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National and Regional Contests
India Contests
ISI B.Math Entrance Exam
2006 ISI B.Math Entrance Exam
2006 ISI B.Math Entrance Exam
Part of
ISI B.Math Entrance Exam
Subcontests
(8)
8
1
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Arithmetic mean of the integers less than and coprime with n
Let
S
S
S
be the set of all integers
k
k
k
,
1
≤
k
≤
n
1\leq k\leq n
1
≤
k
≤
n
, such that
gcd
(
k
,
n
)
=
1
\gcd(k,n)=1
g
cd
(
k
,
n
)
=
1
. What is the arithmetic mean of the integers in
S
S
S
?
7
1
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B.Math - geometry
In a triangle
A
B
C
ABC
A
BC
,
D
D
D
is a point on
B
C
BC
BC
such that
A
D
AD
A
D
is the internal bisector of
∠
A
\angle A
∠
A
. Now Suppose
∠
B
\angle B
∠
B
=
2
∠
C
2\angle C
2∠
C
and
C
D
=
A
B
CD=AB
C
D
=
A
B
. Prove that
∠
A
=
7
2
0
\angle A=72^0
∠
A
=
7
2
0
.
6
1
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B.Math - crossing the river
You are standing at the edge of a river which is
1
1
1
km wide. You have to go to your camp on the opposite bank . The distance to the camp from the point on the opposite bank directly across you is
1
1
1
km . You can swim at
2
2
2
km/hr and walk at
3
3
3
km-hr . What is the shortest time you will take to reach your camp?(Ignore the speed of the river and assume that the river banks are straight and parallel).
5
1
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B.Math - covering with domino
A domino is a
2
2
2
by
1
1
1
rectangle . For what integers
m
m
m
and
n
n
n
can we cover an
m
∗
n
m*n
m
∗
n
rectangle with non-overlapping dominoes???
4
1
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B.Math - derivative
Let
f
:
R
→
R
f:\mathbb{R} \to \mathbb{R}
f
:
R
→
R
be a function that is a function that is differentiable
n
+
1
n+1
n
+
1
times for some positive integer
n
n
n
. The
i
t
h
i^{th}
i
t
h
derivative of
f
f
f
is denoted by
f
(
i
)
f^{(i)}
f
(
i
)
. Suppose-
f
(
1
)
=
f
(
0
)
=
f
(
1
)
(
0
)
=
.
.
.
=
f
(
n
)
(
0
)
=
0
f(1)=f(0)=f^{(1)}(0)=...=f^{(n)}(0)=0
f
(
1
)
=
f
(
0
)
=
f
(
1
)
(
0
)
=
...
=
f
(
n
)
(
0
)
=
0
.Prove that
f
(
n
+
1
)
(
x
)
=
0
f^{(n+1)}(x)=0
f
(
n
+
1
)
(
x
)
=
0
for some
x
∈
(
0
,
1
)
x \in (0,1)
x
∈
(
0
,
1
)
3
1
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B.Math - equation
Find all roots of the equation :-
1
−
x
1
+
x
(
x
−
1
)
2
!
−
⋯
+
(
−
1
)
n
x
(
x
−
1
)
(
x
−
2
)
.
.
.
(
x
−
n
+
1
)
n
!
=
0
1-\frac{x}{1}+\frac{x(x-1)}{2!} - \cdots +(-1)^n\frac{x(x-1)(x-2)...(x-n+1)}{n!}=0
1
−
1
x
+
2
!
x
(
x
−
1
)
−
⋯
+
(
−
1
)
n
n
!
x
(
x
−
1
)
(
x
−
2
)
...
(
x
−
n
+
1
)
=
0
.
2
1
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B.Math - prime
Prove that there is no non-constant polynomial
P
(
x
)
P(x)
P
(
x
)
with integer coefficients such that
P
(
n
)
P(n)
P
(
n
)
is a prime number for all positive integers
n
n
n
.
1
1
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B.Math - Bishop
Bishops on a chessboard move along the diagonals ( that is , on lines parallel to the two main diagonals ) . Prove that the maximum number of non-attacking bishops on an
n
∗
n
n*n
n
∗
n
chessboard is
2
n
−
2
2n-2
2
n
−
2
. (Two bishops are said to be attacking if they are on a common diagonal).