
=
(
1
−
x
)
6
x
4
Suppose that
g
(
x
)
=
f
(
h
(
x
)
)
g(x)=f(h(x))
g
(
x
)
=
f
(
h
(
x
))
for all
x
∈
(
0
,
1
)
x \in (0,1)
x
∈
(
0
,
1
)
.(a) Show that
h
h
h
is a strictly increasing function.(b) Show that there exists a real number
x
0
∈
(
0
,
1
)
x_0 \in (0,1)
x
0
∈
(
0
,
1
)
such that
g
g
g
is strictly decreasing in the interval
(
0
,
x
0
]
(0,x_0]
(
0
,
x
0
]
and strictly increasing in the interval
[
x
0
,
1
)
[x_0,1)
[
x
0
,
1
)
.
2
1
Hide problems
sketch and find minimum of f(x)
Let
f
(
x
)
=
∫
0
1
∣
t
−
x
∣
t
d
t
f(x)=\int_0^1 |t-x|t \, dt
f
(
x
)
=
∫
0
1
∣
t
−
x
∣
t
d
t
for all real
x
x
x
. Sketch the graph of
f
(
x
)
f(x)
f
(
x
)
. What is the minimum value of
f
(
x
)
f(x)
f
(
x
)
?
1
1
Hide problems
a,b,c form right angle triangle as well as 1/a,1/b,1/c
Let
a
,
b
a,b
a
,
b
and
c
c
c
be the sides of a right angled triangle. Let
θ
\theta
θ
be the smallest angle of this triangle. If
1
a
,
1
b
\frac{1}{a}, \frac{1}{b}
a
1
,
b
1
and
1
c
\frac{1}{c}
c
1
are also the sides of a right angled triangle then show that
sin
θ
=
5
−
1
2
\sin\theta=\frac{\sqrt{5}-1}{2}
sin
θ
=
2
5
−
1
3
1
Hide problems
functional equation with two variables
Let
f
f
f
be a function defined on
{
(
i
,
j
)
:
i
,
j
∈
N
}
\{(i,j): i,j \in \mathbb{N}\}
{(
i
,
j
)
:
i
,
j
∈
N
}
satisfying(i)
f
(
i
,
i
+
1
)
=
1
3
f(i,i+1)=\frac{1}{3}
f
(
i
,
i
+
1
)
=
3
1
for all
i
i
i
(ii)
f
(
i
,
j
)
=
f
(
i
,
k
)
+
f
(
k
,
j
)
−
2
f
(
i
,
k
)
f
(
k
,
j
)
f(i,j)=f(i,k)+f(k,j)-2f(i,k)f(k,j)
f
(
i
,
j
)
=
f
(
i
,
k
)
+
f
(
k
,
j
)
−
2
f
(
i
,
k
)
f
(
k
,
j
)
for all
k
k
k
such that
i
<
k
<
j
i <k<j
i
<
k
<
j
.Find the value of
f
(
1
,
100
)
f(1,100)
f
(
1
,
100
)
.
4
1
Hide problems
Trigonometric equation
Find all real solutions of the equation
sin
5
x
+
cos
3
x
=
1
\sin^{5}x+\cos^{3}x=1
sin
5
x
+
cos
3
x
=
1
.
7
1
Hide problems
Bijective counting of partitions and compositions
Q. For integers
m
,
n
≥
1
m,n\geq 1
m
,
n
≥
1
, Let
A
m
,
n
A_{m,n}
A
m
,
n
,
B
m
,
n
B_{m,n}
B
m
,
n
and
C
m
,
n
C_{m,n}
C
m
,
n
denote the following sets:
A
m
,
n
=
{
(
α
1
,
α
2
,
…
,
α
m
)
:
1
≤
α
1
≤
α
2
≤
…
≤
α
m
≤
n
}
A_{m,n}=\{(\alpha _1,\alpha _2,\ldots,\alpha _m) \colon 1\leq \alpha _1\leq \alpha_2 \leq \ldots \leq \alpha_m\leq n\}
A
m
,
n
=
{(
α
1
,
α
2
,
…
,
α
m
)
:
1
≤
α
1
≤
α
2
≤
…
≤
α
m
≤
n
}
given that
α
i
∈
Z
\alpha _i \in \mathbb{Z}
α
i
∈
Z
for all
i
i
i
B
m
,
n
=
{
(
α
1
,
α
2
,
…
,
α
m
)
:
α
1
+
α
2
+
…
+
α
m
=
n
}
B_{m,n}=\{(\alpha _1,\alpha _2,\ldots ,\alpha _m) \colon \alpha _1+\alpha _2+\ldots + \alpha _m=n\}
B
m
,
n
=
{(
α
1
,
α
2
,
…
,
α
m
)
:
α
1
+
α
2
+
…
+
α
m
=
n
}
given that
α
i
≥
0
\alpha _i \geq 0
α
i
≥
0
and
α
i
∈
Z
\alpha_ i\in \mathbb{Z}
α
i
∈
Z
for all
i
i
i
C
m
,
n
=
{
(
α
1
,
α
2
,
…
,
α
m
)
:
1
≤
α
1
<
α
2
<
…
<
α
m
≤
n
}
C_{m,n}=\{(\alpha _1,\alpha _2,\ldots,\alpha _m)\colon 1\leq \alpha _1< \alpha_2 < \ldots< \alpha_m\leq n\}
C
m
,
n
=
{(
α
1
,
α
2
,
…
,
α
m
)
:
1
≤
α
1
<
α
2
<
…
<
α
m
≤
n
}
given that
α
i
∈
Z
\alpha _i \in \mathbb{Z}
α
i
∈
Z
for all
i
i
i
(
a
)
(a)
(
a
)
Define a one-one onto map from
A
m
,
n
A_{m,n}
A
m
,
n
onto
B
m
+
1
,
n
−
1
B_{m+1,n-1}
B
m
+
1
,
n
−
1
.
(
b
)
(b)
(
b
)
Define a one-one onto map from
A
m
,
n
A_{m,n}
A
m
,
n
onto
C
m
,
n
+
m
−
1
C_{m,n+m-1}
C
m
,
n
+
m
−
1
.
(
c
)
(c)
(
c
)
Find the number of elements of the sets
A
m
,
n
A_{m,n}
A
m
,
n
and
B
m
,
n
B_{m,n}
B
m
,
n
.