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National and Regional Contests
India Contests
ISI B.Math Entrance Exam
2008 ISI B.Math Entrance Exam
2008 ISI B.Math Entrance Exam
Part of
ISI B.Math Entrance Exam
Subcontests
(10)
10
1
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B.Math 2008-GCD
If
p
p
p
is a prime number and
a
>
1
a>1
a
>
1
is a natural number , then show that the greatest common divisor of the two numbers
a
−
1
a-1
a
−
1
and
a
p
−
1
a
−
1
\frac{a^p-1}{a-1}
a
−
1
a
p
−
1
is either
1
1
1
or
p
p
p
.
7
1
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B.Math 2008-Combinatorics.
Let
C
=
{
(
i
,
j
)
∣
i
,
j
C=\{ (i,j)|i,j
C
=
{(
i
,
j
)
∣
i
,
j
integers such that
0
≤
i
,
j
≤
24
}
0\leq i,j\leq 24\}
0
≤
i
,
j
≤
24
}
How many squares can be formed in the plane all of whose vertices are in
C
C
C
and whose sides are parallel to the
X
−
X-
X
−
axis and
Y
−
Y-
Y
−
axis?
6
1
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B.Math 2008-fibonacci
Let
(
n
k
)
\dbinom{n}{k}
(
k
n
)
denote the binomial coefficient
n
!
k
!
(
n
−
k
)
!
\frac{n!}{k!(n-k)!}
k
!
(
n
−
k
)!
n
!
, and
F
m
F_m
F
m
be the
m
t
h
m^{th}
m
t
h
Fibonacci number given by
F
1
=
F
2
=
1
F_1=F_2=1
F
1
=
F
2
=
1
and
F
m
+
2
=
F
m
+
F
m
+
1
F_{m+2}=F_m+F_{m+1}
F
m
+
2
=
F
m
+
F
m
+
1
for all
m
≥
1
m\geq 1
m
≥
1
. Show that
∑
(
n
k
)
=
F
m
+
1
\sum \dbinom{n}{k}=F_{m+1}
∑
(
k
n
)
=
F
m
+
1
for all
m
≥
1
m\geq 1
m
≥
1
. Here the above sum is over all pairs of integers
n
≥
k
≥
0
n\geq k\geq 0
n
≥
k
≥
0
with
n
+
k
=
m
n+k=m
n
+
k
=
m
.
5
1
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B.Math 2008-Integer Polynomial
If a polynomial
P
P
P
with integer coefficients has three distinct integer zeroes , then show that
P
(
n
)
≠
1
P(n)\neq 1
P
(
n
)
=
1
for any integer
n
n
n
.
4
1
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B.Math 2008-Pigeonhole
Let
a
1
,
a
2
,
.
.
.
,
a
n
a_1,a_2,...,a_n
a
1
,
a
2
,
...
,
a
n
be integers . Show that there exists integers
k
k
k
and
r
r
r
such that the sum
a
k
+
a
k
+
1
+
.
.
.
+
a
k
+
r
a_k+a_{k+1}+...+a_{k+r}
a
k
+
a
k
+
1
+
...
+
a
k
+
r
is divisible by
n
n
n
.
9
1
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B.Math 2008-System of equations
For
n
≥
3
n\geq 3
n
≥
3
, determine all real solutions of the system of n equations :
x
1
+
x
2
+
.
.
.
+
x
n
−
1
=
1
x
n
x_1+x_2+...+x_{n-1}=\frac{1}{x_n}
x
1
+
x
2
+
...
+
x
n
−
1
=
x
n
1
.......................
x
1
+
x
2
+
.
.
.
+
x
i
−
1
+
x
i
+
1
+
.
.
.
+
x
n
=
1
x
i
x_1+x_2+...+x_{i-1}+x_{i+1}+...+x_n=\frac{1}{x_i}
x
1
+
x
2
+
...
+
x
i
−
1
+
x
i
+
1
+
...
+
x
n
=
x
i
1
.......................
x
2
+
.
.
.
+
x
n
−
1
+
x
n
=
1
x
1
x_2+...+x_{n-1}+x_n=\frac{1}{x_1}
x
2
+
...
+
x
n
−
1
+
x
n
=
x
1
1
8
1
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B.Math 2008-Equations
Let
a
2
+
b
2
=
1
a^2+b^2=1
a
2
+
b
2
=
1
,
c
2
+
d
2
=
1
c^2+d^2=1
c
2
+
d
2
=
1
,
a
c
+
b
d
=
0
ac+bd=0
a
c
+
b
d
=
0
Prove that
a
2
+
c
2
=
1
a^2+c^2=1
a
2
+
c
2
=
1
,
b
2
+
d
2
=
1
b^2+d^2=1
b
2
+
d
2
=
1
,
a
b
+
c
d
=
0
ab+cd=0
ab
+
c
d
=
0
.
3
1
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B.Math 2008-Complex number
Let
z
z
z
be a complex number such that
z
,
z
2
,
z
3
z,z^2,z^3
z
,
z
2
,
z
3
are all collinear in the complex plane . Show that
z
z
z
is a real number .
2
1
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B.Math 2008 , polynomial
Suppose that
P
(
x
)
P(x)
P
(
x
)
is a polynomial with real coefficients, such that for some positive real numbers
c
c
c
and
d
d
d
, and for all natural numbers
n
n
n
, we have
c
∣
n
∣
3
≤
∣
P
(
n
)
∣
≤
d
∣
n
∣
3
c|n|^3\leq |P(n)|\leq d|n|^3
c
∣
n
∣
3
≤
∣
P
(
n
)
∣
≤
d
∣
n
∣
3
.Prove that
P
(
x
)
P(x)
P
(
x
)
has a real zero.
1
1
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B.Math 2008-Integration .
Let
f
:
R
→
R
f:\mathbb{R} \to \mathbb{R}
f
:
R
→
R
be a continuous function . Suppose
f
(
x
)
=
1
t
∫
0
t
(
f
(
x
+
y
)
−
f
(
y
)
)
d
y
f(x)=\frac{1}{t} \int^t_0 (f(x+y)-f(y))\,dy
f
(
x
)
=
t
1
∫
0
t
(
f
(
x
+
y
)
−
f
(
y
))
d
y
∀
x
∈
R
\forall x\in \mathbb{R}
∀
x
∈
R
and all
t
>
0
t>0
t
>
0
. Then show that there exists a constant
c
c
c
such that
f
(
x
)
=
c
x
∀
x
f(x)=cx\ \forall x
f
(
x
)
=
c
x
∀
x